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Toxicological Evaluation of Product Safety
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Toxicological Evaluation of Product Safety
Toxicological Evaluation of Product Safety
Toxicological Evaluation of Product Safety ROBERT P. GIOVACCHINI, Ph.D. * Toxicology was once considered to cover only the study of poisons. Today it...
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Toxicological Evaluation of Product Safety ROBERT P. GIOVACCHINI, Ph.D. *
Toxicology was once considered to cover only the study of poisons. Today it has expanded to include the evaluation of all aspects of hazards to man from substances with which he may come in contact.
DEFINITIONS Virtually all such substances can present a hazard under appropriate conditions and concentrations. Thus the definitions of such terms as "poison," "toxicity," and "hazard" are of fundamental importance in establishing criteria for evaluating product safety. There are basically two types of definitions, which mayor may not coincide at any time. The first is the definition established through accepted scientific usage. The second is the definition adopted for regulatory purposes by governmental laws and regulations. The term "poison" has been used generally by the scientific community to mean a substance that, when used in small amounts, is injurious to health or dangerous to life. The term "toxicity" has been used to describe the degree of poisonous hazard presented by a given substance. Although toxic foods and drugs have been subjected to federal regulations since 1906, the first federal legislation designed speCifically to control poisons was the Caustic Poison Act of 1927. In that statute, Congress avoided defining such terms as "poisons" or "hazard" by simply listing all the chemicals that were considered sufficiently dangerous to regulate. The Federal Hazardous Substances Act of 1960 (FHSA), as amended in 1966 and 1969, has now displaced the earlier legislation, except for foods, drugs, and cosmetics. Under the FHSA, traditional toxicological terms have been given very definite content. A substance is considered "highly toxic" if it produces death in half or more than half of a statis':'Vice President, Medical Evaluations, The Gillette Company Research Institute, Rockville, Maryland
Pediatric Clinics of North America- Vol. 17, No.3, August, 1970
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tic ally significant number of animals after either oral feeding, inhalation, or topical application at a dose level of 50 mg. per kg. or less (oral), 200 parts per million (2 mg. per liter) or less (inhalation), or 200 mg. per kg. or less (topical), or is determined by the Commissioner of Food and Drug to be "highly toxic" on the basis of human experience. A substance is considered "toxic" if oral doses from 50 mg. per kg. to 5 gm. per kg. (and corresponding dose levels in inhalation or topical application) produce death in half or more than half of a statistically significant number of animals or if the Commissioner of Food and Drug determines it on the basis of human experience. A "primary irritant" means a substance that is not corrosive, but is a primary irritant according to available data of human experience or which, when tested by an animal test described in the regulations, gives a prescribed dermal irritancy score. A substance is considered an "ophthalmic irritant" if the available data on human experience indicate that it is an irritant for the eye mucosa, or when tested by the prescribed rabbit eye test, the animal's eyes show, at 24, 48, and 72 hours, discernible opacity or ulceration of the cornea or inflammation of the iris, or a diffuse deep-crimson red of the conjunctivae with individual vessels not easily discernible, or an obvious swelling with partial eversion of the lids. A "corrosive substance" is one that causes visible destruction or irreversible alterations in the tissues at the site of contact. The test for a corrosive substance is whether, by human experience or appropriate animal tests, such tissue destruction occurs at the site of application. A "strong allergic sensitizer" is defined as a substance that produces, by means of an "antibody mechanism," an allergic reaction in a substantial number of persons who come in contact with it. A "photodynamic sensitizer" is a substance that causes an alteration in the skin or mucous membrane so that when these areas are subsequently exposed to ordinary sunlight or equivalent radiant energy an inflammatory reaction will develop. The FHSA definitions and animal screening tests provide a useful outline for undertaking a thorough safety evaluation of a consumer product. But no set of artificial or rigid rules, regulations, and definitions can control the toxicologists' responsibility for determining from animal tests that a proposed new product or ingredient is safe for human use under conditions of recommended use and potential misuse. As the FHSA itself recognizes, evidence of safety or hazard in humans ultimately must prevail over contrary evidence obtained from animal screening tests. The data obtained from poison control centers and physicians are very useful in indicating where product hazards may and may not exist. Accidental ingestion or misuse provides an opportunity for direct human evaluation of a type that would, of course, be wholly improper on an experimental basis. Physicians and hospitals should be urged to report all such results to product manufacturers and to poison control centers, whether or not it involves any hazard to health, in order to develop a better understanding of the correlation between animal screening tests and human experience.
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GENERAL PRINCIPLES All substances are potential toxicological hazards. Balancing a compound's or product's utility against its toxicological hazards is often difficult, particularly because the worst conditions of abuse or misuse cannot always be foreseen. Nor has there been devised any set of animal toxicological, pharmacological, biochemical, or physiological tests which can demonstrate all the possible effects of a new substance or product in the human. In evaluating the toxicological potential of a material or product, the toxicologist must concern himself with not only gastrointestinal absorption but also dermal, ophthalmic and pulmonary absorption and irritation. There is a tendency, due probably to the limited availability of toxicological information and knowledge, to assume that substances which are closely related chemically have similar toxicological properties. While this is often true, it is not universally true. Toxicological evaluation by analogy therefore can be misleading. Dependent upon minor differences of the original chemical structure, intermediate compounds formed as a result of the detoxification process after absorption by the body can produce different and in some cases more toxic intermediates and end products. In order to make the determination that a new product is safe for its intended use, one must first review (1) the known toxicological information on each ingredient, (2) its volume and level of concentration, (3) its potential physiological effects, (4) the combination of intermediates that may form, (5) the method of dispensing, and (6) the extent of proposed human exposure. The above is also true when only one ingredient is involved. The medical, chemical, and pharmacological literature must be critically reviewed for background information. When little or no information on a new ingredient is available from the literature, sufficient toxicological testing must be done to ascertain where the new ingredient fits on the toxicological ladder. After the literature review, a preliminary toxicologic~l judgment can usually be made. Any ingredient with a known problem history should be removed from the formulation and replaced with adequate and less toxic ingredients. If this is not possible, then thought should be given to reducing the potential consumer hazard through adequate labeling and proper containers. The possible effects of removing the ingredients from or detoxifying them in the gastrointestinal tract, if the mixture is accidentally ingested, must also be considered. If it appears that the ingredient will react differently in a mixture, animal testing should be designed to evaluate that ingredient in the specific mixture. In all instances, the final product must be tested using the ingredients that will be used if the product is manufactured for public distribution. Adequate specifications must be set for the ingredients and the product to reduce the problem of batch to batch toxicological variation and nonreproducible biological results. In addition, stability data must be obtained to insure that the product is stable under conditions of heat and cold, and that it has the ability to withstand pathogenic bacteriological contamination.
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STAGES IN TESTING Over the past 100 years academic, government, and industry pharmacologists and toxicologists have developed an armamentarium of animal screening tests which can determine the quantity of test material required per gram of experimental animal body weight to produce untoward or fatal effects. The terms minimum lethal dose (the smallest quantity known to have produced death), median lethal dose (that which is fatal to a given percentage of a species of animals-usually 50 per cent-abbreviated as LD 50 ), and toxic dose (the dose capable of producing functional derangement in the animal or human) have become part of the toxicologist's vocabulary. Once any substance is inhaled, ingested, or topically absorbed, it will become involved with the intricate chemistry of the body. Some substances will have essentially no effect while others may produce functional (dizziness, nausea), biochemical (detected by clinical chemistry studies), or structural (noted on microscopic organ examination) changes. In attempting to recognize the possible toxicological properties of the product and assess risk, three stages of testing are usually involved. They are (1) animal acute screening studies, (2) animal subacute and chronic studies, and (3) human prophetic patch and use tests. The specific product, its proposed use and proposed exposure will dictate the order and type of tests that should be utilized.
ACUTE TOXICITY Five types of animal screening studies are generally used. They are (1) acute oral toxicity, (2) ophthalmic irritancy, (3) percutaneous toxicity,
(4) dermal irritancy, and (5) sensitization tests. If these studies elicit unexpected or unexplained reactions, additional tests to evaluate the specific problem must be instituted. The most frequently used test is the acute oral toxicity test. Here several doses, spaced according to some geometric or logarithmic progression, are given to a species of animals. At least three dose levels are used. The animals are observed and the number of deaths are recorded daily for a period of 14 days following intubation. Gross and microscopic examinations of various organs are performed in many cases. Too little attention has been paid to subjective observations and possibly too much to objective measures. The physician dealing with acute human oral ingestion is dealing with subjective observation and usually does not have the opportunity for confirmatory laboratory tests. Thus, the subjective observations noted on animal studies can provide valuable information. The animals most used in oral toxicity tests are rats and mice. Voluminous toxicological data have been compiled with these animals, but it cannot be assumed that the results obtained in these species are predictive of human responses. For example, the rat and mouse are resistant to certain chemical agents (aromatic amines, phenols) and
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respond, in many cases, differently than other species. Morphine is a stimulant to the mouse and cat and a depressant to the rat, dog, and man. Not only may animal species react differently to the same material or compound, but sex, age, breed, and weight can create differing toxicological responses and LD50'S in the same species. Thus when working with a new substance, more than one species, sex, age, and weight of animal are used. Other types of hazard testing also do not necessarily predict human responses. For example, materials that are minimal dermal irritants in humans, when applied to shaved rabbits' backs, may show either a severe or minimal response. When severe reactions are produced in the rabbit eye by a test material it does not necessarily follow that the same reaction will occur in the human eye. On the other hand, it is generally believed that when a new substance causes no ophthalmic reactions in rabbits it is highly unlikely to cause ophthalmic reactions in humans. Thus, when concern over rabbit ophthalmic results occurs, dog and monkey ophthalmic studies are instituted in order to obtain data more predictive of human reactions. In many situations these species will demonstrate only minimal transient conjunctival irritation rather than the severe and prolonged corneal, irital, or other ophthalmic damage seen in the rabbit eye. Because the test methods now available are far from perfect, close attention and scrutiny must be paid to all effects noted. Generally, when several dissimilar species demonstrate the same effect to a given substance or product, it is very likely that the human will respond similarly. It is important therefore to review the overall results of preliminary animal screening tests and then evaluate the speCific results on which different data were obtained with a multispecies comprehensive battery of tests which can more specifically isolate the toxicological activity. At all times one must attempt to elucidate the substance's mechanism of action. One must determine the nature and manifestations of the damage (organ, cellular).
SUBACUTE AND CHRONIC TOXICITY Subacute and chronic poisoning, in many cases, offers serious diagnostic problems to the physician. Animal studies which help elucidate these toxicological hazards are subacute and chronic ingestion studies, dermal absorption studies, inhalation studies, and in certain cases teratogenic studies. Subacute and chronic toxicity tests are designed to determine the primary toxic effect of the product when it is used at a low dose level a few times or on a daily basis. It is most important in these to consider carefully not only the product's chemical and physiological properties, but also its intended use. Every class of product and every product within each class presents specific problems. While standard chronic test designs are available and acceptable, one should use or design a type of test protocol that will give the most fruitful information on the particular
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product or ingredient, rather than using an all-purpose test which can lead to broad results that do not afford specific conclusions. Other considerations that must be resolved prior to testing are (1) animal species to be used, (2) diet, (3) route and frequency of administration, (4) dosage, (5) number of dose levels, (6) number of treated control and untreated control groups, (7) age, sex, and number of experimental animals, (8) duration of the study, and (9) parameters to be examined (symptomatology, hematology, clinical chemistry and organ function, gross and histopathologic examination). Irrespective of the test design finally employed, it should include one dose level of animals at which signs of toxicity are produced. Since the number of animals exposed to the product can never equal the number of humans who will eventually use it, one cannot expect to detect with any reliability toxic manifestations that may occur in a small minority of human users. Studying large numbers of animals (50 or more per dose level) causes cursory attention to be given to any one. It is far better to study in detail a smaller group (10 to 25) with adequate controls. These controls extend to not only the use of treated control and untreated control animals but also the use of both males and females from a single source and adequate staff to care for the animals and maintain a normal environment for them.
HUMAN PROPHETIC PATCH TESTING Human dermal patch testing aids in eliciting the product or ingredient's potential hazards with respect to (1) primary skin irritancy, (2) contact sensitization, and (3) contact photo-allergic or phototoxic reactions. The particular type of primary skin irritancy test that is run will depend on the intended use of the product and the background information on the propensity of the product or ingredients to be irritants. One can use (1) overnight, semi-occlusive patching, (2) daily patching for 5 or 8 hours for 1 week occlusively, (3) daily occlusive patch testing for 2 weeks, Monday through Friday, and (4) patch testing daily occlusively for 2 weeks, Monday through Friday, a 1 week rest, followed by 4 weeks of product use under label directions, then challenge patches. The latter is a combination irritancy-contact sensitization test. Primary irritancy patch tests should never be run without adequate controls which should include a product that has had extensive safe commercial use. Further, the test design should be such that minimal erythema is produced in some of the test subjects with the control product. Tests such as those described by Brunner, Kligman, and Rostenberg are useful in evaluating potential contact sensitization. The specific test used should be dependent upon the propensity of the product to produce sensitization and its potential use. Curwen and Jillson have described a diagnostic test which, with the following modification, is used to elicit photosensitization potential. The
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test subjects' backs are exposed daily to the proposed new ingredient or product for a minimum of 14 days. During this period the test sites are also exposed to a source of ultraviolet light (11J2 minutes) at least five times (day 1,4,7,10, and 14). Following a short rest period of 4 or 5 days, the diagnostic photo-patch test is conducted.
ANTIDOTAL EVALUATION After a review of the toxicological data the product should be given a relative toxicological rating based on the possible amount that could be ingested, inhaled, or absorbed based on the body weight or surface area-dosage relationship. If antidotal procedures are required they should be developed and tested in appropriate animal species for their efficacy. Generally this involves deciding (1) should the substance or product be removed from the gastrointestinal tract, (2) what antidote, chemical or physiological, should be given, (3) what supportive therapy should also be instituted, and (4) which should be done first. Some thought should also be given to repeating acute oral toxicity studies in young animals. The proposed antidotal procedure as evaluated in animals should be made available, either in writing or by telephone, to qualified individuals and organizations on request. This information should be available for the products that are considered practically nontoxic as well as those that are considered to have some hazard. To the physician both types of information are valuable and may, in many cases, prevent the rigors of antidotal procedures when they are not required.
CONCLUSIONS Toxicological studies are involved biological research programs. It is impossible to set out a specific set of tests and rules that one should follow. Each evaluation must be designed with the background literature, the ingredients, the length of exposure, and the type of use in mind. This is especially true with respect to potential misuse. Thus, throughout its life, the product must always be under scrutiny. New products and new ingredients can produce different toxicological effects and different pharmacological modes of action, based on different chemical structures not only of the ingredients but also of the products of their interactions. Toxicological testing can aid in the replacing of more toxic ingredients with less toxic ones. We can never achieve the use solely of nontoxic ingredients, because under appropriate conditions any substance can produce toxicological effects. Toxicological testing can warn the manufacturer and the physician of the possible side effects which could occur under certain conditions of use and misuse. This, in association with proper labeling, packaging, and education of the consumer, can contribute to safer consumer use of products.
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REFERENCES Brunner, M. J., and Smiljanic, A.: Procedure for evaluating skin sensitizing power of new materials. Arch. Dermatol., 66:703, 1952. Code of Federal Regulations. Part 191, Chapter 1, Title 21, revised as of January 1, 1969. Washington, U.S. Government Printing Office, 1969. Curwen, W. L., and Jillson, O. F.: Light hypersensitivity. J. Invest. Dermatol., 34:207, 1960. Kligman, A. M.: The identification of contact allergens by human assay. J. Invest. Dermatol., 47:369, 1966. Lichtfield, J. T., Jr., and Wilcoxon, F.: A simplified method of evaluating dose-effect experi· ments. J.Pharmacol. Exper. Therap., 96:99-113, 1949. Rostenberg, A.: Predictive procedures for eczematous hypersensitivity. A.M.A. Arch. Ind. Health, 20:9, 1959. Wei!, C. S.: Tables for convenient calculation of median-effective dose (LD,. or ED,.) and instructions in their use. Biometrics, 8:249-263, 1952. 1413 Research Boulevard Rockville, Maryland 20850
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Ms Sql Server
Network Administrator
Responsibilities included creating users, groups and managing profile using Active Directory Service.
- Installing and configuring and patching Linux, Windows 2008 and Windows 2012 Servers.
- Configuring and Installation of Cisco Switches by implementing VLan over OSPF.
- Configured client workstation consisting Windows Desktop version (XP, 7, and 8) as well as MacBook's Pros.
- Designed, implemented, and supervised VPN Administration though the process of a VPN concentrator for remote desktop support and administration.
- Installing, upgrading and patching Oracle Database Servers, MS SQL Database Servers over various operating systems.
- Maintained, troubleshooting, and repaired hardware and computer peripherals including Printers, Desktops, Scanners, Wireless devices.
- Designed and implemented and maintain company core Wi-Fi policy and procedures.
- Assisted in configuration of PDX system utilizing VOIP.
Candidate Info
years in workforce
years at this job
Business / Computer Science
Information Technology Project Management
IT Supervisor / Senior Network Administrator, Information Technology Department
Managed IT Help Desk, Electronic Document Interchange and System Administration.
- Balanced vendor engagement, project management and operational duties.
- Maintained accurate records of all maintenance, inventory, and security measures.
- Monitored and tested network performance, supplied statistics and reports to management.
- Configured, tested and troubleshot routers, firewalls, LAN/WAN/VLAN/Wireless network.
- Administered Intrusion Detection Systems, VoIP devices and Active Directory.
- Upgraded and maintained servers, server room infrastructure, desktops and peripherals.
- Participated in on-call rotation including off-hours and weekends.
Candidate Info
years in workforce
years at this job
Electronics
Network Administrator / Back Office Support
Network administrator/hardware/software/VoIP support for debt settlement company
- Performed AR/AP functions and created financial reports for management
- Interfaced with the clients to resolve any inquiries or complaints. Audited sales and customer service call recordings to ensure SOP standards
- Wrote sales and employee performance reports in Excel and Word.
- Published web articles for company websites to generate sales leads
Candidate Info
years in workforce
years at this job
Psychology
Sr. Network Administrator
Responsible for the repair of hardware/software configurations, support of all laptops, desktops, and servers to ensure full functionality. Installation of network & phone wiring, including Cisco switches and hubs as needed. Also responsible for plant access control system and video surveillance equipment.
- Central contact between this division and Dell; responsible for RMA process and/or equipment replacement parts ordering.
- Installation of network & phone wiring, Cisco switches and hubs as needed.
- Created and installed IOS modifications to block all known "file share" usage.
- Installed and configured new managed CISCO switches (3550's, 2600's, & 3600's).
- Installed and configured Ghost, SUS, and SMS servers.
- Supported Avaya phone system (adds, moves, deletes).
- Selected, negotiated, and managed contracting firm to rewire the entire building w/CAT6 & Fiber.
Candidate Info
years in workforce
year at this job
Management Information Systems
Design
Network Administrator
Administered duties for three state sites. Appointed Lead Technician.
- Managed Help Desk support staff in a 24/7 environment.
- Deployment and administrative experience with Cisco Equipment
- Managed databases and active directories. Supported 300+ employees accessing mail via Microsoft Exchange.
- Scheduled team meetings and project documentation. Set up presentation equipment and video conference.
- Implemented best practices for desktop/laptop, server support.
- Configured and maintained servers, computers, printers, scanners and other equipment.
- Maintain inventory, track equipment usage and monitor condition.
Candidate Info
years in workforce
years at this job
Outsourced Network Administrator
Managed 30 Server/100 user Network with two direct reports.
- Designed application and OS rollout for 1500 PC's using over 30 applications.
- Designed external company Web Page and managed all VPN connections through Microsoft ISA Server.
- Compiled and organized a common problem/solutions MIS FAQ page on internal Intranet.
- Performance Monitoring and Seagate Crystal Reporting to upper management.
- Offsite-administrator of 32 Windows NT/2002/2003 networks.
- Performed Sarbanes Oxley network auditing of Network, including Active Directory and Wireless Access Points.
Candidate Info
years in workforce
years at this job
Network Administrator / Database Developer
Administered servers and network for main office and remote facilities
- Provided technical support for main office and remote facilities
- Installed and troubleshot hardware and software
- Developed and maintained databases and created programs in Microsoft Access
- Administered company Web sites and implemented online databases
- Produced instructional videos, help files, and documentation for end users and provided training
Candidate Info
years in workforce
years at this job
Mn Computer Technical Support Specialist
Helpdesk Desk Level 3 and Network Administrator
Responsible for technical troubleshooting within an enterprise environment, including system crashes, slow-downs and data recoveries. Engaged and tracked Priority 1 issues, with responsibility for the timely documentation, resolution and closure of trouble tickets.
- Network: Built and maintained a multilayered LAN/Wan, using Extreme switches, Watchguard firewall's, and Adtran edge routers.
- Phillips Company: Offsite sales office of USDP. Responsible for day to day IT operations and workload for 30 remote workers and 10+ office staff.
- Researched and implemented a Helpdesk ticketing portal to aid in tracking pc related issues and trends.
- Directed program of the pc/server lifecycle. Purchased, set up, maintained, replaced, and recycling of all pc's and servers.
- Installation of an enterprise wide VOIP phone system.
Candidate Info
years in workforce
years at this job
Computer Technology
Applied Science
Network Administrator
Administered and supported seven Windows 2003 Active Directory servers, three Windows 2000 servers, two Windows 2003 Terminal servers and two Novell 4.11 servers
- Ran load balancing on Windows 2003 Terminal Servers
- Installed, configured and troubleshot Citrix 1.8, SonicWall 330 Firewall, SonicWall Tele3 Firewall, and Sonic Wall VPN
- Installed, configured and maintained Brightstor Arcserve backups
- Maintained computers and peripherals for 150 users at our Illinois, Indiana and Mexico sites
- Managed Microsoft Exchange Server 5.5 for three sites
- Company products included Microsoft Windows XP, Windows 2000, Office 2003, and Office 2002
Candidate Info
years in workforce
years at this job
Information Technology
Telecommunications Systems Management
Network Administrator
Handles technical troubleshooting within a fast paced environment, including system crashes, operator training, network administration and maintenance.
- On call to provide networking/desktop support and perform account maintenance tasks.
- Recognized for teamwork, flexibility and work excellence in providing IT support to staff and guests.
- Implemented complex network design in small office setting ensuring secure use.
- 99.9% network uptime.
- Trained others in use and administration of (BYOD) Bring Your Own Device network environment.
Candidate Info
years in workforce
months at this job
Computer Networking Technology
Information Technology
Network Administrator
Prove technical support and manage non-secure internet protocol routing network (NIPR), Secure Internet Protocol Routing Networks (SIPR), Combined Enterprise Regional Exchange System (Centrix), and communications issues
- Remotely installed and troubleshot technical issues in timely fashion and in accordance with designated priority as outlined in standard operation procedures (SOP)
- Analyze system requirements, prepared work plans, conducted design reviews, performed validation testing and drafted documentation and conducted training classes.
- Escalation support and following of STIG and Regional procedures
Candidate Info
years in workforce
years at this job
Computer Science
Computer Science
Network Administrator/manager
Managed and monitored network and internal/external site communications.
- Active Directory Administrator: Responsible for site maintenance and security for North America.
- Antivirus Administrator: Managed Symantec and Checkpoint corporate antivirus solutions.
- MS Exchange 2003/2007 Administrator: Maintained company email systems for North America. Managed mail flow, SPAM filtering (using Surf Control), pocket PC connections, Outlook support, and security (using Symantec Mail Security) for 1200 mailboxes. Researched and implemented the company Sherpa Archiving and eDiscovery solutions. Co-implemented and maintained an Exchange DR site using Double Take software replication. Project lead for Exchange 2007 migration.
- Manager for three employees. Accountable for day-to-day workload as well as yearly performance reviews.
- MS SMS Administrator: Sole implementation and maintenance of the corporate MS SMS 2003 sites.
- Web security Administrator: Controlled access to external websites using SurfControl. Provided daily access reporting.
Candidate Info
years in workforce
year at this job
Network Administrator
Configure, troubleshoot, provide maintenance, install, upgrade firmware, and monitor performance for more than 1500 Cisco network routers and switches on three separate enclaves (unclassified/classified).
- Monitoring, maintenance and modeling support for the Enterprise Spectrum Network Management system using CA SPECTRUM OneClick.
- Add, remove, modify and maintain more than 15,000 Cisco IP phones using Cisco Unified Call Manager Administration.
- Ensure outages and corrective actions are reported to Theater Network Operations Center.
- Documentation and mapping of Network diagrams using Microsoft Visio.
- Configured 802.1x port-based Network Access Control (PNAC) at the access layer and distribution layer, providing an authentication mechanism for devices requesting to connect to the LAN.
- Increased network redundancy and throughput by converting multiple trunk links into individual Port Channels.
- Utilize BMC Remedy for tracking, troubleshooting and resolution of over 500 Tier II/III network-related incidents.
Candidate Info
years in workforce
years at this job
Helpdesk Technician, and Network Administrator
Responsible for 2-11 personnel, all off ship communications, and the ship wide LAN during 12 hour shifts.
- Lead network administrator for more than 10 servers varying from Windows to UNIX.
- Trained new personnel on network administration, telecommunications, EKMS and helpdesk procedures.
- Managed a shipboard helpdesk and completed trouble calls in a professional and timely fashion for 300+ users.
- SharePoint 2003 System architect adept at designing, implementing, customizing, upgrading and migrating SharePoint Portal Sites.
- Experienced in Microsoft SQL Server relational databases, and development/maintenance.
Candidate Info
years in workforce
years at this job
Associate of Science
System/network Administrator
Assisted Network Administrator in IT department with responsibility for managing internal network infrastructure supporting hundreds of users.
- Worked on Micros 9700 Hospitality Management System software and hardware.
- Provided system administration and helpdesk support to include software and hardware installations, maintenance, upgrades, troubleshooting, user account administration, and support under stringent deadlines in a fast-paced, 24X7 environment.
- Built workstations and servers; configured laptops; and troubleshot LAN, and WAN access issues.
- Served as Assistant Administrator on major projects and implementations.
Candidate Info
years in workforce
years at this job
Network Administrator
Implemented Cisco Unified Communications Manager (CallManager), to improve the quality of voice and video communications. This reduced costs dramatically and also made other activities easier, such as: remote training for staff members, better communication with branches nationwide, better communications with customers and stakeholders, etc.
- Provided QoS to control the traffic of information across the network, by reserving resources, setting priorities to share data, voice and video, and monitoring the performance of the network.
- Provided maintenance to the datacenter, which included Windows NT and Windows Server 2008 servers and also worked setting up and maintaining virtual machines (VMWare). This also included daily back up of the data stored in the servers and databases.
- Firewall configuration and implementation of McAfee to ensure internet security.
- Designed an emergency and recovery plan in case of disaster.
Candidate Info
years in workforce
years at this job
Electrical Engineering
Network Administrator, Operations Manager
Developed and implemented a network system consisting of 6 computers integrated into one server. Conducted network stress tests and software capability tests by maximizing software traffic using more than one computer. Maintain company's computers, software updates and installations, as required.
- Designed network from the ground up when company started in 2001
- Trained employees on software and hardware use
- Assisted in customer relations acquiring customers for business
- Assisted in daily, weekly accounting and payroll tasks
Candidate Info
years in workforce
years at this job
Bachelor of Science
Network Administrator
Created and maintained our WSUS server
- Managed the IT Department
- Oversee and assisted in the conversion from Fiserv to Symitar
- Supervised and support the Help Desk ticketing system
- Maintained the Active Directory
- Installed and supported the servers
Candidate Info
years in workforce
months at this job
Associate of Science
Network Administrator
Responsible for installing and supporting network hardware, software, and applications for PC's, laptops and servers
- Implement ideas that enhance the effectiveness of technologies for users
- Performed preventive maintenance on workstations, laptops and servers
- Responsible for network evaluations, troubleshooting various network problems and implementing software and hardware upgrades
- Provided escalated IT support including hardware troubleshooting, backup and recovery, email support and application support
- Setup and maintained LAN connections and wireless networks
- Scheduled, coordinated and deployed server updates and preventative maintenance
- Evaluate the needs of the company regarding purchase of new computer equipment
Candidate Info
years in workforce
years at this job
Electronics & Communication Engineering
Electrical Engineering
Network Administrator
Configuring and installing network infrastructure and telecommunications systems
- Configured CISCO switches, GSM Gateways, call centers and telephone systems
- Lead a team of six technicians on planning and implementing the installation of a call center system
- Provided full system support and maintenance
- Fulfilled administrative responsibilities such as dealing with vendors, clients and providing training for call center agents
- VHDL (VHSIC Hardware Description Language)
- Asterisk (Open source PBX based on LINUX OS)
Candidate Info
years in workforce
years at this job
Biomedical Electrical Technician
Network Administrator
Constructed virtual infrastructure including virtual pooling and machines in a VMWare environment.
- Created new user accounts and virtual machines as needed.
- Maintained active directory and exchange as well as virtual systems in an environment that included Microsoft Server 2008 machines and VMWare VSphere 5.0, with Wyse end terminals.
- Installed setup and maintained the office helpdesk software, SysAid, to allow for a quick resolution of office network issues.
- Troubleshoot and repair user application, hardware, and server level issues.
- Install, Upgrade And repair office applications such as Microsoft Office, QSHR reporting, CCE.
Candidate Info
years in workforce
months at this job
Electrical Engineering Technology
Network Administrator
[company name] was a nationwide network of approximately 60 inbound and outbound call centers. The network hub was located in Rosedale Maryland. As a Network Administrator with Answernet Networks I maintained and troubleshoot all network related issues to include local telecom and system related issues.
- Maintained and troubleshoot multiple VM platforms across 12 ESXi machines.
- Managed user and station profiles through an Active Directory environment
- Managed and performed backups of corporate IMAP email accounts
- Performed software updates on Spectrum hardware
- Performed needed changes to LAN environment to include but not limited to: joining and removing operator stations to the domain, troubleshooting station outages, troubleshooting user profile issues, imaging operator stations using Novell imaging software
- Created and maintained up to date documentation of pfSense firewall build
- Managed and troubleshoot pfSense firewall rules to maintain security continuity
Candidate Info
years in workforce
months at this job
Desktop & Network Administrator
Provided network administration support for business owners managing business networks.
- Worked with Active Directory, Exchange servers, SBS servers, and Hyper-V setups.
- Managed servers on a daily basis, 2008r2, and 2011 SBS 2012 r2 servers monitored logs fixed errors. Desktops, laptops and printers.
- Setup new accounts in Active Directory (AD) and exchange 2007 and 2010
Candidate Info
years in workforce
years at this job
Computer Information Systems
Business
Network Administrator
IWATSU PBX and voicemail networks Programmer
- Network administrator of server 2003 and XP workstations as well as printers, scanners, and fax
- Implemented VoIP system with various WAN providers.
- Provided network wiring for voice, data, and video in larger healthcare facilities and individual customer sites.
- Designed VoIP systems and provided ongoing training and support of state of the art voicemail and communication devices, i.e. Smartphone technology.
Candidate Info
years in workforce
years at this job
Information Technology
Network Administrator
Work on a team of four system administrators to support over forty business networks from ten to fifty users
- Responsible for securing networks for businesses that do not have security practices in place
- Configure and maintain Dell SonicWalls for clients and manage them using Dell GMS
- Implement site-to-site VPN's using IKE, DES, and SHA for authentication and encryption
- Actively optimize networks to run more efficiently to maintain 100% uptime
- Multitask between issues in a fast paced and constantly changing environment
- Write procedures to improve troubleshooting and increase operational efficiency
Candidate Info
year in workforce
year at this job
Certificate
English Translation
Network Administrator
The company was one of the largest refrigerated warehouses in the country with racked spaces leased for storing food such as meat, fruit and vegetables.
- The network included multiple wireless routers and the vendors used a fat-client application to connect to the hosted application on the company's server
- The industrial level scale used the network to send data to the main server
- The fat-client applications' communication established through an RRAS console
- Provided support and maintanence for the network infrastructure
- Windows 2000, Windows XP, McAfee AntiVirus, Outlook Express, Email Contacts, connecting a legacy Novel network to the Windows
Candidate Info
years in workforce
years at this job
Psychology
Network Administrator
- Resolved hardware, software, and application problems for networked, computer systems, and other resources
- Monitored computer workstations, servers, and application databases for performance and resource utilization
- Reported hardware failures to the appropriate vendors via telephone or on-line systems
- Trained end-users in operating systems, computer language, applications, utilities and hardware operation, one-on-one, in-person and using remote desktop technology
- Analyzed problems, and designed technical solutions for enterprise or specific end-users
- Helped design and transition to a far more efficient, less expensive, backup, archive, and disaster recovery system in a multi-location, geographically diverse organization
- Organized and lead hardware and software training classes (video conferencing, newly adopted enterprise applications)
- Provided one-on-one help with smart phones and useful apps (iOS, Android)
- Created technical documents and training materials for end-users
Candidate Info
years in workforce
year at this job
Network Administrator
- Managed team of 70-120 employees and contractors responsible for IT in past 10 years. Recruit, hire, train and supervise staff, or participate in staffing decisions.
- Managed an asset management inventory project that covered over 1700 Lowes Store utilizing over 100 techs from various agencies all over US.
- Provided a store/customer perspective/assessment, of current technological capabilities for the identified stores.
- Addressed and resolved Project issues and identified solutions to meet productivity, quality and customer goals.
- Ability to work with multiple customers, and able to build and maintain effective team and customer relationships.
- Managed project plans to meet target deliverables.
- Provide users with technical support for computer problems.
Candidate Info
years in workforce
years at this job
Information Systems
Network+
A+
Network Administrator
- Installed, configured, and managed Exchange 2010, Exchange 2013, SharePoint Server 2013, Lync Server 2013, Dynamic CRM Server 2011, and a File and Print Server on Windows Server 2012 using Hyper-V.
- Performed weekly backups and test recovery procedures using Veritas and Microsoft backup solutions.
- Installed and managed SysAid asset tracking and management system.
- Configured Cisco Aironet WAP enterprise wireless infrastructure.
Candidate Info
years in workforce
years at this job
Art
Information Technology Infrastructure
Master of Arts
Network Administrator / IT Manager
- Identified and stabilized business critical infrastructure during aggressive growth within the company.
- Managed and mentored IT staff to provide business focused support in a high demand environment.
- Provided immediate expert level support to business services including but not limited to application and web development, high volume call center platform, compliance inquires, and infrastructure spanning multiple geographic locations.
- Supported virtual and physical data center that provides hosted applications to internal and external business partners.
Candidate Info
years in workforce
years at this job
Computer Networking Systems
Network Administrator
Implements new hardware installations for company LANs across ten cities and maintains each topology
- Expert in terminating fiber patch and MM runs from workstations to IDF and MDFs using Corning kits
- Monitors and resolves issues if Wi-Fi access points go offline by either physical connection or reboot
- Builds new runs to patch panels, layer 2 and layer 3 switches for LAN expansion
- New installation and part replacement for network printers and routinely cleans units per PM schedules
- Works diligently and professional using field trade tools and with any level of supervision
Candidate Info
years in workforce
month at this job
High School Diploma
Network Administrator
Manage Server 2003, server 2008, server 2012, Exchange 2007, Exchange 2010 and Exchange 2013 environments, Lync 2010, Lync 2013
- Administrate Terminal Server and Hyper-V environments
- Deploy Laptops, Desktops, Thin Clients and printers
- Install patch panels, network drops, rack mounted servers, server racks and components
- Provide remote support to customers with issues involving Microsoft office 2003, Office 2007, Office 2010, Office 2013, Gmail, In-house application, web browsing software, Apple products, VPN clients, and web applications
- Create and manage support documentation for IT systems
- Provide onsite support for projects, network issues, etc.
Candidate Info
years in workforce
years at this job
Economics
Sales / Marketing
Network Administrator
To provide helpdesk support to students and staff in maintaining a smooth flow of information from user machines to the network/servers.
- Maintaining a network of about 1200 plus users. Responsible to create their network accounts, assign network permissions and provide them support with their IT needs.
- Constantly looking for new techniques and technologies to be more efficient and productive.
- Providing help and support to users on different databases running on our servers.
- Maintaining an inventory of the computers and other technology items and troubleshoot/repair these devices if needed.
- Maintaining a backup strategy of all databases running on different servers to secure sensitive data.
- Arranging for the required technologies for students/teachers in class rooms, library media center and computer labs.
Candidate Info
years in workforce
years at this job
Economics
Network Administrator
Performed duties as Network Administrator including troubleshooting and repair of hardware, software and communication equipment.
- Compilation and analysis of monthly reports including Trend Analysis and Efficiency Reports
- Maintain operational NETSEC in a SCIF environment.
- Conducted training to incoming personnel as well as inter-department cross-training and briefing.
- Perform administrative duties including monthly counseling, scheduling, and maintaining training records.
- Enforced compliance with organizational standards including security protocols and Standard Operating Procedures.
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Network Administrator Duties and Responsibilities
From our analysis of job postings, Network Administrators perform a variety of duties in their quest to keep computer systems working properly. Key responsibilities frequently include the following:
Maintenance Network administrators examine the day-by-day performance of the organization's network and back up its data. They also install network hardware and software and teach people to use it.
Troubleshooting Whether the whole server goes down or an individual is having difficulty accessing email, the Network Administrator examines the problem and uses his or her expertise to rectify the situation.
Ensuring security Network Administrators add users, deal with passwords, delete old accounts, construct firewalls and take other protective measures to ensure data isn't compromised and unauthorized users cannot gain access to the system.
Evaluating Network Administrators look at data to figure out how well the system is working and accomplishing needs. Findings may result in them seeking improved methods/equipment and consulting with vendors.
Network Administrator SkillsWhile a love of computers and technology is at the heart of becoming a Network Administrator, a variety of other qualities are also central to developing into a top-notch professional. Network Administrators should be thorough in their work and committed to security in order to protect information. They also must be persistent problem solvers capable of drawing from their knowledge in both traditional and creative ways. Employers seek Network Administrators who excel at things such as:
- Adapting to situations, since problems don't always follow a textbook format
- Attending to details in order to maintain efficiency and spot potential pitfalls
- Handling problems calmly because leaders and employees tend to get irate when technology goes awry
- Strategically analyzing situations (such as "Is it just one person who is having trouble logging on or multiple employees? Do those affected have anything in common, such as the same type of devise or geographic location?")
- Juggling multiple tasks as needed
- Prioritizing job responsibilities to ensure the most essential matters get handled first
- Aspiring to stay on top of the field through additional training as needed since technology oftentimes changes quickly
- Communicating in language that non-tech people can understand
- Working well with others to understand problems, acquire pertinent information, assign tasks and keep those affected updated
Network Administrator Education and TrainingMost Network Administrators hold a bachelor's degree or higher. Common majors include computer network and systems administration, computer science, information science, computer engineering and electrical engineering. Employers sometimes require Network Administrators to get a certificate showing their competency with a specific product, such as Microsoft or Cisco. Hands-on experience also grabs a hiring manager's attention, so an internship can pay off significantly when searching for a job. Since technology advances quickly, Network Administrators should plan on continuing their education throughout their career.Employers will often pay for these classes and conferences.
Network Administrator SalaryThe median annual salary for Network Administrators, categorized by the BLS as "Network and Computer Systems Administrators," is $79,700. Network Administrators in the 10th percentile earn about $48,800 a year, and the highest paid make in excess of $127,600 a year. Network Administrators in Maryland, District of Columbia and New Jersey make the highest median salaries in the U.S. - $106,000, $96,290 and $94,870, respectively.
Network Administrator Resources
We checked the Web to find the top industry resources to help you explore a career as a Network Administrator. From relevant books to industry groups, this list offers plenty of sources of helpful information.
Excellence in IT: Achieving Success in an Information Technology Career by Warren C. Zabloudil - For people looking to rise to the top in the field, this book offers advice on subjects such as keeping up with new developments, combating stress and offering top-notch service.
Network Professional Association - The website for this group contains a plethora of resources for Network Administrators of all career levels, such as insightful interviews with successful Network Administrators, articles on topics of interest, job listings and assistance in finding your local chapter.
Networks and Systems Professionals Association - Another great reference point for prospective Network Administrators is NaSPA's website. The group has been around for more than 30 years and continues to advance with the times, just like technology itself.
Association for Women in Computing - Since 1978, this organization has been helping women advance their computing careers. Mentoring, networking and continuing education are among the topics covered on its website.
Technology Jobs Network - With more than 118,000 members, this LinkedIn group contains a variety of computer professionals who likely can answer your questions about becoming a Network Administrator or steer you in the right direction.
Create your own professional resume in just minutes. | https://www.jobhero.com/resume/examples/data-systems-administration/network-administrator |
Interpret financial information to make more informed decisions.
8 weeks, excluding
orientation
6–8 hours per week,
entirely online
Weekly modules,
flexible learning
1
A practical finance toolkit that you can use to assess the performance and health of your organization and engage meaningfully in financial conversations and planning.
2
An understanding of the factors impacting financial decision-making, empowering you to make more informed choices in your company or department.
3
The ability to engage meaningfully with your company’s financials to make more informed, strategic business decisions.
Module 1:
Understand Financial Decision-Making
Explore how the financial decisions of a company contribute to its objectives.
Module 2:
Assess the Financial Health of a Company
Learn how to use financial ratios to determine how healthy a company's finances are.
Module 3:
Estimate the Free Cash Flow
Learn how to prepare cash flow projections for a company.
Module 4:
Explore Financial Markets and Financial Instruments
Examine the role of financial markets and the various financing options available to companies.
Module 5:
Estimate the Cost of Capital
Understand the method used to calculate the cost of capital.
Module 6:
Putting it Together: Enterprise Value
Explore the term ‘enterprise value of a company’ and learn how to calculate it.
Module 7:
Evaluate Investment Opportunities
Learn how to evaluate investment opportunities to make investment decisions.
Module 8:
Understand Bias in Financial Decision-Making
Examine the influence of bias on financial decision-making.
Fatma Sonmez-Leopold
Assistant Teaching Professor of Finance at the Martin J. Whitman School of Management
Dr. Sonmez-Leopold has earned a B.S. in mathematics from Hacettepe University in Ankara-Turkey, two M.Sc. degrees in Mathematics and Engineering Management/Industrial Engineering from Middle East Technical University in Ankara-Turkey and a Ph.D. in Finance from the University of Toronto, where she was an assistant professor of finance at the Smith School of Business at Queen’s University.
Her teaching and research interests include empirical asset pricing, investments, corporate finance, behavioral finance, and household finance. She has worked on the pricing impact of idiosyncratic volatility, systematic risk and leverage, and has published her work in a number
of international scholarly journals.
This Syracuse University online short course is delivered in collaboration with GetSmarter, a brand of 2U, Inc. Join a growing community of global professionals who have already benefited from the opportunity to:
Gain verifiable and relevant competencies and earn invaluable recognition from an international selection of universities, entirely online and in your own time
Enjoy a personalized, people-mediated online learning experience created to make you feel supported at every step
Experience a flexible but structured approach to online education as you plan your learning around your life to meet weekly milestones
Want to know more?
Enter your information below to access the course brochure from GetSmarter. | https://short-courses.syracuse.edu/presentations/lp/syracuse-university-finance-for-non-financial-managers-online-short-course/?utm_source=teachdotcom&utm_medium=minisites&utm_campaign=fpt_financeshortcourse |
Science Curriculum Intent
Science teaching at Stoneydelph Primary School aims to give all children a strong understanding of the world around them whilst acquiring specific skills and knowledge to help them to think scientifically, to gain an understanding of scientific processes and also an understanding of the uses and implications of Science within the wider world, their own experiences and to prepare and equip them for the world of today and tomorrow.
We seek to encourage and support all children to develop and foster a natural curiosity and enthusiasm to explore the world around them and to question and discuss their learning in an open and safe manner.
Science Impact Statement
A range of extra-curricular activities are arranged for pupils in our provision throughout each academic year. Parents are invited to share in these experiences with their children.
In addition, we measure the impact of our curriculum through the following methods:
- A reflection on standards achieved against a progressive ‘essential skills’ ladder from Cornerstones;
- A carefully mapped Science curriculum;
- A celebration of learning during ‘Science Weeks’ which demonstrates progression across the school;
- Pupil discussions about their learning. | https://www.stoneydelph.staffs.sch.uk/web/science/520167 |
INTRODUCTION
============
The identification of genetic markers that impact the phenotype of an individual is an important step towards identifying the genetic basis of disease. Replicated findings of such associations have become increasingly common ([@gks1449-B1]). However, a formidable remaining challenge is finding the mechanisms through which these identified markers act to ultimately drive phenotypic variation. The epigenome is now recognized as playing a critical role in developmental processes and is also likely to be involved in ultimately determining phenotypic traits ([@gks1449-B2]). For example, DNA methylation of CpG dinucleotides can exert regulatory influence on gene expression levels, which in turn can influence phenotype ([@gks1449-B3]). Methylation levels vary between cell types ([@gks1449-B4]), between individuals ([@gks1449-B5]), and they are known to be influenced by both environmental and genetic factors ([@gks1449-B6]). Importantly, methylation has been implicated in a wide range of diseases, including cancers, autism-spectrum disorders ([@gks1449-B2]), as well as several auto-immune diseases ([@gks1449-B7]). It, therefore, stands to reason that finding and characterizing the genetic determinants of methylation could yield insight into mechanisms of disease and the functional consequences of genetic variation.
Genetic sequence has been implicated as a determinant of DNA methylation in a number of contexts. Individuals who are heterozygous at a gene locus can exhibit allele-specific methylation that is dependent on DNA sequence and leads to differential gene expression patterns between the alleles (i.e. allele-specific gene expression) ([@gks1449-B8]). Hellman and Chess ([@gks1449-B12]) found that individuals who shared more parental chromosomes (i.e. are more related) tend to exhibit more similar methylation patterns. Single-nucleotide polymorphisms (SNPs) may also disrupt CpG dinucleotides (i.e. causing them to no longer be CpG), thereby preventing methylation there or at neighbouring loci ([@gks1449-B8]). Several methylome-wide studies have identified individual SNPs that are correlated with specific methylation loci ([@gks1449-B13]).
Despite these findings, the extent to which differences in stretches of *cis*-DNA sequence (i.e. multivariate SNP signal) explain differences in methylation of a given CpG dinucleotide between individuals, and, correspondingly, to what extent methylation is deemed heritable when estimated from SNP data from unrelated individuals, remains unclear. Recently, Bell *et al.* ([@gks1449-B14]) used the differences in correlation between monozygotic and dizygotic twins to estimate the heritability of methylation in blood samples, finding a genome-wide mean heritability of 18%. Twin-based analyses are important in shedding light on an upper bound of heritability, but yield no information as to the mechanism of action underlying heritability, a critical piece of the story. Before this, several more focused studies have been conducted on examining the heritability of methylation in particular contexts, such as between cell divisions of cancer cells ([@gks1449-B16]), for a particular gene ([@gks1449-B17]) or for the major histocompatibility complex region in a twin study focused only on immune cells ([@gks1449-B18]).
Herein, we identify 'heritable methylation loci'---those loci for which *cis*-SNPs explain more of the phenotypic variance than expected by chance---within the human methylome, in four distinct brain regions across 150 unrelated individuals from publicly available data. Our goals were to investigate what role stretches of *cis*-DNA sequence plays in influencing methylation, what is an optimum definition of *cis* (i.e. locality) in this context, whether the additive effects of measured SNPs could explain the twin-based estimates of heritability previously reported and whether CpG dinucleotides with heritable methylation were more likely to be within or neighbouring particular classes of genes or genomic features.
MATERIALS AND METHODS
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Individual SNP data and chromosomal coordinates were downloaded from dbGAP Study Accession phs000249.v1.p1. Normalized methylation levels across four brain regions \[cerebellum (CRBLM), frontal cortex (FCTX), caudal pons (PONS) and temporal cortex (TCTX)\] from 150 individuals were obtained from the Gene Expression Omnibus (GEO) database (accession GSE15745). This data profiled methylation levels of 27 578 CpG loci assayed using an Illumina HumanMethylation27 BeadChip. Methylation locus chromosome coordinates were obtained from GEO (GPL8490). SNP data for the same individuals were generated from tissue collected in the cerebellum brain region. All SNPs missing in \>1% of the individuals, or those whose minor allele frequency was \<0.01 were discarded. All individuals missing \>5% of their SNP data were removed. Several methylation loci and individual samples were removed because of data quality concerns \[see [Supplementary Information](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1) of Gibbs *et al.* ([@gks1449-B13])\].
Initially, we found that our estimates of heritability were significantly correlated with the number of SNPs within the methylation probe region. Thus, to avoid erroneously identifying methylation loci as heritable from such artefacts, we filtered out any methylation loci whose respective probe overlapped a SNP with minor allele frequency ≥0.05 (using the highest reported minor allele frequency from dbSNP, and the list of probe SNPs as provided by Illumina). This filter further removed 5816 methylation loci, leaving 21 000 methylation loci for our analysis. Individual covariate data were obtained from [Supplementary Table S1](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1) from Gibbs *et al.* ([@gks1449-B13]) and converted to a 1-of-(M-1) encoding for discrete variables. [Table 1](#gks1449-T1){ref-type="table"} reports the final number of individuals and SNPs for each of the four brain regions. Table 1.Number of individuals and SNPs used in analyses for each of the four brain regionsRegionNumber of individualsNumber of SNPsCRBLM106495 788FCTX132495 873PONS124495 870TCTX125495 866
Identification of heritable methylation loci
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We used linear mixed models (LMMs) to assess the narrow-sense heritability of each methylation locus ([@gks1449-B19]). Let the vector ***y****~i,t~* of length *N* represent the methylation levels of locus *i* at brain region *t* across *N* individuals. Using LMMs, we can decompose the variance associated with ***y****~i,t~* as the sum of a linear additive genetic () and residual () component, where *X* is the *N×Q* matrix of *Q* individual covariates (gender, age, post-mortem interval, region source and methylation assay batch) and offset term, *β* is the *Q×1* vector of covariate effects, *I* is the *N×N* identity matrix and *K* is the realized relationship matrix (RRM) ([@gks1449-B20]) of size *N×N*. Note that *K* factors as , where *W* of dimension *N×s* contains the *s* SNPs in our window local to the gene and that when *s\<N*, parameter estimation and computation of the log likelihood becomes extremely efficient ([@gks1449-B21]). We used the method of Lippert *et al.* ([@gks1449-B21]) to compute restricted maximum likelihood estimates of and . Narrow-sense heritability for a particular methylation locus *i* in brain region *t* was then estimated as ([@gks1449-B19])
To compute a *P*-value for whether a methylation locus, ***y****~i,t~*, was heritable---that is, to compute the significance of the genetic variance component in the model---we set to obtain the null model, and then used a modified likelihood ratio test, which accounted for the fact that the parameter being tested was on the boundary of the allowed space in the null model ([@gks1449-B22],[@gks1449-B23]). That is, in the null model, and in the alternative model because it is a variance parameter. However, on checking the calibration of *P*-values by way of permutation tests, we discovered the *P*-values to be conservative, owing partly to the small sample size, but also to the approximation of the null distribution in this case \[*e.g.* ([@gks1449-B24])\], and thus used the permutation-based *P*-values instead (using 420 000 permutations of the individuals in the methylation data, and using the same permutations for each methylation locus). We defined a 'heritable locus' as one in which the *P*-value of association was smaller than a significance level of 0.05 after Bonferroni correction. Note that this test can be viewed as a test for association between the SNPs in the set and the phenotype in question, and it has been used in a similar manner in ([@gks1449-B25],[@gks1449-B26]).
Determining an optimum *cis* window size
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To find an optimal window size across all methylation loci for inclusion of *cis*-acting SNPs, we systematically varied the window size through 10 kb, 50 kb, 100 kb, 500 kb and 1 Mb, use of the entire chromosome that the locus fell on, and all SNPs assayed in the genome. We then deemed the optimum window to be the one yielding the largest number of heritable methylation loci, where we identified heritable loci by the permutation strategy previously described, but limiting the number of permutations to 10 000 for each window size for computational efficiency. Note the final set of heritable loci we report is based on the 420 000 permutations. We used a window that was symmetric around the methylation locus of interest. That is, we defined a window of size *X* kb centred at a methylation locus at position *i* as the DNA sequence within the region kb, inclusive. Closely related individuals can be problematic when estimating heritability ([@gks1449-B27]) because of confounding owing to shared environmental factors. For example, Visscher and colleagues require removal of all individuals with RRM similarity \>0.05 ([@gks1449-B27]). In our data set, no two individuals were related this closely; thus, we did not filter any individuals by this criterion. Furthermore, a univariate scan of various methylation loci, randomly chosen, did not show significant deviation of the genomic control factor ([@gks1449-B28]), , from 1.0, suggesting that hidden confounders were not present in this data set.
When scanning *cis* window sizes, we restricted our comparison with the 15 179 methylation loci for which we could find at least one SNP within each of the window sizes considered.
Assigning methylation loci to gene sets
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We first assigned methylation loci to genes, based on proximity, and then assigned genes to gene sets. Methylation loci were assigned to their closest neighbouring genes as reported by the Illumina HumanMethylation27 BeadChip annotation files. Next, we associated genes to gene sets, considering all genes that were associated with at least one methylation locus under study. Gene sets were obtained from the Gene Ontology (GO) ([@gks1449-B29]), which yields gene sets organized by biological process, from the Molecular Signatures Database (MSigDB) ([@gks1449-B30]) that defines canonical biological pathways and from the Pharmacogenomics Knowledgebase (PharmGKB) ([@gks1449-B31]) that defines known pathway targets of drugs. GO annotations for humans were obtained from GO on 14 May 2012. We tested the 2464 GO sets for which there were between 20 and 500 member genes, inclusive. Canonical pathway definitions from MSigDB version 3.0 were used, totalling 880 gene sets. All pathways for which at least one drug was known to target it were downloaded from PharmGKB on 21 July 2012, totalling 263 gene sets. In total, there were 3607 sets tested.
Computing correlation of heritable loci with open chromatin regions and known regulatory elements
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We explored whether heritable loci were enriched for loci lying in open chromatin regions or in known regulatory elements. To do so, we used Fisher's exact test (FET) ([@gks1449-B32]), using our results of which loci were deemed heritable, in conjunction with external data sources which could be used to annotate the loci. In particular, we obtained open chromatin regions from data published by the encyclopedia of DNA elements (ENCODE) Project Consortium ([@gks1449-B33]). Briefly, the University of North Carolina at Chapel Hill has collected formaldehyde-assisted isolation of regulatory elements (FAIRE) evidence of open chromatin and has made this data available through the University of California, Santa Cruz Genome Browser ([@gks1449-B34]), from which we obtained it on 19 November 2012. In particular, we obtained all 273 110 open chromatin annotations for normal human astrocyte cells (cell type NH-A), the only cell type relevant to brain tissue that was available at this time, and used the LiftOver tool to map the coordinates to build hg18. For computing the overlap of heritable loci with known regulatory elements, we used publicly available data obtained from the ORegAnno database containing 23 206 known regulatory elements ([@gks1449-B35]) downloaded on 19 November 2012. The majority (17 744 of 23 206, or 76%) of regulatory elements stored in ORegAnno are binding sites of CTCF; therefore, we restricted the regulatory elements to CTCF sites only. Determination of overlap between methylation loci and genomic annotations was computed using the BEDTools software ([@gks1449-B36]).
Gene set enrichment testing
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We performed gene set enrichment testing using FET, which tests whether the proportion of heritable methylation loci belonging to a gene set is larger than that expected by chance. We hypothesized that the FET *P*-values may be inaccurate because FET treats loci as independent and, therefore, does not account for correlated loci ([@gks1449-B2]). Thus, we computed permutation-based *P*-values (using permutations of individuals) for the FET and found that the closed-form FET *P*-values were inflated. Consequently, we used the permutation-based *P*-values, from 200 000 permutations of the individuals, calling those with Bonferroni-corrected *P*-values \<0.05 as significant.
Identification of genes preferentially expressed in brain regions using the same individuals
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To identify genes that were preferentially expressed in each brain region (those expressed more highly in that region as compared with other regions), we used the matching gene expression data from our publicly available data set (GEO accession GSE15745). For this analysis, only individuals for whom all four brain regions were profiled (and were done so within the same batch) were kept, leaving 122 individuals. For each probe and each individual, the ranks of the probe intensities across the four brain regions were computed. Then, for each brain region and each probe, the ranks across all individuals were summed, resulting in a matrix of Rx4 summations, one for each of the four brain regions and each of the R probes. By the central limit theorem, each summation of ranks is normally distributed with mean 305.0 and variance 203.333, as there are 122 terms in each sum, and each term is sampled from a distribution with mean 2.5 and variance 1.667, assuming all ranks (1, 2, 3, 4 because we have four tissues) are equally likely. Probes were then mapped to genes using the Illumina probeset information file, and only those genes assayed by exactly one probe were retained. Finally, using FET, we measured correlation between whether a methylation locus was heritable and whether the gene associated with the locus was preferentially expressed in the relevant brain region. Only the set of genes both profiled in the expression data, and linked to a methylation locus, were considered.
Identification of genes preferentially expressed in brain regions using independent data
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To identify genes that were highly expressed in brain tissue in general, we downloaded the raw gene expression profiles collected by Su *et al.* ([@gks1449-B37]) for multiple cell types from GEO accession GSE1133. We used the robust multi-array average algorithm in Bioconductor ([@gks1449-B38]) with R version 2.15.1 to both pre-process the array data and map probes to gene Entrez ID ([@gks1449-B39]) using an updated annotation file hgu133ahsrefseqcdf_15.1.0. We kept only samples of normal tissues and cell types, leaving 73 samples profiled in duplicate. We then performed a one-sided Wilcoxon rank sum test to identify preferential expression in brain cell types relative to all other profiled cell types. Similarly to the previous section, FET was used to look for associations between a methylation locus being heritable, and whether the gene associated with that locus was preferentially expressed.
RESULTS
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The number of heritable loci depended on the window size for defining *cis*-acting SNPs
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To find an optimal window size across all methylation loci for inclusion of *cis*-acting SNPs, we centred a window symmetrically around each methylation locus, extending the size of this window through 10 kb, 50 kb, 100 kb, 500 kb and 1 Mb, and we also tried the entire local chromosome, as well as the entire genome. We then deemed the optimum window to be the one yielding the largest number of heritable methylation loci among those loci which had at least one SNP for every window size. As shown in [Figure 1](#gks1449-F1){ref-type="fig"}a, a window size of 50 kb led to the highest number of heritable methylation loci. After more permutations to obtain more accurate *P*-values (see 'Materials and Methods' section), we found 654, 812, 600 and 636 heritable methylation loci for FCTX, TCTX, PONS and CRBLM, respectively. Although the number of heritable loci is similar for both the 50- and 100 kb windows, it is clear that using too large of a window (e.g. the entire genome), or too small of a window (e.g. ≤10 kb), dramatically reduced the number of heritable loci. Figure 1.Number of heritable methylation loci in the four brain regions: TCTX, FCTX, CRBLM and PONS. (**a**) Number of methylation loci passing a Bonferroni-corrected *P*-value threshold of 0.05, as a function of DNA sequence window size, when using only methylation loci analysed for all window sizes (so as to make them comparable). (**b**) Histogram of the number of SNPs found within the 50 kb window of each of the 21 000 methylation loci.
We believe that our loss of power to detect heritable loci when the window size was extended beyond 50 kb is related to the loss of power we observed when using LMMs to correct for confounding variables in genome-wide association studies ([@gks1449-B40],[@gks1449-B41]), although we now have a better understanding of this effect ([http://research.microsoft.com/apps/pubs/default.aspx?id=178646](http://research.microsoft.com/apps/pubs/?id=178646)). In particular, in the present context, most SNPs influencing a methylation locus are expected to be physically near to the locus (i.e. are *cis*-acting); therefore they can be captured by a relatively small window such as the 50 kb window we identified in [Figure 1](#gks1449-F1){ref-type="fig"}a. Below this window size, many influential SNPs are likely to be missed, causing a downwards bias in the estimate of and, therefore, of heritability. With increasing window sizes, more and more extraneous SNPs are included in the RRM, causing an increase in the variance of the estimate of heritability. This bias-variance trade-off is perhaps best understood in light of the fact that an LMM with no fixed effects, using genetic similarities constructed from a set of SNPs, is equivalent to a form of linear regression of those SNPs on the phenotype. Thus, using extraneous SNPs in the estimation of the RRM is equivalent to using them as additional covariates in this form of linear regression, which increases the variance of the estimate of , diminishing our power to detect heritable loci (). Therefore, in our analysis, as we included more and more SNPs up to and including a window which contained most influential SNPs (i.e. the 50 kb window), the downwards bias on heritability decreased (and the estimate of heritability increased). As we went beyond this optimal window size, an increasing proportion of extraneous SNPs were included in the RRM, up until the point where the variance of the estimate of heritability almost completely diminished our power to detect significantly heritable loci. This bias-variance effect would be mitigated by a larger sample size.
[Figure 1](#gks1449-F1){ref-type="fig"}b illustrates the number of SNPs included in the local sequence window, for all methylation loci, at the selected optimal 50 kb window size. Our locality result is similar to that found by Price *et al.* ([@gks1449-B42]), where it was found that heritability of gene expression was primarily because of SNPs at *cis* loci. In the univariate SNP-methylation association analysis of Bell *et al.* ([@gks1449-B14]), they examined SNPs within 100 kb, but found that most associations were actually within a few kilobases, whereas Gibbs *et al.* ([@gks1449-B13]) reported finding a peak at 45 kb. However, as noted in ([@gks1449-B19]), use of a stringent, multiple-testing correction to select significantly associated univariate SNPs, as done in these two studies, is likely to miss much of the weaker signal that the LMM can capture. Thus, it is not surprising that our analysis finds an optimal local window which is slightly larger than what one might have speculated from stringent univariate analyses.
We also found that heritable methylation loci tended to have larger number of SNPs within their windows than non-heritable loci: for CRBLM, the median number of SNPs in the 50 kb window was nine versus seven, whereas for all other tissues, the median number was eight versus seven (all *P* \< 10^−8^, Wilcoxon rank-sum test). This result suggests that with more SNPs, there is more power to uncover heritable methylation loci. The number of SNPs in each window are provided in [Supplementary Table S1](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1).
[Figure 2](#gks1449-F2){ref-type="fig"} illustrates the distribution of estimated narrow-sense heritability over all 21 000 methylation loci for all four regions (region-specific distributions were similar to one another). The mean estimated heritability of all methylation loci deemed heritable (aggregated across all four brain regions) was 29.9%, indicating the extent to which local sequence alone can account for variation in methylation at those loci. Across all loci (including those not deemed heritable), the mean estimated heritability was 2.8%. Figure 2.Narrow-sense heritability estimates over all methylation loci in all four brain regions. Loci were divided based on whether they were located in (**a**) CpG islands (15 469 loci) or (**b**) not in CpG islands (5531 loci). Within each plot, loci are then further grouped based on whether they were identified as heritable. The smoothed histograms were constructed using density estimation with a Gaussian kernel with default parameters in R, and the y-axis is scaled to a maximum of 1. The number of individuals used in this analysis is reported in [Table 1](#gks1449-T1){ref-type="table"}.
Concordance of heritable loci across brain regions and with eQTL and methQTL
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We next assessed the extent to which heritable methylation loci were shared across regions when using the 50 kb window size. We found that 181 loci were heritable across all four regions with mean estimated heritability of 41.4%, whereas 207 loci were heritable across at least three regions ([Figure 3](#gks1449-F3){ref-type="fig"}a). The estimated narrow-sense heritability shows generally good agreement among FCTX, TCTX and PONS ([Figure 3](#gks1449-F3){ref-type="fig"}b). [Supplementary Table S2](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1) reports the list of all methylation loci, their estimated heritability and the significance of association with their 50 kb cis-sequence window. Figure 3.Concordance of heritable loci across the four brain regions and with eQTL and mQTL from Gibbs *et al.* (**a**) Number of heritable loci found to be overlapping in each of the four regions, using the 50 kb window size. (**b**) Correlation of the estimated narrow-sense heritability for each methylation locus, between tissue regions, using only the 1451 loci that were significant in at least one region. (**c**) Breakdown of heritable methylation loci according to whether a locus was also found to have at least one *cis*-mQTL in the Gibbs study---'common' refers to a locus we identified as heritable and for which Gibbs found at least one mQTL; 'Quon-specific' means the locus was found to be heritable but did not have an mQTL in the Gibbs study; and 'Gibbs-specific' means Gibbs *et al.* found at least one mQTL for a locus that we did not find to have heritable methylation. (**d**) Percentage of heritable loci for which at least one eQTL was reported by Gibbs *et al.* for the gene nearest to the heritable methylation locus (and where the eQTL was within the 50 kb window of the heritable locus), as compared with the number for all methylation loci. An asterisk indicates significant to a threshold of 0.05, as determined by a FET.
We compared the set of heritable loci to the set of methylation loci identified by Gibbs *et al.* as being associated with at least one *cis*-methylation quantitative trait locus (methQTL). We found that on average, 43% of each tissue's set of heritable loci was identified as being associated with at least one *cis*-methQTL in the Gibbs study, indicating that we identified overlapping but distinct loci from that of Gibbs *et al.* ([Figure 3](#gks1449-F3){ref-type="fig"}c). We also identified on average 54% more methylation loci (with *cis* association) than did Gibbs *et al.* in their univariate scan. Note that their multiple-testing burden was larger because they also looked for trans-methQTLs. We next cross-referenced our list of heritable loci with the expression quantitative trait loci (eQTLs) reported by Gibbs *et al.* (first restricting the set of eQTLs to those within the 50 kb window of the 21 000 methylation loci and whose target gene is the same gene as the one we assigned to the respective methylation locus). We observed that in three of the four tissues (all but PONS), the heritable methylation loci were enriched for genomic regions containing cis-eQTLs \[[Figure 3](#gks1449-F3){ref-type="fig"}d; *P =*1.05 × 10^−3^ (FCTX), *P =*3.16 × 10^−3^ (TCTX), *P =*0.076 (PONS), *P =*0.0202 (CRBLM); FET\].
To explore the relationship between heritable methylation loci in each of the four brain regions and levels of gene expression in these brain regions, we again used the expression data corresponding to our samples, now to identify genes preferentially expressed in each region---genes expressed higher in that region than in others (see 'Materials and Methods' section). We found that the genes assigned to heritable loci identified in the frontal cortex and cerebellum brain regions were significantly depleted in genes preferentially expressed in that region \[*P =*0.024 (FCTX), *P =*6.20 × 10^−5^ (CRBLM), *P =*0.55 (TCTX), *P =*0.90 (PONS), FET\].
For a more general investigation of heritable loci and brain-specific expression, we obtained genome-wide expression profiles for 73 different cell types so as to identify those genes preferentially expressed in the brain compared with all other tissues. We found that heritable loci identified in three of the four brain regions (frontal cortex, temporal cortex and cerebellum) were significantly depleted near genes expressed more highly in the brain compared with other tissues \[*P =*4.64 × 10^−4^ (FCTX), *P =*4.44 × 10^−3^ (TCTX), *P =*2.32 × 10^−4^ (CRBLM), *P =*0.19 (PONS), FET\]. These results suggest that heritable loci are not regulating genes highly expressed in either brain-specific regions or whole-brain tissue, both of whom may be critical to brain function.
Heritable methylation loci were enriched for genomic locations containing regulatory elements
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To assess the potential role of heritable methylation loci in gene regulation, we checked to see whether our heritable loci lay in regions previously annotated with genomic features that are indicative of gene regulatory elements. Using data from the ENCODE project (see 'Materials and Methods' section), we found that the heritable loci for all four brain regions were enriched in open chromatin regions \[*P =*9.43 × 10^−3^ (CRBLM), *P =*0.02 (PONS), *P =*0.0122 (FCTX), *P =*0.018 (TCTX) FET\]. Furthermore, when comparing our heritable loci with known CTCF binding sites \[by way of ORegAnno ([@gks1449-B35]), see 'Materials and Methods' section\] we also found significant enrichment for overlap between the heritable loci and these regulatory elements \[*P =*0.031 (CRBLM), *P =*0.035 (PONS), *P =*0.035 (FCTX), *P =*0.027 (TCTX), FET\]. CTCF is implicated in both diverse genomic regulatory functions (activation, repression, insulation) and the global organization of chromatin architecture ([@gks1449-B43]). Furthermore, DNA methylation of CTCF's binding site is the best understood mechanism for modulating CTCF binding ([@gks1449-B43]). As an example, methylation of CpG loci within the CTCF binding site eliminates binding of CTCF *in vivo* and has been demonstrated to disrupt its regulatory activity at the methylated binding site ([@gks1449-B44]). These results suggest that heritable loci may play a regulatory role in the expression of neighbouring genes by modulating binding and activity of regulators, such as CTCF.
We also investigated whether those methylation loci found to be heritable favoured any particular position relative to the nearest transcription start site (TSS). We found the heritable methylation loci for each brain region were enriched for loci lying outside of CpG-islands (all *P* \< 1.84 × 10^−4^, FET). Furthermore, as illustrated in [Figure 4](#gks1449-F4){ref-type="fig"}, we found that the heritable loci in the PONS tissue region were preferentially located downstream of the TSS relative to other methylation loci (median position relative to TSS was 72 versus −2 bp, *P =*2.6 × 10^−3^, Wilcoxon rank-sum test); we did not find similar preferences for the other three tissue regions (all other *P* \> 0.57). Heritable loci located much farther downstream from the TSS indicate possible genetic influence over alternative splicing events ([@gks1449-B45]). Figure 4.Relative position of heritable and non-heritable loci identified in the PONS tissue region with respect to the TSS of the gene to which they were closest. The x-axis has been thresholded at a distance of 2 kb.
Genes proximal to heritable methylation loci are involved in a variety of processes
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One of the primary roles of DNA methylation is to control gene expression of particular genes. We next identified whether heritable loci seemed to be controlling any specific classes of genes. To do so, we first assigned methylation loci to genes, based on proximity (see 'Materials and Methods' section). We then performed a gene set enrichment analysis on all genes assigned to heritable methylation loci, using 3607 gene sets from the Gene Ontology (GO) Process hierarchy, canonical biological pathways from the Molecular Signatures Database (MSigDB) and drug-targeted pathways from the Pharmacogenomics Knowledgebase (PharmGKB). [Supplementary Table S3](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1) shows which methylation loci are assigned to which gene sets.
Among the gene sets found significant ([Figure 5](#gks1449-F5){ref-type="fig"}), two involved neurotransmitters (agmatine and dopamine), another involved neurotransmitter transporters (SLC transporters) and another involved nicotinamide salvaging (an anti-inflammatory pathway), suggesting candidate epigenetic mechanisms through which genotype may play an important role in drug efficacy. Other gene sets associated with heritable methylation loci involved regulation of energy production and the immune system. [Supplementary Table S4](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1) reports the results of the enrichment analysis on all categories tested. Enrichment tests were performed using only genes that were assigned to at least one of the 21 000 methylation loci assayed. This 'background' set of genes was not itself significantly enriched for any specific brain functions, although it was enriched for 55 GO categories (of the 2464 tested) across a variety of processes ([Supplementary Table S5](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1)). Figure 5.Gene set enrichment of the heritable loci in each of the four brain regions. (**a**) A black rectangle indicates significant enrichment () for the specified set and brain region combinations. (**b**) Network illustration of gene sets found to be enriched for heritable loci. Each node represents one gene set, whereas each edge represents an overlap of at least one methylation locus between the two gene sets. The size of each node is proportional to the number of methylation loci assigned to the respective gene set, and the width of each edge is proportional to the gene set coherence, defined as the number of loci in the overlap divided by the smaller size of the two gene sets. The legend depicts the minimum and maximum node sizes, as well as the edge width corresponding to the minimum gene set coherence (0.05) and maximum gene set coherence (1.0).
DISCUSSION
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Epigenetic mechanisms, such as DNA methylation, play a critical role in controlling the gene expression programme of cells, which in turn is thought to have significant impact on phenotype ([@gks1449-B3]). Epigenetic markers, therefore, represent a potential mechanism through which genetic variation can affect phenotype. Herein, we examined how *cis*-DNA sequence influences methylation across the human genome in four phenotypically normal brain regions from unrelated individuals. We found that between 3 and 4% of the tested loci were heritable with respect to an empirically selected optimal *cis* DNA window of size 50 kb. Furthermore, the heritable loci were shown to be enriched in open chromatin regions, and also enriched in locations of known binding sites of CTCF, suggesting a functional role for at least some of these heritable loci in disrupting or modulating binding of transcription factors, such as CTCF. Also, genes associated with heritable loci in some of the brain regions were enriched in several pathways, including those involved in neurotransmitter processing, regulation of energy production and the immune system. None of the enriched gene sets are clearly brain region specific, suggesting the heritable loci we identified may be heritable in a wide range of tissues rather than brain-specific.
The number of heritable methylation loci depended on how large of a window of SNPs was considered local. We found that a window size of 50 kb was optimum in achieving a maximal number of heritable loci across all regions, and informs on a window in which the mechanistic action through which SNPs alter CpG methylation could be investigated. As the window size was extended beyond an optimal range, we hypothesize that the variance in the estimate of heritability became extremely high (especially with such a small cohort), and, therefore, that our ability to detect significance was diminished (<http://research.microsoft.com/apps/pubs/?id=178646>).
Our estimates of heritability were less than that reported in the twin-based study of Bell *et al.* ([@gks1449-B14]), who found a mean genome-wide heritability of 18% from blood samples, as compared with our 3%. In Gervin *et al.* ([@gks1449-B18]), heritability of the major histocompatibility complex region in cultured lymphocyte cells was investigated using a twin-based approach and was found to be low (2--16%). The discrepancy between our estimates and the twin-based estimates could be explained by unmeasured SNPs ([@gks1449-B46]), the cohort or tissue in which measurements were performed, the upwards bias of twin-based studies ([@gks1449-B47]) and limited sample size. Further studies should shed more light on this issue.
There are a number of reasons to suspect that the fraction of CpG dinucleotides whose methylation status is heritable is larger than what we have reported here. First, our study only included individuals with phenotypically healthy brains, and we expect that analysis of a wider range of tissues may uncover genetic dependencies that are tissue or condition specific. Second, our Bonferroni correction of *P*-values is likely ignoring weakly heritable loci. Third, use of more dense SNP and methylation assays will allow for a more refined exploration of the genetic basis of methylation. Finally, if heritable loci were tissue specific, we would lose power to detect them when analysing mixed tissues as we have here.
SUPPLEMENTARY DATA
==================
[Supplementary Data](http://nar.oxfordjournals.org/lookup/suppl/doi:10.1093/nar/gks1449/-/DC1) are available at NAR Online: Supplementary Tables 1--5.
FUNDING
=======
Microsoft Research. Funding for open access charge: Microsoft Research.
*Conflict of interest statement*. J.L., C.L. and D. H. own stock in Microsoft.
Supplementary Material
======================
###### Supplementary Data
The authors thank Jonathan Carlson and Carl Kadie for help with tools to manage and analyse the data, Jim Jernigan and the MSR HPC team for cluster support and anonymous reviewers for their suggestions. Division of Aging Biology and the Division of Geriatrics and Clinical Gerontology, NIA; Intramural Research Program, NIA.
| |
|July 15, 1786|
PLATE III.
A Ring chart.
M. Bourignon de Saintes, Correspondent of the Royal Society of Medicine, for several Academies, and of MONSIEUR's Museum, has made us the honor of sending us Research on Rings and other Jewels of women of antiquity, and gave us permission to put it into our Books. We will not fail to use it, to express our gratitude to him, and to get our Subscribers to see the similarities between past eras and our own.
"Wearing rings," says M. Bourignon de Saintes, "dates back to the furthest times past. The Chaldeans, the Babylonians, the Persians, and the Greeks wore rings. The Sabines also had them in the times of Romulus: theirs were similar to those of the Greeks. From the Sabines, they passed to the Romans.
"Rings were of gold, silver, copper, iron, or glass. Some were made of a simple metal, others of a mixed or alloy metal; for sometimes iron or silver were plated, or surrounded gold with iron. The first were simple and of a common metal; following that, they were made in silver and gold, and soon no others were worn, or at least not unless they were gilded.
"A ring distinguished free men from slaves, in the beginnings of the Republic: the citizens wore them of glass. The right of wearing them in gold only belonged to Senators who had satisfied some ambassadorship in a foreign Country. This practice was then permitted to other Senators, and became at last the proper and distinctive sign of Knights. In this era citizens and freedmen wore silver rings, and slaves those of iron; but after the ruin of the Republic, the gold ring was no more than a weak distinction which was accorded even to freedmen. Augustus was the first to whom they were obliged for this honorable favor: Septimus Severus extended it to simple soldiers.
"The ancients singularly varied the manner of wearing their rings. The Hebrews placed them on the right hand, the Greeks on the fourth finger of the left hand, the Gauls and ancient Britons on the middle finer. Before they were adorned with precious stones, when the face was engraved even on the plaque of the ring, they were worn indifferently on either hand; but Fashion having made a rule for the practice, they were put first on the fourth finger, then on the index, the little finger, and successively on all the fingers, except the middle one. After art had added stones, they were worn on the left hand, and by a sought-after delicacy, on the right hand.
"The Greeks and Romans multiplied them gradually, until they were worn with several on each joint of each finger; they had pushed luxury and magnificence to the point of having winter and summer fashions, and others that were only worn on birthdays. Seneca declaimed much against the vanity of women, who wore one or two inheritances on their fingers.
"Marriage rings were round and plain, that is to say without any stone, and they were of iron. In later times they were gold, enriched with precious stones. The ring which the fiancé gave to his intended, was a mark of the engagement that he contracted with her, and of the power that he gave her to rule his household. Religion sanctified further this practice; for the benediction of the nuptial ring is found in the ancient Liturgies.
"Some Authors trace the origin of wedding rings to the Hebrews; they base this on a passage from Exodus. Leon of Modena however supports that the Hebrews never used the nuptial ring."
About us, we imagine that the Rings that are worn today, have only given birth to the practice of wearing the nuptial ring, whether this ring came from the Hebrews, or it only came from the earliest times of our Monarchy. We do not remember having read in our History any passage which gives its origin. This ring could be subject to as many variation, in material, as those of the ancients; it is that which we are ignorant of. But, still, we think that it is to that which rings owe their birth. The custom which the French have of taking the hand of a young person on whom a ring is seen, be it for congratulating her on her marriage, be it for the right to admire a beautiful hand, and of complimenting it, must give to women the desire of parading richness or taste. They will have first worn a single enclosed diamond, then two, then a great number of the same ring, then finally a great number of rings.
This practice may have been adopted by men, that is what stuns us. Have these Messieurs pretended to admire their hands, and to kiss them? The great Bayard may have worn a ring with the device, without fear and beyond reproach, that he had made for the brave Knights who followed in his exploits to wear, it was a mark that a man could show on his hand, that Bayard had chosen. Our Great Men may thus wear a ring with a device as just as that of Bayard, and give a matching one to each man in whom they have recognized merit, and that they will have adopted, this is what we will applaud; but our men wear rings, our Turcarets, over-all, have their fingers covered by them ... We do not perceive that we should speak of Fashion, and that we take censure for it. What matter? these reflections will be put with the number of complaints, and Fashion will not go less in its train.
It must be confessed however that if it is a bad thing, this bad has something agreeable. It is useful at least, in that it is not the smallest branch of commerce, and, to consider it under this point of view, is necessary. Also we only criticize it a little, because we respect its effects. Wear on, Messieurs, wear a great number of them; it is the Fashion, and it is a lasting Fashion; it has as much taste as richness. Those made today have a pretty form, and this can still justify the Fashion.
They are very-large now, very different from those that were worn less than two years ago, which were only often composed of a large enclosed stone. A large diamond, a large brilliant stone is put in the middle of an oval, squared, lozenge, plain squared, eight-sectioned squared composition stone. In the middle of this composition stone, the diamond is surrounded with other fine stones, or roses, or it is alone. The composition stone is surrounded with diamonds, or roses, or pearls; or it is nude, it is all plain. If the stone in the middle is not rather large, two smaller ones are put at the two ends of the setting: it is surrounded as well with other diamonds. Most often the setting is dotted with little diamonds mounted in little stars, and these are called Rings au Firmament. If the stone is rather strong, it is put alone in the middle of the composition stone, and it is still dotted with little stars in diamond.
The lozenge settings have four sharp points, or are little marked, and hardly form long ovals. The squared are long, or are perfect squares, and are all with cut angles.
The compositions stones have a green, Sky blue, violet, puce, yellow, or grey ground.
In the place of white stones in the middle of the setting, colored stones are placed, in observing the unity, as there should be, of the composition stone of the setting with the surrounded stone. It is necessary that that of the setting matches the surrounded stone in a manner which flatters the eye, the only judge of taste. These latter ones are called, Rings à l'Enfantement. If they are large, they are for women, as for men.
You can make your choice from the Rings drawn in the Ring Chart represented in the IIIrd Plate; the colors seem to match well.
This Ring Chart is drawn from the Shop of M. Moricand, Merchant Jeweler, residing in the place Dauphine, no. 30. All sorts of works in jewelry are found in his Shop, such as Mirzas, Medallions, Crosses à la Jeannette, all in brilliants, diamonds, and roses. The Curious will also find there all sorts of colored Stones, fine, crude, or cut, Stones engraved with reliefs and in hollows, fine Pearls, together with antique Stones, of the best kind. Those who ask it of him, are sure of being very-well served.
One can also write, for Rings, to M. Granché, at the Little Dunkirk. He possesses a very-rich assortment. | https://www.mimicofmodes.com/2014/03/cabinet-des-modes-17e-cahier-3e-figure.html |
Location technology specialist, TomTom, and the University of Amsterdam (UvA), have announced the launch of a new public-private research lab.
Atlas Lab will focus on using Artificial Intelligence (AI) for developing advanced, highly accurate and safe high definition (HD) maps for self-driving vehicles. The lab is part of ICAI, the national Innovation Centre for AI, based in the Amsterdam Science Park.
In collaboration with TomTom, the UvA is embarking on research on the use of AI for creating HD maps suitable for all levels of autonomous driving. Theo Gevers, one of the Scientific Directors at Atlas Lab, comments: "At the UvA we are already doing research on automated recognition of items in images and videos. Yet the recognition of items and creation of HD maps in highly complex situations like a moving car, is still a huge challenge. This collaboration with TomTom provides an extra dimension to new and challenging AI-research."
For the next five years, five PhD students will work in the Atlas lab on projects contributing to automated recognition of items like traffic signs, 3D-localization of vehicles and combining LIDAR (light detection and ranging) laser and camera images. For retrieving data, mobile mapping vans equipped with sensors, like LIDAR-systems and cameras, are being used.
"TomTom is pushing the boundaries of the use of AI for making HD maps for self-driving cars," said Harold Goddijn, TomTom's CEO. "We need groundbreaking research into AI technology, which is why we're collaborating with UvA's world-leading AI department on this initiative. This will move us a step closer to an autonomous future with safer roads, free of congestion and emissions."
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Part 2 examines how the freewheeling modernism that had shocked audiences in the first two decades of the century came under state control. Initially, many practitioners thought the totalitarian regimes would be good for music and the arts. What followed in Germany was a ban on music written by Jews, African-Americans and communists, while in the Soviet Union there was a prohibition on music the workers were unable to hum. After the cataclysm of the 1940s, a new generation of composers - Boulez, Stockhausen, Xenakis, Nono, Ligeti - turned their back on what they saw as the discredited music of the past and tried to reinvent it from scratch. Or, at least, from serialism, which became as much of a straitjacket as totalitarianism's strictures had been. But from this period of avant-garde experimentation, which many listeners found baffling and terrifying, came some of the most influential and radical musical innovations of the century.
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Series concludes with the focus shifting to the United States in the post-war years of the 1950s and beyond. Beginning with arguably the most notorious work of 20th century classical music, John Cage's 'silent' composition 4'33", it looks at how a series of maverick Americans re-invented the sound of classical music into a more simple form, bringing back harmonies and rhythms that made it increasingly popular with audiences across the world. It also examines how this music found its way into a spiritual realm, with the strain of pared-down religious composition that came to be known as 'holy minimalism'. From the Maverick concert hall in Woodstock, New York to an Orthodox cathedral in Estonia to a car park in Peckham, south London, the story is told by a stellar line-up of contributors including Philip Glass, Steve Reich, John Adams, Arvo Pärt and John Tavener.
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Mechanical Marvels: Clockwork Dreams
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Professor Simon Schaffer presents the amazing and untold story of automata - extraordinary clockwork machines designed hundreds of years ago to mimic and recreate life. The film brings the past to life in vivid detail as we see how and why these masterpieces were built. Travelling around Europe, Simon uncovers the history of these machines and shows us some of the most spectacular examples, from an entire working automaton city to a small boy who can be programmed to write and even a device that can play chess". All the machines Simon visits show a level of technical sophistication and ambition that still amazes today. As well as the automata, Simon explains in great detail the world in which they were made - the hardship of the workers who built them, their role in global trade and the industrial revolution and the eccentric designers who dreamt them up. Finally, Simon reveals that to us that these long-forgotten marriages of art and engineering are actually the ancestors of many of our most loved modern technologies, from recorded music to the cinema and much of the digital world.
Downloaded
2013 Technology
Alex Winter explores the downloading revolution; the kids that created it, the bands and the businesses that were affected by it, and its impact on the world at large. He focuses on the advent of digital media sharing, including the rise of game-changing company Napster and controversial pioneers Shawn Fanning and Sean Parker. The digital revolution ultimately created a technology paradigm shift and upended the music industry. Audiences will hear insight from well known music artists and figures within the music industry including: The Beastie Boys' Mike D, Noel Gallagher, Henry Rollins, former Sony Music Chairman, Don Ienner, former record producer and Island Records founder Chris Blackwell and Hilary Rosen, former CEO of the Recording Industry Association of America.
A Leap of Faith
2013 History
With optional Hebrew subtitles. This episode explores how the spread of the Enlightenment brought ghetto walls around Europe crashing down and allowed Jews to join the wider fabric of modern life in Europe in unprecedented ways. This Jewish renaissance saw Giacomo Meyerbeer and Felix Mendelssohn to establish the enduring tradition for Jewish musical prodigies. However the integration of Jewish talent into the mainstream of European culture and commerce eventually stirred up ancient prejudice, expressed in the new fashion of Romantic nationalism and the pseudo-science of anti-semitism.
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LWC Counseling Professor Receives Third Fulbright Award
Dr. Daya Singh Sandhu has devoted a good part of his research trying to lower the suicide rates in his native country of India, the “suicide capital of the world.” And Sandhu was recently selected to receive the 2017-18 Fulbright-Nehru Academic and Professional Excellence Award.
It’s the third Fulbright award for Sandhu, who is an LWC counseling professor and the college’s director of research. In November, Sandhu began a six-month fellowship to research suicide prevention and awareness at Punjabi University in Patiala, India.
“There is a need for an awakening about depression in India,” said Sandhu. “There is no suicide awareness in India. Secrecy and stigma are associated with mental health issues and depression.”
Sandhu used his first Fulbright award in 2002 to create awareness about suicide. His goal was to change the dialogue about behavior and emotional illness in India. When Sandhu returned in 2010 thanks to a second Fulbright award, his focus was on suicide prevention and he conducted a cross-cultural study for people from ages 19-24.
“This time I’ll be taking a more focused look at how two cultures and two nations view the problem of suicide in different ways,” he said. “There is a big difference in the way individuals cope with daily problems in Eastern and Western society.”
Sandhu said that although suicide is a world-wide problem, the reasons for committing suicide and the ways in which college students commit suicide are different across cultures and countries.
“Suicide is rising among college students,” said Sandhu. “There is intense pressure on academic achievement in India. And the values system is different. The focus is not on individual success but on improving the status of an entire family. Such pressures are often compounded by relationship problems or the lack of job opportunities.”
Sandhu’s interest in suicide awareness and prevention be- came “a professional mission” after two close friends lost sons to suicide.
“If you can save one person’s life or one parent from experiencing a tragedy like this, I think you have done something,” he said.
As part of his research, Sandhu has discovered an acute shortage of trained mental health professionals in India.
“Psychology is an established field in India but no one is practicing,” said Sandhu. “What I’m trying to do is introduce the topic of professional counseling as a discipline so the people can start practicing.”
In 2010, Sandhu worked with the Fulbright Commission, Guru Nanak Dev University in Amritsar, India, and the United States-Indian Educational Foundation to found the Indian Association of Mental Health Counselors. He has also developed programs to train more mental health professionals and open counseling centers throughout India.
“India is the suicide capital of the world,” Sandhu said. “Depression seems to be everywhere and resources are limited. We have to change the way that people cope with their problems.”
Sandhu grew up in rural northwest India and is pioneer in multicultural counseling. He has published nearly two dozen books, more than 50 articles and provided editorial work to several professional journals. As LWC’s School of Professional Counseling Director of Research, he is responsible for guiding faculty research and working with graduate students in the college’s doctoral program. Sandhu holds six graduate degrees, including a doctor of counselor education.
Sandhu will chronicle his time and work in India, which will span from November 2017 to May 2018. | https://www.lindsey.edu/news/story.cfm?id=1095 |
Master Biomedical Engineering
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---
abstract: 'Inspired by the definition of generalized wreath product of permutation groups, we define the generalized wreath product of graphs, containing the classical Cartesian and wreath product of graphs as particular cases. We prove that the generalized wreath product of Cayley graphs of finite groups is the Cayley graph of the generalized wreath product of the corresponding groups.'
address: 'Università degli Studi Niccolò Cusano - Telematica Roma - Via Don Carlo Gnocchi, 3 00166 Roma, Italia Tel.: +39 06 45678350 Fax: +39 06 45678379'
author:
- Alfredo Donno
title: Generalized wreath products of graphs and groups
---
[: 05C76, 20B25, 20E22.]{}
Introduction
============
The idea of constructing new graphs starting from smaller component graphs is very natural. Products of graphs were widely studied in the literature for their theoretical interest in Combinatorics, Probability, Harmonic Analysis, but also for their practical applications. Standard products include the Cartesian product, direct product, strong product, lexicographic product [@imrich; @sabidussi; @cartesian] (see also the beautiful handbook [@imrichbook]). In [@annals], the zig-zag product was introduced in order to produce constant-degree expanders of arbitrary size (see the surveys [@expander; @lubotullio] for definition, properties and further references on expander graphs). The zig-zag product and the simpler replacement product play also an important role in Geometric Group Theory, since it turns out that, when applied to Cayley graphs of two finite groups, they provide the Cayley graph of the semidirect product of these groups [@groups; @alfredo1; @expander; @communications], with a suitable choice of the corresponding generating sets. An analogous result holds for the classical wreath product of graphs (Theorem \[proofwreath\], Section \[section2\]).\
Inspired by the paper [@bayleygeneralized], where the definition of generalized wreath product of permutation groups is given as a generalization of the classical direct and wreath product of permutation groups, we define the [*generalized wreath product of graphs*]{} (note that in [@erschler] a different notion of generalized wreath product of graphs is presented). It is remarkable that, with a particular choice of the generating sets, our construction of the generalized wreath product applied to Cayley graphs of finite groups gives the Cayley graph of the generalized wreath product of the groups (Theorem \[theoremlast\], Section \[section3\]), providing a strong generalization of Theorem \[proofwreath\].
Preliminaries {#section2}
=============
Let us start by recalling the definition of Cayley graph of a finitely generated group with respect to some symmetric generating set. We denote by $1_G$ the identity element of a group $G$.
Let $G$ be a group generated by a finite set $S$, and suppose that $S$ is symmetric, i.e., if $s\in S$, then also $s^{-1}\in S$, and that $1_G \not\in S$. The [*Cayley graph*]{} $Cay(G,S)$ of $G$ with respect to $S$ is the graph whose vertex set is $G$, and where two vertices $g$ and $g'$ are adjacent (we will use the notation $g\sim g'$) if there exists a generator $s\in S$ such that $gs=g'$. The graph $Cay(G,S)$ is clearly a connected regular graph of degree $|S|$.
Note that we assume $1_G \not\in S$ in order to avoid loops in the graph $Cay(G,S)$.
Let us recall the definition of Cartesian product (see, for instance, [@cartesian], or [@imrichbook; @imrich], where a more general construction containing it as a particular case is introduced) and wreath product of graphs [@erschler].
\[definitioncartesianproduct\] Let $\mathcal{G}_1=(V_1, E_1)$ and $\mathcal{G}_2=(V_2,E_2)$ be two finite graphs. The *Cartesian product* $\mathcal{G}_1
\square \mathcal{G}_2$ is the graph with vertex set $V_1\times
V_2$, where two vertices $(v_1,v_2)$ and $(w_1,w_2)$ are adjacent if:
1. either $v_1= w_1$ and $v_2\sim w_2$ in $\mathcal{G}_2$;
2. or $v_2=w_2$ and $v_1\sim w_1$ in $\mathcal{G}_1$.
It follows from the definition that if $\mathcal{G}_1$ is a $d_1$-regular graph on $n_1$ vertices and $\mathcal{G}_2$ is a $d_2$-regular graph on $n_2$ vertices, then the graph $\mathcal{G}_1\square \mathcal{G}_2$ is a $(d_1+d_2)$-regular graph on $n_1n_2$ vertices. Notice also that the graphs $\mathcal{G}_1\square \mathcal{G}_2$ and $\mathcal{G}_2\square
\mathcal{G}_1$ are isomorphic.
\[defierschler\] Let $\mathcal{G}_1=(V_1, E_1)$ and $\mathcal{G}_2=(V_2,E_2)$ be two finite graphs. The *wreath product* $\mathcal{G}_1\wr
\mathcal{G}_2$ is the graph with vertex set $V_2^{V_1}\times V_1=
\{(f,v) | f:V_1\to V_2, \ v\in V_1\}$, where two vertices $(f,v)$ and $(f',v')$ are connected by an edge if:
1. ([*edges of the first type*]{}) either $v=v'=:\overline{v}$ and $f(w)=f'(w)$ for every $w\neq \overline{v}$, and $f(\overline{v})\sim f'(\overline{v})$ in $\mathcal{G}_2$;
2. ([*edges of the second type*]{}) or $f(w)=f'(w)$, for every $w\in
V_1$, and $v\sim v'$ in $\mathcal{G}_1$.
It follows from the definition that, if $\mathcal{G}_1$ is a regular graph on $n_1$ vertices with degree $d_1$ and $\mathcal{G}_2$ is regular graph on $n_2$ vertices with degree $d_2$, then the graph $\mathcal{G}_1\wr \mathcal{G}_2$ is a $(d_1+d_2)$-regular graph on $n_1\cdot n_2^{n_1}$ vertices.
The wreath product of graphs represents a graph-analogue of the classical wreath product of groups (Theorem \[proofwreath\]). To show that, we need to recall the basic definition of semidirect product of groups. Let $A$ and $B$ be two finite groups, and suppose that an action by automorphisms of $B$ on $A$ is defined, i.e., there exists a group homomorphism $\phi: B \to Aut(A)$. For every $a\in A$ and $b\in B$, we denote by $a^b$ the image of $a$ under the action of $\phi(b)$ and, similarly, we denote by $a^B =
\{a^b\ |\ b\in B\}$ the orbit of $a$ under the action of the group $B$.
The *semidirect product* $A\rtimes B$ is the group whose underlying set is $A\times B = \{(a,b) \ |\ a\in A,b\in B\}$, and whose group operation is defined by $$(a_1,b_1)(a_2,b_2) = (a_1a_2^{b_1},b_1b_2), \qquad \mbox{for all }
a_1,a_2\in A, b_1,b_2\in B.$$
It is easy to check that the identity of $A\rtimes B$ is given by $(1_A,1_B)$, where $1_A$ and $1_B$ are the identity in $A$ and $B$, respectively, and that $ (a,b)^{-1} =
((a^{-1})^{b^{-1}},b^{-1})$, for all $a\in A, b\in B$. Note that the subgroup $A\times \{1_B\}$ of $A\rtimes B$ is isomorphic to $A$, it is normal in $A\rtimes B$ and the action of $B$ on $A$ by conjugation coincides with the original action of $B$ on $A$. In formulas, we have $ (1_A,b)(a,1_B)(1_A,b)^{-1} = (a^b,1_B)$, for all $a\in A, b\in B$.
Let $A$ and $B$ be two finite groups. The set $B^A = \{f:A\to B\}$ can be endowed with a group structure with respect to the pointwise multiplication: $(f_1f_2)(a) = f_1(a) f_2(a)$. The [*wreath product*]{} $A\wr B$ is the semidirect product $B^A\rtimes A$, where $A$ acts on $B^A$ by shifts, i.e., if $f\in B^A$, one has $$f^a(x) = f(a^{-1}x), \quad \mbox{for all } a,x\in A.$$
We introduce some notation. If $f\in B^A$ and $A = \{a_1, a_2,
\ldots, a_{n_A}\}$, then we write $f=(f_1,f_2, \ldots, f_{n_A})$, where we denote by $f_i$ the element $f(a_i)\in B$, for each $i=1,\ldots, n_A$. In particular, an element of $B^A\times A$ will be written as $((f_1,\ldots, f_{n_A}), a)$.
\[proofwreath\] Let $A$ and $B$ be two finite groups and let $S_A$ and $S_B$ be symmetric generating sets for $A$ and $B$, respectively. Then $$Cay(A,S_A)\wr Cay(B,S_B) = Cay(A\wr B, S),$$ where $S$ is the generating set of $A\wr B$ given by $$S = \{((s_b,1_B,\ldots,1_B),1_A), ((1_B,\ldots, 1_B),s_a)\ |\
s_a\in S_A, s_b\in S_B\}.$$
It is easy to check that $S$ is a symmetric generating set of $A\wr B = B^A\rtimes A$. More precisely, any element $(f,a)=((f_1,
\ldots, f_{n_A}), a)\in B^A\rtimes A$, with $f_i\in B$ and $a\in
A$, can be decomposed as $$((f_1,1_B, \ldots, 1_B),1_A)((1_B, f_2,1_B,\ldots, 1_B),1_A)\cdots
((1_B, \ldots, 1_B,f_{n_A}),1_A)((1_B,\ldots,1_B),a).$$ Now if $a = \prod_{k=1}^r s_k^{m_k}$, with $s_k\in S_A$ and $m_k\in \mathbb{N}$, then one has $$((1_B,\ldots,1_B),a) = \prod_{k=1}^r ((1_B,\ldots,1_B),s_k)^{m_k}.$$ Similarly, if $f_1 = \prod_{h=1}^{l}s_h^{m_h}$, with $s_h\in S_B$ and $m_h\in \mathbb{N}$, it holds: $$((f_1,1_B, \ldots, 1_B),1_A) = \prod_{h=1}^l ((s_h,1_B, \ldots,
1_B),1_A)^{m_h}.$$ Finally, we have $$((1_B,\ldots, 1_B,f_i,1_B,\ldots,1_B),1_A) =
((1_B,\ldots,1_B),a_ia_1^{-1})((f_i,1_B,\ldots,1_B),a_1a_i^{-1})$$ and so we conclude that $S$ generates $A\wr B$.\
What we have to prove now is that an edge in the graph product $Cay(A,S_A)\wr Cay(B,S_B)$ corresponds exactly to the multiplication by an element of $S$ in $A\wr B$.\
Consider an edge of the first type in $Cay(A,S_A)\wr
Cay(B,S_B)$: such an edge connects the vertices $((f_1,
\ldots,f_k, \ldots, f_{n_A}),a_k)$ and $((g_1, \ldots,g_k,\ldots,
g_{n_A}), a_k)$. By definition, it must be $f_i = g_i$ for every $i\neq k$, whereas $f_k$ and $g_k$ are vertices adjacent in $Cay(B,S_B)$. It follows that there exists $s_k\in S_B$ such that $f_k s_k = g_k$. Then one gets $((g_1, \ldots,g_k,\ldots,
g_{n_A}), a_k)$ from $((f_1, \ldots,f_k, \ldots, f_{n_A}),a_k)$ by multiplying by $((s_k,1_B,\ldots,1_B),1_A)$. In fact: $$((f_1,\! \ldots\!,\!f_k,\! \ldots\!,\!
f_{n_A}\!),\!a_k)\cdot((s_k,\!1_B,\!\ldots\!,\!1_B),\!1_A\!)\! =\!
((f_1,\! \ldots\!,\!f_k,\! \ldots\!,\!
f_{n_A})\cdot(s_k,\!1_B,\!\ldots\!,\!1_B)^{a_k}\!,\! a_k\cdot 1_A)$$ $$=((f_1,\! \ldots\!,\!f_k,\! \ldots\!,
f_{n_A})\cdot(\!1_B,\!\ldots\!,\!
1_B,\!\underbrace{s_k}_{k\tiny{\mbox{-th
place}}}\!,\!1_B,\!\ldots\!,\!1_B), a_k) \!=\! ((g_1, g_2,\!
\ldots\!,g_k,\!\ldots\!, g_{n_A}),a_k).$$ This implies that edges of the first type correspond to multiplication by elements of the form $((s_b,1_B,\ldots,1_B),1_A)$, with $s_b\in S_B$.\
Consider now an edge of the second type in $Cay(A,S_A)\wr
Cay(B,S_B)$: such an edge connects the vertices $((f_1, \ldots,
f_{n_A}),a_k)$ and $((g_1, \ldots, g_{n_A}), a_h)$. By definition, it must be $f_i = g_i$ for every $i=1,\ldots, n_A$, whereas $a_k$ and $a_h$ are vertices adjacent in $Cay(A,S_A)$. It follows that there exists $s_a\in S_A$ such that $a_k s_a = a_h$. Then one gets $((g_1, \ldots, g_{n_A}), a_h)$ from $((f_1, \ldots,
f_{n_A}),a_k)$ by multiplying by $((1_B,\ldots,1_B),s_a)$. In fact: $$((f_1,\ldots, f_{n_A}),a_k)\cdot((1_B,\ldots,1_B),s_a) =
((f_1,\ldots, f_{n_A})\cdot(1_B,\ldots,1_B)^{a_k}, a_k\cdot s_a)=$$ $$((f_1, \ldots, f_{n_A})\cdot(1_B,\ldots, 1_B), a_h) =
((g_1,\ldots, g_{n_A}),a_h).$$ This ensures that edges of the second type correspond to multiplication by elements of the form $((1_B,\ldots,1_B),s_a)$, with $s_a\in S_A$.
Consider the graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ in Fig. \[duek2\].
(200,20)=0,15mm A=(50,20)B=(150,20)C=(250,20)D=(350,20)
(A)[$\bullet$]{}(B)[$\bullet$]{}(C)[$\bullet$]{}(D)[$\bullet$]{}
(A,B)(C,D)
(42,30)[$u_1$]{}(142,30)[$v_1$]{}(242,30)[$u_2$]{}(342,30)[$v_2$]{} (95,-3)[$\mathcal{G}_1$]{}(295,-3)[$\mathcal{G}_2$]{}
Then the wreath product $\mathcal{G}_1\wr \mathcal{G}_2$ is the octagonal graph in Fig. \[ottagono\].
(200,180)=0,11mm A=(250,420)B=(390,360)C=(450,220)D=(390,80)E=(250,20)F=(110,80)G=(50,220) H=(110,360)
(A)[$\bullet$]{}(B)[$\bullet$]{}(C)[$\bullet$]{}(D)[$\bullet$]{} (E)[$\bullet$]{}(F)[$\bullet$]{}(G)[$\bullet$]{}(H)[$\bullet$]{}
(A,B)(B,C)(C,D)(D,E) (E,F)(F,G)(G,H)(H,A)
(750,200)[$\mathcal{G}_1\wr \mathcal{G}_2$]{}
(170,440)[$((u_2,u_2),u_1)$]{}(-78,350)[$((v_2,u_2),u_1)$]{}
(-135,215)[$((v_2,u_2),v_1)$]{}(-70,70)[$((v_2,v_2),v_1)$]{}
(170,-15)[$((v_2,v_2),u_1)$]{}(405,70)[$((u_2,v_2),u_1)$]{}
(460,205)[$((u_2,v_2),v_1)$]{}(405,355)[$((u_2,u_2),v_1)$]{}
If we regard the graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ as the Cayley graphs of two cyclic groups of two elements, then the generators given by Theorem \[proofwreath\] have order $2$, but they do not commute: the wreath product $\mathcal{G}_1\wr\mathcal{G}_2$ is the Cayley graph of the wreath product of these groups, which is isomorphic to the dihedral group of $8$ elements.
Now let $\mathcal{G}_3$ be a triangular graph. In Fig. \[grande24\] the wreath product $\mathcal{G}_3\wr \mathcal{G}_1$ is represented.
(350,160)=0,11mm
AA=(10,200)BB=(90,200)CC=(50,271) (AA)[$\bullet$]{}(BB)[$\bullet$]{}(CC)[$\bullet$]{}
(AA,BB)(BB,CC)(CC,AA)
(-50,230)[$\mathcal{G}_3$]{}(750,230)[$\mathcal{G}_3\wr
\mathcal{G}_1$]{}
A=(310,340)B=(310,60)C=(360,10)D=(640,10)E=(690,60)F=(690,340)G=(640,390) H=(360,390)I=(360,340)L=(360,60)M=(640,60)N=(640,340)O=(420,280) P=(420,230)Q=(420,170)R=(420,120)S=(470,120)T=(530,120)U=(580,120) V=(580,170)Z=(580,230)X=(580,280)Y=(530,280)W=(470,280)
(A)[$\bullet$]{}(B)[$\bullet$]{}(C)[$\bullet$]{}(D)[$\bullet$]{} (E)[$\bullet$]{}(F)[$\bullet$]{}(G)[$\bullet$]{}(I)[$\bullet$]{}(M)[$\bullet$]{}(N)[$\bullet$]{} (H)[$\bullet$]{}(L)[$\bullet$]{}(O)[$\bullet$]{}(T)[$\bullet$]{} (P)[$\bullet$]{}(U)[$\bullet$]{}(Q)[$\bullet$]{}(V)[$\bullet$]{} (R)[$\bullet$]{}(Z)[$\bullet$]{}(Y)[$\bullet$]{}(Z)[$\bullet$]{} (S)[$\bullet$]{}(X)[$\bullet$]{}(W)[$\bullet$]{}
(A,B)(A,H)(A,I)(B,L) (B,C)(C,L)(C,D)(D,E) (D,M)(E,M)(E,F) (F,G)(F,N)(G,N)(G,H) (H,I)(I,O)(O,P)(O,W)(P,Q) (Q,R)(Q,S)(R,L) (R,S)(S,T)(T,U)(T,V) (U,M)(U,V)(V,Z)(Z,Y)(Z,X) (N,X)(X,Y)(Y,W)(P,W)
Note that the graph $\mathcal{G}_3\wr \mathcal{G}_1$ is obtained in [@alfredo1] as the replacement product of the $3$-dimensional Hamming cube with a triangular graph.
Let $\mathcal{G}_1=(V_1, E_1)$ and $\mathcal{G}_2=(V_2,E_2)$ be two finite graphs. The *lexicographic product* $\mathcal{G}_1 \circ \mathcal{G}_2$ is the graph with vertex set $V_1\times V_2$, where two vertices $(v_1,v_2)$ and $(w_1,w_2)$ are adjacent if:
1. either $v_1\sim w_1$ in $\mathcal{G}_1$;
2. or $v_1=w_1$ and $v_2\sim w_2$ in $\mathcal{G}_2$.
It follows from the definition that if $\mathcal{G}_1$ is a $d_1$-regular graph on $n_1$ vertices and $\mathcal{G}_2$ is a $d_2$-regular graph on $n_2$ vertices, then the graph $\mathcal{G}_1\circ \mathcal{G}_2$ is a $(d_1n_2+d_2)$-regular graph on $n_1n_2$ vertices.\
Sometimes the lexicographic product of graphs, whose automorphism group contains the wreath product of the automorphism groups of the factors, is called wreath product. It has nothing to do with the wreath product of Definition \[defierschler\].
Generalized wreath product of graphs {#section3}
====================================
Before introducing the notion of generalized wreath product of graphs (Definition \[defimine\]), we recall the definition of poset block structure and generalized wreath product of permutation groups introduced in [@bayleygeneralized]. We will follow the same notation for the action to the right presented there. See also [@ischia2008; @orthogonal; @lumpability], where the Gelfand pairs associated with the action of a generalized wreath product of groups on a poset block structure are studied, in connection with Markov chain Theory.
Let $(I,\preceq)$ be a finite poset, with $|I| = n$. For every $i\in I$, the following subsets of $I$ can be defined:
- $A(i)=\{j\in I : j \succ i\}$ and $A[i] = A(i) \sqcup \{i\}$;
- $H(i)=\{j\in I : j \prec i\}$ and $H[i] = H(i) \sqcup \{i\}$.
A subset $J\subseteq I$ is said [*ancestral*]{} if, whenever $i
\succ j$ and $j\in J$, then $i\in J$. Note that by definition $A(i)$ and $A[i]$ are ancestral, for each $i\in I$. The set $A(i)$ is called the ancestral set of $i$, whereas the set $H(i)$ is called the hereditary set of $i$.\
For each $i\in I$, let $X_i$ be a finite set, with $|X_i|\geq 2$. For $J\subseteq I$, put $X_J = \prod_{i\in J}X_i$. In particular, we put $X = X_I$. If $K\subseteq J \subseteq I$, let $\pi^J_K$ denote the natural projection from $X_J$ onto $X_K$. In particular, we set $\pi_J = \pi^I_J$ and $x_J=x\pi_J$, for every $x\in X$. Moreover, we will use $X^i$ for $X_{A(i)} = \prod_{j\in A(i)}X_j$ and $\pi^i$ for $\pi_{A(i)}$.\
Let $\mathcal{A}$ be the set of ancestral subsets of $I$. If $J\in \mathcal{A}$, then the equivalence relation $\sim_J$ on $X$ is defined as $$x \sim_J y \quad \Longleftrightarrow \quad x_J = y_J,\qquad
\mbox{for } x,y \in X.$$
A [*poset block structure*]{} is a pair $(X,\sim_{\mathcal{A}})$, where
1. $X = \prod_{(I,\preceq)}X_i$, with $(I,\preceq)$ a finite poset and $|X_i| \geq 2$, for each $i\in I$;
2. $\sim_{\mathcal{A}}$ denotes the set of equivalence relations on $X$ defined by all the ancestral subsets of $I$.
For each $i\in I$, let $G_i$ be a permutation group on $X_i$ and let $F_i$ be the set of all functions from $X^i$ into $G_i$. For $J\subseteq I$, we put $F_J = \prod_{i\in J}F_i$ and set $F =
F_I$. An element of $F$ will be denoted $f = (f_i)_{i\in I}$, with $f_i \in F_i$.
For each $f\in F$, the action of $f$ on $X$ is defined as follows: if $x = (x_i)_{i\in I}\in X$, then $$\begin{aligned}
x f = y,\quad \mbox{where }y = (y_i)_{i\in I}\in X \quad
\mbox{and }\ y_i = x_i(x\pi^i f_i), \quad \mbox{for each }i\in I.\end{aligned}$$
It is easy to verify that this is a faithful action of $F$ on $X$, i.e, if $xf = xg$ for every $x\in X$, then $f=g$. Therefore $(F,X)$ is a permutation group, called the [*generalized wreath product of the permutation groups $(G_i,X_i)_{i\in I}$*]{} and denoted $\prod_{(I,\preceq)}(G_i,X_i)$.
An automorphism of a poset block structure $(X,\sim_{\mathcal{A}})$ is a permutation $\sigma$ of $X$ such that, for every equivalence relation $\sim_J$ in $\sim_{\mathcal{A}}$, $$x \sim_J y \qquad \Longleftrightarrow \qquad (x \sigma)\sim_J (y
\sigma), \qquad \mbox{for all } x, y \in X.$$
The following fundamental results are proven in [@bayleygeneralized]. We denote by $Sym(X_i)$ the symmetric group acting on $X_i$.
The generalized wreath product of the permutation groups $(G_i,
X_i)_{i\in I}$ is transitive on $X$ if and only if $(G_i, X_i)$ is transitive for each $i\in I$.
Let $(X, \sim_{\mathcal{A}})$ be the poset block structure associated with the poset $(I,\preceq)$. Let $F$ be the generalized wreath product $\prod_{(I,\preceq)}Sym(X_i)$. Then $F$ is the automorphism group of $(X,\sim_{\mathcal{A}})$.
Given $f\in F$ and $J\subset I$ ancestral, a map $f_J: X_J \to
X_J$ is defined such that $f \pi_J = \pi_J f_J$. The following lemma holds.
\[bayleyproduct\] Let $f,h\in F$. Then $fh = t$, with $$t_i = f_i\cdot f_{A(i)}h_i, \qquad \mbox{for every }i\in I,$$ where the product of $f_i$ and $f_{A(i)}h_i$ is pointwise.
Let $x\in X$. We have: $$\begin{aligned}
(xfh)_i &=& (x f)_i (x f \pi^i h_i)\\
&=& x_i(x\pi^i f_i) (x \pi^i f_{A(i)}h_i)\\
&=& x_i(x\pi^i(f_i\cdot f_{A(i)}h_i))\\
&=& x_i(x\pi^it_i).\end{aligned}$$
If $(I,\preceq)$ is a finite poset, with $\preceq$ the identity relation (Fig. \[figure13\]), then the generalized wreath product is the permutation direct product. In this case, we have $A(i) = \emptyset$, for each $i\in I$, so that an element $f$ of $F$ is given by $f = (f_i)_{i\in I}$, where the function $f_i$ is identified with an element of $G_i$, so that its action on $x_i$ does not depend on any other coordinate of $x$.
(300,30) (80,20)[$\bullet$]{}(110,20)[$\bullet$]{}(140,20)[$\bullet$]{}(150,23) (160,23)(170,23)(180,23)(190,23) (200,23)(210,23)(220,20)[$\bullet$]{} (78,8)[$1$]{}(110,8)[$2$]{}(140,8)[$3$]{}(220,8)[$n$]{}
If $(I,\preceq)$ is a finite chain (Fig. \[figure14\]), then the generalized wreath product is the classical permutation wreath product $(G_1,X_1)\wr(G_2,X_2)\wr \cdots \wr(G_n,X_n)$. In this case, we have $A(i) = \{1,2,\ldots,i-1\}$, for each $i\in I$, so that an element $f\in F$ is given by $f = (f_i)_{i\in I}$, with $$f_i:X_1 \times \cdots \times X_{i-1}\longrightarrow G_i$$ In other words, the action of $f$ on $x_i$ depends on its ancestralcoordinates $x_1, \ldots, x_{i-1}$.
(250,110) A=(125,100)B=(125,80)C=(125,60) D=(125,55)E=(125,50)F=(125,45) G=(125,40)H=(125,35)I=(125,15)
(A)[$\bullet$]{}(B)[$\bullet$]{}(C)[$\bullet$]{}(H)[$\bullet$]{} (I)[$\bullet$]{} (D)(E)(F)(G) (A,B)(B,C) (H,I) (129,97)[$1$]{}(129,77)[$2$]{}(129,57)[$3$]{}(130,32)[$n-1$]{}(129,13)[$n$]{}
Inspired by the definition of generalized wreath product of permutation groups, we introduce here the notion of generalized wreath product of graphs.\
Let $\mathcal{S},\mathcal{T}$ be two sets. Given two functions $f,g:\mathcal{S}\longrightarrow\mathcal{T}$, we will use the notation $f\equiv g$ to say that $f(x)=g(x)$ for every $x\in
\mathcal{S}$. Similarly, we will write $f\equiv g$ in $A\subset
\mathcal{S}$ to say that $f(x) = g(x)$ for every $x\in A$.
\[defimine\] Let $(I,\preceq )$ be a finite poset, with $|I|=n$, and let $\mathcal{G}_i=(V_i,E_i)$ be a finite graph, for every $i\in I$. The [*generalized wreath product*]{} of the graphs $\{\mathcal{G}_i\}_{i\in I}$ is the graph $\mathcal{G}$ with vertex set $$V_\mathcal{G} = \{(f_1,f_2, \ldots, f_n)\ |\ f_i : \prod_{j\in
A(i)}V_j\to V_i, \ \mbox{for each }i\in I\}$$ and where two vertices $f=(f_1,f_2, \ldots, f_n)$ and $h=(h_1,h_2,
\ldots, h_n)$ are adjacent if there exists $i\in I$, with $A(i) =
\{i_1, \ldots, i_p\}$, such that:
1. $f_j\equiv h_j$, for every $j\neq i$;
2. $f_i\equiv h_i$ in $\prod_{j\in A(i)}V_j\
\setminus\ \{(f_{i_1}, \ldots, f_{i_p})\}$, and $f_i(f_{i_1},\ldots, f_{i_p})\sim h_i(f_{i_1},\ldots, f_{i_p})$ in $V_i$.
The elements $f_{i_l}$, for $l=1,\ldots, p$, are defined recursively, starting from indices whose ancestral set in $(I,
\preceq)$ is empty; more precisely, they represent vertices of $V_{i_l}$ obtained by evaluating the functions $f_{i_l}$ on $(f_{j_1},\ldots, f_{j_m})$, where $A(i_l) = \{j_1,\ldots, j_m\}$, and so on.
In other words, $f=(f_1,\ldots, f_n)$ and $h=(h_1,\ldots, h_n)$ are adjacent if there exists $i\in I$ such that $f_j\equiv h_j$ for each $j\neq i$ and $f_i$ coincides with $h_i$, except when evaluated on the $p$-tuple $(f_{i_1}, \ldots, f_{i_p})$, where $A(i) =\{i_1, \ldots, i_p\}$. Note also that, if $A(i)=\emptyset$, the condition $(2)$ means that it must be $f_i\sim h_i$ in $\mathcal{G}_i$.
We have $|V_\mathcal{G}| = \prod_{i=1}^n |V_i|^{\Pi_{j\in
A(i)}|V_j|}$, where we put $\Pi_{j\in A(i)}|V_j|=1$ if $A(i)=\emptyset$. Moreover, if $\mathcal{G}_i$ is a $d_i$-regular graph for every $i\in I$, then $\mathcal{G}$ is a regular graph of degree $\sum_{i\in I}d_i$.
Consider the case where $|I|=4$, with the poset $(I,\preceq)$ and the graphs $\mathcal{G}_i$ represented in Fig. \[figure15\].
(300,60) A=(40,50)B=(60,30)C=(80,50) D=(60,0)E=(140,30)F=(170,30) G=(245,40)H=(230,15)I=(260,15)
(A)[$\bullet$]{}(B)[$\bullet$]{}(C)[$\bullet$]{}(D)[$\bullet$]{} (E)[$\bullet$]{}(F)[$\bullet$]{}(G)[$\bullet$]{} (H)[$\bullet$]{}(I)[$\bullet$]{}
(0,30)[$(I,\preceq)$]{}(115,50)[$\mathcal{G}_1=\mathcal{G}_2=\mathcal{G}_3$]{}(280,30)[$\mathcal{G}_4$]{}
(A,B)(C,B)(D,B) (E,F)(G,H)(H,I)(I,G)
(36,55)[$1$]{}(76,55)[$2$]{}(65,25)[$3$]{}(65,-3)[$4$]{}(137,16)[$a$]{}(167,16)[$b$]{}(242,45)[$c$]{}(227,3)[$d$]{}(257,3)[$e$]{}
In this case, $A(1)=A(2)=\emptyset$, $A(3)=\{1,2\}$ and $ A(4)=\{1,2,3\}$, so that $$V_\mathcal{G} = \left\{(f_1,f_2,f_3,f_4)\ |\ f_1\in V_1,\ f_2\in
V_2,\ f_3: \{a,b\}^2\to \{a,b\},\ f_4:\{a,b\}^3\to
\{c,d,e\}\right\}.$$ The function $f_3$ will be represented as a $4$-tuple of elements in $\{a,b\}$, whereas $f_4$ will be an $8$-tuple of elements in $\{c,d,e\}$ (coordinates are ordered lexicographically). We have $|V_\mathcal{G}| = 2\cdot 2\cdot 2^4\cdot 3^8$. Consider, for instance, the vertex $f=(a,b,(b,b,a,a),(c,e,d,c,e,d,e,e))\in
V_\mathcal{G}$. Its $5$ neighbors in $\mathcal{G}$ are the vertices:
1. $(b,b,(b,b,a,a),(c,e,d,c,e,d,e,e))$, since $f_1=a\sim b$ in $\mathcal{G}_1$;
2. $(a,a,(b,b,a,a),(c,e,d,c,e,d,e,e))$, since $f_2=b\sim a$ in $\mathcal{G}_2$;
3. $(a,b,(b,a,a,a),(c,e,d,c,e,d,e,e))$, since $f_3(f_1,f_2) = f_3(a,b) =b\sim a$ in $\mathcal{G}_3$;
4. $(a,b,(b,b,a,a),(c,e,d,d,e,d,e,e))$ and $(a,b,(b,b,a,a),(c,e,d,e,e,d,e,e))$, since\
$f_4(f_1,f_2, f_3(f_1,f_2)) = f_4(a,b,b) =c\sim d,e$ in $\mathcal{G}_4$.
If $(I,\preceq)$ is the poset $(I,\preceq_1)$ (resp. $(I,\preceq_2)$) in Fig. \[figure12\], one obtains the classical Cartesian product of Definition \[definitioncartesianproduct\] (resp. the classical wreath product of Definition \[defierschler\]).
(300,45) A=(60,25)B=(90,25) C=(225,40)D=(225,10)
(A)[$\bullet$]{}(B)[$\bullet$]{}(C)[$\bullet$]{}(D)[$\bullet$]{}
(-10,20)[$(I,\preceq_1)$]{} (260,20)[$(I,\preceq_2)$]{} (57,11)[$1$]{}(87,11)[$2$]{}(230,37)[$1$]{}(230,7)[$2$]{} (C,D)
In [@generalizedcrested], finite posets are used to define products of finite Markov chains, called generalized crested products, and to develop their spectral analysis. Notice that this construction generalizes the crested products of Markov chains introduced in [@crested].
We are going to prove that the generalized wreath product of Cayley graphs of finite groups is the Cayley graph of the generalized wreath product of the groups.
Let $(I,\preceq)$ be a finite poset, with $|I|=n$, and let $G_i$ be a finite group, for each $i\in I$. Let $S_i$ be a symmetric generating set for $G_i$ and consider the Cayley graph $\mathcal{G}_i=Cay(G_i,S_i)$. In order to see the correspondence, we regard the group $G_i$ as a permutation group on itself, acting on its elements by right multiplication (according with the notation of [@bayleygeneralized]). Definition \[defimine\] can be reformulated as follows.
\[defirevisited\] Let $(I,\preceq )$ be a finite poset, with $|I|=n$, and let $\mathcal{G}_i=Cay(G_i,S_i)$, where $G_i$ is a finite group and $S_i$ is a symmetric generating set of $G_i$, for all $i\in I$. We construct the graph $\mathcal{G}$ with vertex set $$V_{\mathcal{G}} = \{(f_1,f_2, \ldots, f_n)\ |\ f_i : \prod_{j\in
A(i)}G_j\to G_i, \ \mbox{for each }i\in I\}$$ and where the vertices $(f_1,f_2, \ldots, f_n)$ and $(h_1,h_2,
\ldots, h_n)$ are adjacent if there exists $i\in I$, with $A(i) =
\{i_1, \ldots, i_p\}$, such that:
1. $f_j\equiv h_j$, for every $j\neq i$;
2. $f_i\equiv h_i$ in $\prod_{j\in A(i)}G_j\
\setminus\ \{(f_{i_1}^{-1}, \ldots, f_{i_p}^{-1})\}$, and the vertices $f_i(f_{i_1}^{-1},\ldots, f_{i_p}^{-1})$ and $h_i(f_{i_1}^{-1},\ldots, f_{i_p}^{-1})$ are adjacent in $\mathcal{G}_i$.
Also in this case, the elements $f_{i_l}$, for $l=1,\ldots, p$, are defined recursively, so that they represent elements of the group $G_{i_l}$ obtained by evaluating the functions $f_{i_l}$ on $(f_{j_1}^{-1},\ldots, f_{j_m}^{-1})$, where $A(i_l) =
\{j_1,\ldots, j_m\}$.
The following theorem is a strong generalization of Theorem \[proofwreath\].
\[theoremlast\] The generalized wreath product $\mathcal{G}$ of the graphs $\{\mathcal{G}_i=Cay(G_i,S_i)\}_{i\in I}$ is the Cayley graph of the generalized wreath product $G$ of the groups $\{G_i\}_{i\in
I}$, with respect to the generating set $$S = \{\overline{f_i}=({\bf 1}_1, \ldots, {\bf 1}_{i-1},
\overline{s_i}, {\bf 1}_{i+1}, \ldots, {\bf 1}_n),\ i\in I\},$$ where $\overline{s_i}$ is a function taking the value $s_i\in S_i$ on $(1_{G_{1_1}}, \ldots, 1_{G_{i_p}})$, with $A(i) =
\{i_1,\ldots, i_p\}$, and the value $1_{G_i}$ elsewhere, whereas ${\bf 1}_q$ is the constant function taking the value $1_{G_q}$ on $\prod_{u\in A(q)}G_u$, for each $q\neq i$.
Generalizing the argument developed in the proof of Theorem \[proofwreath\], one can check that $S$ is a generating set of the group $G$. Hence, we have to show that an adjacency in the generalized wreath product of the Cayley graphs can be obtained by multiplication by an element of $S$.\
Suppose that the vertices $f = (f_1,f_2, \ldots, f_n)$ and $h = (h_1,h_2, \ldots, h_n)$ are adjacent, i.e., there exists $i\in I$ satisfying the conditions of Definition \[defirevisited\]. Without loss of generality, we can suppose that $$A(i) = \{l_1< \cdots <l_r <k_1<\cdots <k_s\}, \qquad \mbox{with
}r+s=p,$$ with $A(l_i) = \emptyset$, $A(k_j) = \{m_{j,1}, \ldots,
m_{j,t_j}\}$ and $A(k_j)\subset A(k_{j+1})$. In particular, observe that it must be $t_j\leq p$ and $A(k_j)\subset A(i)$. By definition, it must be $f_j\equiv h_j$, for $j\neq i$; moreover, we have $f_i\equiv h_i$ in $\prod_{j\in A(i)}G_j\ \setminus\
\{(f_{i_1}^{-1}, \ldots, f_{i_p}^{-1})\}$, and there exists $s_i\in S_i$ such that $f_i(f_{i_1}^{-1},\ldots, f_{i_p}^{-1}) =
g_i$ and $h_i(f_{i_1}^{-1},\ldots, f_{i_p}^{-1}) = g_is_i$. We have to show the following identity: $$\begin{aligned}
\label{identity}
(f_1,f_2,\ldots, f_i, \ldots, f_n)\overline{f_i} =
(h_1,h_2,\ldots, h_i, \ldots, h_n).\end{aligned}$$ The identity is clearly true for every coordinate $j\neq i$. We use Lemma \[bayleyproduct\] in order to verify it for the coordinate $i$. Let $x=(x_1,\ldots,x_n)\in \prod_{i\in
I}G_i$. Observe that, for every $c=1,\ldots,r$, we can put $f_{l_c} = g_{l_c}$, for some $g_{l_c}\in G_{l_c}$, since $A(l_c)=\emptyset$.\
The action of $(f_1,f_2,\ldots, f_i, \ldots,
f_n)\overline{f_i}$ on $x_i$ is given by $$x_i\cdot(f_i(x_{l_1}, \ldots, x_{l_r}, x_{k_1}, \ldots,
x_{k_s}))\cdot$$ $$\cdot((x_{l_1}g_{l_1}, \ldots,
x_{l_r}g_{l_r},x_{k_1}(f_{k_1}(x_{m_{1,1}},\ldots,x_{m_{1,t_1}})),\ldots,
x_{k_s}(f_{k_s}(x_{m_{s,1}},\ldots,x_{m_{s,t_s}})))\overline{s_i}).$$ Now if $$(x_{l_1},\!\ldots\!, x_{l_r},x_{k_1},\!\ldots\!,x_{k_s})\! =\!
(g_{l_1}^{-1}\!,\! \ldots\!, g_{l_r}^{-1}\!,
(f_{k_1}(f_{m_{1,1}}^{-1}\!,\!\ldots\!,f_{m_{1,t_1}}^{-1}))^{-1}\!,\!\ldots\!,
(f_{k_s}(f_{m_{s,1}}^{-1}\!,\!\ldots\!,f_{m_{s,t_s}}^{-1}))^{-1})\!,$$ then the argument of $\overline{s_i}$ is $(1_{G_{l_1}}, \ldots,
1_{G_{l_r}}, 1_{G_{k_1}}, \ldots, 1_{G_{k_s}})$, so that one gets $$x_i(f_1,f_2,\ldots, f_i, \ldots, f_n)\overline{f_i}= x_ig_is_i$$ and so $(f \overline{f_i})_i = h_i$.\
Otherwise, the argument of $\overline{s_i}$ is different from $(1_{G_{l_1}}, \ldots, 1_{G_{l_r}}, 1_{G_{k_1}}, \ldots,
1_{G_{k_s}})$, so that one gets $x_i(f_1,f_2,\ldots, f_i, \ldots,
f_n)\overline{f_i} = x_ig_i1_{G_i}=x_ig_i$ and so $(f
\overline{f_i})_i = f_i= h_i$.
Acknowledgement {#acknowledgement .unnumbered}
===============
I would like to express my deepest gratitude to Fabio Scarabotti and Tullio Ceccherini-Silberstein for their continuous encouragement. A part of this work was developed during my stay at the Technische Universität of Graz, and I want to thank Wolfgang Woess and Franz Lehner for several useful discussions. This research was partially supported by the European Science Foundation (Research Project RGLIS 4915).\
I would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.
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D’Angeli, D., Donno, A.: The lumpability property for a family of Markov chains on poset block structures. [*Adv. in Appl. Math.*]{} [**51**]{}, 367–391 (2013), Issue 3.
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Hammack, R., Imrich, W., Klavžar, S.: Handbook of product graphs. Second edition. [*Discrete Mathematics and its Applications (Boca Raton)*]{}. CRC Press, Boca Raton, FL (2011).
Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their application. [*Bull. Amer. Math. Soc. (N.S.)*]{} [**43**]{}, 439–561 (2006), no. 4.
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| |
In his work with CREW, Eric Joris seeks a mixture of theatre and technology. Through the use of ‘immersive’ technologies CREW explores the boundaries of the aesthetic experience. The company’s productions seem to transform the spectator into the protagonist. But in whose story? The creations constantly negotiate this question with the spectator. The answer involves taking a walk along the boundaries of our senses.
EUX is based on the idea of the ‘doppelgänger’ which has been interpreted since the Romantic Movement as an eerie encounter between the individual and his own ego. Through the use of technological protheses, the spectator’s body is duplicated in a mirror image that appears to be more real than the ‘true image’. This blurring of the boundary between fiction and reality confuses the spectator. Is the world he experiences around him a performance of his own creation or is it precisely the other way around? In EUX the visitor is submerged in a virtual world: the project offers an ‘immersive’ experience for one person at a time.
EUX is the result of a study and creation residency at La Chartreuse (Centre National des Ecritures de Spectacles, Villeneuve lez Avignon) in April 2008. For this production Joris worked with the young French writer Eli Commins and others. Following this the project was developed further at various locations.
|Eric Joris will give a demonstration lecture on 13 and 14/02, 22:00
|
In English, 90 min., free entrance, booking by telephone obligatory: 02/201 59 59. | https://kaaitheater.be/en/agenda/eux |
BACKGROUND OF THE INVENTION
An airbrush couples to a pressurized air source and a media source, e. g., a liquid ink or paint media. The media intermixes with an air stream as it exits the airbrush in an atomized spray. Air brushes are versatile and popular tools in many fields. An airbrush may be modified in operation according to a variety of parameters to accomplish a variety of objectives. The amount of air pressure, amount of media introduced into the air stream, and the rate of movement for the airbrush in relation to the workpiece affect the resulting work, i.e., affect how the media appears as it comes to rest on the workpiece.
The atomized media spray follows a conic pattern as it emerges from the airbrush nozzle. This conic pattern naturally carries significant overspray, especially when using relatively high air pressure. Given this conic spray pattern and inherent overspray, the airbrush naturally lacks an ability to focus the media spray. As a result, airbrush work typically includes a given amount of overspray and "fuzzy" rendering. Fine lines and distinct edges typically can only be achieved by use of shielding, e. g., a template held between the airbrush and the workpiece to prevent overspray on the workpiece.
Fine line detail work, therefore, presents a challenge in airbrush design and use. Some airbrushes produce fine line detail. Unfortunately, the artist must move the airbrush quickly to preserve fine line detail as the media strikes the workpiece. If the artist does not move the airbrush quickly, fine line detail is lost in excess media buildup and overspray on the workpiece. Accordingly, only the most skilled airbrush artists can produce any meaningful fine line detail. Even such skilled artists, however, cannot make extensive and practical use of fine line detail due to the need to always move the airbrush rapidly. Because of such limitation, i.e., the requirement that the airbrush move at significant speed relative to the workpiece, fine line detail in airbrush work is generally not possible. Even with a limited ability to render fine line detail work by moving the airbrush at significant speed, conventional airbrushes have not and cannot allow elaborate or intricate fine line detail work.
A relatively expensive type of airbrush, i.e. the "turbine" model reciprocating needle by Paache AB, provides a degree of fine line detail work without requiring that the airbrush move rapidly. The cost, typically six to eight times that of an introductory-level conventional airbrush, makes this airbrush unavailable to most airbrush artists.
Accordingly, there remains need for an affordable airbrush capable of providing fine line detail but at significantly lower speeds. By allowing fine line detail at lower speeds, i.e., speed of the airbrush relative to the workpiece, a greater number of artists can make use of fine line detail in their work and a greater variety of airbrush work becomes possible at even greater detail than previously possible using conventional airbrush designs. The subject matter of the present invention provides a fine line detail rendering ability for a conventional or typical airbrush moving at relatively low speed in relation to the workpiece.
SUMMARY OF THE INVENTION
In accordance with the present invention, a conventional airbrush receives an attachment coupling to its spray outlet nozzle. The conventional airbrush output, i.e., atomized media, enters a chamber of the attachment. The flow of atomized media within the chamber is then diverted. The bulk of atomized media passes from the chamber through a bypass outlet. The remaining portion of atomized material, however, passes through an outlet nozzle at the distal end of the chamber as a fine line spray. In essence, the present invention focuses the atomized material more closely along parallel lines as opposed to diverging lines, i.e., as opposed to a conic pattern as found in conventional airbrush design. When applied to a workpiece this focused fine line spray provides fine line detail rendering without requiring rapid movement of the airbrush in relation to the workpiece.
The subject matter of the present invention is particularly pointed out and distinctly claimed in the concluding portion of this specification. However, both the organization and method of operation of the invention, together with further advantages and objects thereof, may best be understood by reference to the following description taken with the accompanying drawings wherein like reference characters refer to like elements.
BRIEF DESCRIPTION OF THE DRAWINGS
For a better understanding of the invention, and to show how the same may be carried into effect, reference will now be made, by way of example, to the accompanying drawings in which:
FIG. 1 illustrates in section in side view partially a conventional airbrush and an adapter attached thereto according to a preferred embodiment of the present invention.
FIG. 2 illustrates partially and in side view a modified form of the airbrush and adapter illustrated in FIG. 1 including a flexible extension hose.
FIGS. 3A-3J illustrate various geometric shapes proposed for use under the present invention.
FIG. 4 illustrates line detail rendered under the present invention as compared to a similar rendering using a conventional airbrush.
FIG. 5 illustrates in section a second embodiment of the present invention.
FIGS. 6A-6C illustrate angular variation in three conventional air brush structures and an elastic tubular gasket used in mounting the air brush attachment of FIG. 5 thereto.
FIG. 7 illustrates dimensions suggested for the air brush attachment of FIG. 5.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
The preferred embodiment of the present invention is an adapter which couples to a conventional airbrush. The adapter contemplated under the present invention couples to the airbrush through a variety of coupling mechanisms and serves as an attachment for the airbrush when performing fine line detail work. Because fine line detail work is not always desired, the adapter may be removed for conventional use of the airbrush. It will be understood, however, that the adapter described herein could be incorporated into and made integral with an airbrush dedicated to fine line detail work.
FIG. 1 illustrates in section a fine line detail attachment 10 coupled to a conventional airbrush 12 (partially illustrated in FIG. 1). Conventional airbrush 12 includes a nose 14 and outlet nozzle 16. Attachment 10 couples to a variety of airbrush devices, and the specific mounting mechanism for attachment 10 relative to a particular airbrush may vary according to the particular nose 14 structures involved. As illustrated, attachment 10 slides onto and engages nose 14.
Attachment 10 includes a generally tubular body 18 defining a main chamber 20 having an open proximal end 20a. Open end 20a of chamber 20 slides over outlet nozzle 16 and onto nose 14 of airbrush 12. A secure and air-tight seal 22 is thereby established between attachment 10 and nose 14 of airbrush 12. As a result, the atomized media spray 24 emerging from outlet nozzle 16 enters directly into chamber 20 of attachment 10. Attachment 10 diverts some of spray 24 as spray flow 24a through a bypass outlet 26. Spray 24a exiting chamber 20 by way of bypass outlet 26 may be diverted through a conduit 27 to a bottle or reservoir 29 as shown schematically in FIGS. 1 and 2. Media accumulating in reservoir 29 may then be recycled as desired. The remaining portion of spray 24, i.e., spray flow 24b, passes through a conduit 30 at the distal end of chamber 20 and into attachment nozzle 32. As illustrated in FIG. 1, the upstream end of conduit 30, i.e., receiving spray flow 24b, extends partially into chamber 20 and forms an annular dam 31. Annular dam 31 improves separation between spray flow 24a and spray flow 24b. More particularly, annular dam 31 passes spray flow 24b as atomized spray traveling through chamber 20. The remaining spray flow 24b, especially that portion encountering the inner walls of chamber 20, does not enter conduit 30 and flows eventually out of chamber 20 via bypass outlet 26. In this manner, spray flow 24b is an atomized spray entering chamber 33 of nozzle 32. The spray flow 24b then emerges from the outlet 36 of nozzle 32 as fine line detail spray 34.
Fine line detail spray 34 applies a highly detailed, high resolution application of media to a given workpiece. Such fine line detail results even when airbrush 12 and attachment 10 move at slow speed relative to the workpiece. Accordingly, the artist produces fine line detail without moving the airbrush at high speed. Allowing fine line detail at low speed, i.e., speed of the airbrush relative to the workpiece, greatly expands the ability of the artist to produce greater detailed work and thereby improves the artist's final work product whenever fine line detail enhances the resulting product. Furthermore, because the present invention provides fine line detail using a relatively inexpensive attachment for a relatively inexpensive airbrush, a broader range of artists have access to fine line detail in their work.
The precise nature of fine line detail spray 34 varies according to several factors. For example, the media used, atomized media spray 24, volume of chamber 20, size of bypass outlet 26, size of nozzle 32 orifice 36, and air pressure applied to airbrush 12.
With respect to air pressure, it is suggested that when using attachment 10 the air pressure applied to airbrush 12 be lowered. Typical air pressure applied to airbrush 12, i.e., in normal use without attachment 10, is 15 to 20 psi.. When using attachment 10, however, air pressure should be turned down to between 5 and 15 psi. for best results.
FIG. 2 illustrates a useful addition to the arrangement of FIG. 1, namely a flexible extension hose 40 coupling nozzle 32 and the down stream end of conduit 30. Hose 40 may be of sufficient length to allow an artist to hold with one hand the distal end 40a of hose 40 and nozzle 32 in the fashion of a writing instrument, i.e., a pencil or pen, and render fine line detail as if using a conventional writing instrument. The other hand operates airbrush 12 controls. FIG. 2 illustrates schematically a hand 42 holding hose 40 and nozzle 32 relative to a workpiece 44 and applying spray 34 to produce on workpiece 44 a fine line 46. The configuration of FIG. 2 allows the artist to hold aside airbrush 12 along with the remainder of attachment 10 as attached thereto while holding and manipulating the distal end 40a of hose 40 and nozzle 32 as illustrated in FIG. 2.
FIGS. 3A-3J show a variety of shapes which may be used in implementing nozzle orifice 36, individually orifices 36a-36j. Without discussing each shape in detail, it will be understood that a great variety of shapes, including shapes not specifically shown in FIGS. 3A- 3J, may be used in implementation of the present invention. Furthermore, nozzle 32 may be configured for easy removal and mounting relative to attachment 10, or hose 40, to allow rapid interchange between various ones of nozzles 32a- 32j as illustrated in FIGS. 3A-3J. Because spray 34 emerging from nozzle 32 follows more closely straight lines than a conic pattern, nozzle 32 may be held stationary and produce on a workpiece a geometric pattern corresponding to the shape of a particular orifice 36. For example, FIG. 3G illustrates a star-shaped orifice 36g. By holding nozzle 32g stationary relative to a workpiece and activating airbrush 12, one produces a fine detail star-shaped pattern on the workpiece. By selecting, for example, the slot-shaped orifice 36e of nozzle 32e shown in FIG. 3e, one produces calligraphy-like alphanumeric renditions. Heretofore, producing any stylized writing with an airbrush has generally been virtually impossible. With the attachment of the present invention, however, such airbrush calligraphy becomes possible.
FIG. 4 is a photocopy of an actual rendition using a conventional airbrush and an airbrush equipped with the attachment 10 of the present invention. In FIG. 4, line 50 was produced using a conventional airbrush and line 52 was produced using the same airbrush equipped with an attachment 10. The airbrush used was an Eclipse model airbrush from Iwata operating at 20 pounds per square inch (psi). Line 50 was drawn during a 25 second interval pulling the airbrush in a straight line across the workpiece. Line 52 was drawn during a 35 second interval pulling nozzle 32 across the workpiece in a straight line. Despite the fact that line 52 was drawn at even slower speed than line 50, line 52 presents fine line detail with significantly thinner width and significantly less overspray. As may be appreciated, artists making use of an attachment 10 under the present invention and having an ability to improve fine line detail as represented in the difference between lines 50 and 52 will appreciate that extremely high detail airbrush work can be accomplished at very low speeds.
FIG. 5 illustrates in section a second embodiment of the present invention, an attachment 10' including a body 18' adapted for coupling to the nose 14 of an air brush and including a nozzle 32' emitting a fine line detail spray. Body 18' defines chamber 20' with a generally conic surface 19. Chamber 20' includes a bypass outlet 26' and an annular dam 31' providing access to the inner chamber 33' of nozzle 32'. 0-ring 35 mounts within an annular groove 37 of body 18' and nozzle 32' mounts by friction at o-ring 35 and establishes a gas-tight seal thereat. The distal end of nozzle 32' defines orifice 36'. As may be appreciated, a plurality of different nozzles 32', i.e., each with a distinct shape for orifice 36', may be provided to allow ah air brush artist to quickly switch between selected orifices 36' of distinct geometric shape.
Attachment 10' receives and establishes a gas-tight seal relative to an air brush nose 14 at the conic surface 19. More particularly, and with reference to FIGS. 6A-6C, a flexible silicon tube gasket 39 surrounds nose 14 of a variety of different types of air brushes. In FIGS. 6A-6C, air brush noses 14a-14c correspond to Iwata air brushes models HP-B, HP-C, and HP-BCS, respectively. As illustrated, each of these air brushes have a different angular geometry, i.e., an angular relationship between a central axis 41 and a line 43 following generally along the corresponding nose 14. These angular relationships vary between 6.333 degrees and 10. 347 degrees as illustrated in FIGS. 6A-6C. With reference to FIG. 5, a corresponding angular relationship between conic surface 19 and central axis 41 is selected to handle this range of air brush configurations, i.e. , an 8 degree relationship between central axis 41 and conic surface 19. By placing the tubular silicon gasket 39 over an airbrush nose 14 and inserting this assembly into chamber 20 to engage the conic surface 19, attachment 10' mechanically mounts to an air brush and a gas-tight seal 22' is established. As a result, spray 24 emerging from the air brush is diverted in chamber 20', with a first portion passing annular dam 31' and entering nozzle 32' and a second portion leaving attachment 10' at the bypass orifice 26' as described above.
As with the first embodiment of the present invention, attachment 10' can be fitted with an extension hose to separate nozzle 32' and body 18' if desired.
While the present invention may be implemented according to a variety of specific designs including shape and dimension selections, applicant provides as FIG. 7 a specific set of dimensions for a specific embodiment of the present invention. It will be understood, however, that the present invention is by no means limited to such specific dimensions and that the present invention may be implemented and practiced across a variety of geometric configurations and dimensional choices for the various parts and the like. Accordingly, it will be understood that FIG. 7 is provided herewith only as a specific teaching of a preferred form of the present invention at the time of filing the present application.
The dimensional representation in FIG. 7 is at the time of filing the application believed to be a working embodiment of the present invention and provided as a teaching of the best-mode of the invention as of the time of filing. At the time of filing, however, this particular design has not yet been actually constructed and tested.
Thus, an attachment for an airbrush has been shown and described. The attachment converts a conventional airbrush into an apparatus capable of producing high detail fine line work at slow relative speed between nozzle 32 and the workpiece. The attachment focuses the atomized spray into a confined spray pattern and produces very little overspray. As a result, no need for templates or other such shielding apparatus arises as the attachment of the present invention allows direct application of media to the workpiece in fine line detail limited only by the imagination and skill of the artist.
It will be appreciated that the present invention is not restricted to the particular embodiment that has been described and illustrated, and that variations may be made therein without departing from the scope of the invention as found in the appended claims and equivalents thereof. | |
One may be wondering why nutrition in adolescence is included in the series of Nutrition-School-on-the-Air: First 1000 Days. But before any prejudice, let us first look back at a basic biology lesson: the human life cycle. In the human life cycle, adolescence (13-19 years old) is the period of transition between childhood, specifically the primary school stage (5-12 years old) and adulthood (36-55 years old). Big changes happen during this stage such as cognitive, social, emotional, sexual, and most especially, physical changes in a person. The adolescence stage in one life’s cycle is a very important window of opportunity wherein one’s poor nutritional practices and behaviors can be corrected and their nutritional status can also be improved. This is also the stage where any nutritional deficiency that the person acquires or experience can have a significant or major impact on their health or the health of their future children which is why it is considered as a critical period. Optimum nutrition in the body is needed in ensuring full growth potential is achieved.
According to the 2015 National Nutrition Survey (NNS) conducted by the Food and Nutrition Research Institute of the Department of Science and Technology (FNRI-DOST), 25% of adolescents in the country are stunted while 14.4% of them are wasted. An alarming 10.8% of adolescents were also recorded to be overweight or obese which is higher than the national prevalence rate of 8.3%. This can only mean that a significant number of adolescents in the country are not getting the optimum nutrition they need to reach their full growth potential, which also puts them in a vulnerable position from diseases and infection.
Nutrition in the adolescent stage should be given a focus on, especially for females as this is the time to prepare the body for the nutritional demands that will happen during pregnancy and lactation in later life. In making sure that the adolescent female is getting the optimum nutrition, there is a bigger possibility that during her pregnancy later in her adult life, complications during pregnancy can be avoided, including fetal/infant complications. Poor nutritional choices made during the adolescent stage can greatly impact one’s adult life which then can affect one’s pregnancy.
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Nutritional needs of women during lactation increased in response to breast milk production. Thus, “Nutrition during Lactation” was the main focus of the 9th episode of the NNC-DOH Nutrition School on the Air (NSOA) First 1,000 (F1K) Days radio magazine program on 23 February 2021.
NSOA F1K radio program co-anchor Ms. Rose Anne M. Cuyco, Nutrition Officer II of NNC-Region III, and special guest Ms. Margarita Santos-Natividad, Nutritionist-Dietitian IV of DOH Central Luzon Center for Health Development discussed the importance of nutrition during lactation to ensure that lactating mothers will keep themselves healthy and well-nourished as well as continue to provide nourishment and care to their young children through breastfeeding.
NO II Cuyco emphasized that during the first six months after delivery, the baby is fed only on breast milk, and that the baby depends on the mother’s nutrient requirements. Thus, lactating mothers are advised to eat a healthy and balanced diet during this period.
Ms. Natividad added that nutritional demands during lactation are high and can have a negative impact on both mother and baby if they are not met. The daily diet of a lactating mother will be adequate provided that food selection and preparation is appropriate.
Additional 500 calories per day is required for a lactating mother. The additional calories can be met by eating, for example, extra three (3) slices of fish per day or one cup of milk and peanut butter sandwich. Simply eating more of the usual balanced diet will enable lactating mother to meet higher energy demand while breastfeeding.
The additional protein requirements during lactation can be met by consuming protein rich foods such as one piece of egg or 25 grams of cheese or 175 grams of milk. Calcium is essential during lactation because it is required for milk production. An intake 500 ml of milk or milk products per day must be taken in addition to eating calcium rich foods, such as green leafy vegetables like malunggay and fish.
The additional food to be eaten during lactation is needed to replenish the energy that is lost through breastfeeding. Lactating mother should eat regularly to increase food intake and meet all nutritional needs by eating a variety of foods from the three food groups – the go, grow and glow foods.
Read more: Nutrition School on the Air F1K Radio Program Featured Nutrition during Lactation
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In culmination of the weeklong celebration of the 2020 National Human Rights Consciousness Week with the theme “Karapatan at Pagbangon sa Lahat ng Panahon: Recover Better – Stand Up for Human Rights,” the Commission on Human Rights Regional Office 3 (CHR-RO3) held a Virtual Awarding Ceremonies for CHR-RO3 Allies and Partners on 9 December 2020.
During the ceremony, the National Nutrition Council-Region III (NNC-Region III) was presented with the Human Rights Champion Award, along with other partner agencies and organizations to recognize their continuous commitment in upholding human rights, especially during the COVID-19 pandemic and after the onslaught of numerous natural disasters.
During the ceremony, the partner agencies and organizations recognized as Human Rights Champion, were each given the chance to express their gratitude and commitment to the agency’s advocacy. Ms. Antonette Gail Garcia, Nutrition Officer I of NNC-Region III, who attended on behalf of Regional Nutrition Program Coordinator Ana Maria B. Rosaldo, congratulated CHR-RO3 for the successful celebration of the Human Rights Awareness Week for the year 2020 and thanked the agency for the invitation and for presenting such “prestigious” award to NNC-Region III. NNC-Region III expressed its utmost support to the endeavor, advocacies, and future undertakings of the CHR-RO3. NNC-Region III also reiterated its role in championing for human rights linked in its very purpose and goal.
“The NNC which main goal is improving the nutritional status in the country and ultimately end malnutrition, continuously advocates for the protection of human rights, especially since the right to adequate food and nutrition, and freedom from malnutrition is a basic human right.”
Read more: National Nutrition Council-Region III Recognized as Human Rights Champion
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The National Nutrition Council Region III, an active member of the Regional Sub-Committee for the Welfare of Children (RSCWC) Central Luzon joined the LCPCng Gumagalaw Online Caravan for the Provinces of Nueva Ecija and Pampanga on 24 November 2020 via zoom teleconference where the agency shared the nutrition situation of children in Central Luzon. Other members of the RSCWC such as the Department of Health, Philippine National Police, Department of Social Welfare and Development, Department of Education also shared the situation of children in Central Luzon of their respective sector.
The online caravan was part of the month-long celebration of the 28th National Children’s Month celebration for the year of 2020, with the theme “Sama-samang Itaguyod ang Karapatan ng mga Bata sa Panahon ng Pandemya”. In ensuring the functionality of the Local Councils for the Protection of Children (LCPC), the caravan aimed to identify issues and concerns on children’s welfare and rights during COVID-19 pandemic, and to strengthen collaboration between and among LCPC key actors and RSCWC Members.
NNC-Region III presented the nutrition situation of children in the provinces of Nueva Ecija and Pampanga based on the results of the 2019 Expanded National Nutrition Survey (ENNS) of the Department of Science and Technology- Food and Nutrition Research Institute (FNRI-DOST).
Read more: NNC-Region III Joins the LCPCng Gumagalaw Online Caravan
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Every year, the NNC and the DOH enjoin everyone to support the celebration of national Goiter Awareness’ Week, with the theme “Goiter Sugpuin, Isip Patalinuhin, Iodized Salt Gamitin!” This is to create greater awareness among Filipinos to consume iodine-rich foods to prevent not only goiter but all types of Iodine-Deficiency Disorders (IDD).
What is Iodine or “yodo”? Why is it important? Iodine is an essential nutrient needed by the body in minute amount. It is an integral part of the thyroid hormone responsible for the regulation of body temperature, metabolic rate, reproduction, growth, nerve, and muscle function.
Iodine is an essential micronutrient for humans. Just like other minerals, it is needed by the body to regulate body temperature, metabolic rate, reproduction, growth, nerve and muscle function. Inadequate intake of iodine has a wide range of serious risks such as mental retardation, reduced IQ points, deaf-mutism, and dwarfism; goiter or enlargement of the thyroid gland, miscarriage and giving birth to abnormal babies. Based on results of studies, an average of 13.5 IQ points is lost due to deficiency in iodine. Both children and adults are affected by IDD.
Bakit inilalagay sa asin?
One cost-effective strategy to address IDD is the addition of iodine or potassium iodate to salt. One of the many ways to ensure iodine supply in your body is through the use of iodized salt in preparing meals. Iodized salt is salt fortified with iodine at levels above the natural state. It is food grade salt, fit for human consumption, and contains the prescribed level of iodine. Iodized salt is just like ordinary salt used to season and make food taste good. It does not make food taste bad or bitter. Iodized salt is not always fine salt o “pinong asin”. It is any salt, whether rock (coarse or “magaspang”), fine or those available in the market using the takal system. Just like any other food, iodized salt can be bought in groceries, supermarkets, sari-sari stores and even in health centers nationwide.
This salt fortification strategy was made initially in Central Luzon, particularly in the Province of Bulacan using an iodizing machine although it was short-lived because of the cost of maintenance of the machine. In Olongapo City, the Barangay Nutrition Scholars of Barangay Sta. Rita had their salt iodized manually through the initiative of a non-government organization under the Urban Basic Services Program in the city. Another one was in Barangay Mabayuan, Olongapo City which was initiated and initially funded by barangay officials. Meanwhile, a barangay cooperative or Samahan ng mga Magsasaka at Mangingisda in Barangay Libaba, Palauig, Zambales had engaged also in manual salt iodization in 2015. The product is labelled as “Bagong Sikat” iodized salt, which is in operation until today.
Maging matalino, mag-iodized salt tayo!
Author: NO III Angelita M. Pasos 29 January 2021
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2020: A challenging yet fulfilling year despite the COVID-19 pandemic in the country. This year marks the busiest days of the 5-member RO3 Team of the National Nutrition Council (NNC)-Region III as the team embraced the challenge of nutrition advocacy with elected Local Chief Executives (LCEs) who Chairs the provincial, city and municipal nutrition committees on the WHYs of investing in nutrition and in integrating nutrition in their Local Development Plans and Annual Investment Programs (AIPs).
There are seven (7) compelling reasons on why Local Government Units (LGUs) need to put high priority to nutrition and make significant investments in nutrition. These are as follow: 1) Alarming nutrition problem; 2) Embarrassment for a middle-income country; 3) Cost of malnutrition to the economy; 4) Negative impact of malnutrition on development; 5) Returns to investment in nutrition very high; 6) Intervention to address malnutrition exists; and 7) Nutrition is a child right and an integral part of the United Nation (UN) Sustainable Development Goals (SDGs) and the Philippine Development Plan (PDP). Supportive policies for LGUs to invest in nutrition exist.
The “very important persons” we are targeting in this endeavor are the undernourished children, pregnant and lactating women, and the nutritionally at-risks families in Central Luzon. The LCEs and the members of the local nutrition committees up to the barangay level are also considered our “very important partners” in ensuring the integration of nutrition in their 2021 AIP, for the continuing effort to secure the outcome targets of the Philippine Plan of Action for Nutrition (PPAN) 2017 – 2022 in the current and post-pandemic situation. There are 32 priority provinces identified under the Human Development and Poverty Reduction Cluster (HDPRC) in the country.
The national team of planning facilitators from the Nutrition Policy and Planning Division (NPPD), and the PHL 04 team conducted a series of meetings with the NNC-Region III team and members of the regional technical working group of the Regional Nutrition Committee of Central Luzon to form as regional team of planning facilitators (RTPF).
The Province of Nueva Ecija led by Dr. Josefina J. Garcia, as the acting Provincial Nutrition Action Officer and Ms. Iluminada R. de Guzman, District Nutrition Program Coordinator, together with the NNC-Region III Team organized four (4) batches of NNC online workshops for the Province of Nueva Ecija from August to September 2020 as follow: First batch – August 17-19; second batch – August 25-27; third batch – September 8-10; and fourth batch – September 16-18, 2020. All four (4) batches were participated by the provincial/city/ municipal nutrition action officers, health officers, planning and development officers and budget officers, and few staff from the nutrition office.
A three-day workshop to integrate nutrition in the AIP was adapted for delivery through online platforms as a mitigating measure following the suspension and limitations in the conduct of mass gatherings such as workshops and trainings due to the COVID-19 pandemic experienced across the country. | https://nnc.gov.ph/regional-offices/luzon/region-iii-central-luzon |
We have a return guest Jason Ten-Pow. Jason was on our podcast last year September and he has returned. Jason is the son of immigrants, moved to Canada with his family when he was seven years old. His passion for customer experience was sparked as a teenager while working behind the meat counter of a carnival-themed grocery store in Toronto, Ontario. From there, Ten-Pow co-ran a niche computer technology company, Visionary Enterprises, that built and installed computers and networks.
This venture taught him the basics of running a business and his commitment to customer service sparked the confidence to found ONR, his CX consulting firm in 2001. As the founder and president of ONR, Ten-Pow has expanded his lifelong passion for creating unbreakable customer relationships into an organization with more than 20 years of experience helping renowned brands evolve their customer success stories.
Questions
- What is Blockchain? what are you talking about? So, could you share with us a little bit about what Blockchain is and how that even can impact customer experience?
- Could you give me in real life terms like, I’m a business; let’s say, for example, I own a retail outlet, how does Blockchain affect me, I’m selling stuff online, I have a retail store where customers can come in face to face. What does that mean for my customers?
- Who do you see adopting Blockchain in terms of customer?
- Could you expound for us as it relates to data transparency and consumer loyalty?
- Could give them maybe one or two CX tips that you think will allow them to really connect with their customers, build better and stronger and deeper relationships. What would those two tips be for 2022?
- Could you share with our listeners, what’s the one thing that’s going on in your life right now that you’re really excited about? Either something you’re working on to develop yourself or your people.
- Where can listeners find you online?
- Do you have a quote or a saying that during times of adversity or challenge, you’ll tend to revert to this quote, it kind of helps to get you back on track if for any reason you get derailed? Do you have one of those?
Highlights
What is Blockchain?
Me: So, we’re having Jason back on our podcast. As I mentioned, he was here with us last year September talking about his book Unbreakable: A proven process for building unbreakable relationships with customers. And today he’s here to share with us a little bit about his release that was released earlier this month. The title of the article was Wider Blockchain Adoption Will Impact Customer Experience. And so, my question to you Jason is for those persons that are listening to me are probably saying to themselves, what is Blockchain? what are you talking about? So, could you share with us a little bit about what Blockchain is and how that even can impact customer experience?
Jason stated that Blockchain can be many things, but at its core, it’s the ability of information to be transferred either by the customer or by a product. So, information is tagged and carried along a pathway that can be picked up and shared, but it’s also a very secure way of sharing information. And at its core, the value or the benefit for businesses is that it allows them to acquire a lot more information about their customers and more holistic information about the customer. And for the customer, the benefit is they can have a much better understanding of the product itself, where it was created and where it was manufactured and how it ended up in their hands.
So, Blockchain is really about a safe way of transmitting information back and forth amongst various sources. And the benefit for CX is that it allows businesses to have to acquire much deeper knowledge about its customers. And for the customer, it allows the customer to understand the product they’re purchasing, and where it came from in a much more deep and meaningful way.
And this is exceptionally important today because, well, you think of movements such as ESG, which is Environmental Impact, Social Responsibility and Governance, which is very important considerations for customers that are purchasing a product, it’s good to have an understanding of who’s manufacturing, where this product is from, and if it was manufactured in an environmentally, with minimally environmental impact, and in a socially responsible manner. And so, these are very important bits of information that are being transformed.
How Does Blockchain Affect a Retail Outlet Owner?
Me: All right, now, you kind of gave us the book definition just now of Blockchain. Could you give me in real life terms like, I’m a business; let’s say, for example, I own a retail outlet, how does Blockchain affect me, I’m selling stuff online, I have a retail store where customers can come in face to face. What does that mean for my customers?
Jason stated that what that means for your customers is that you have a lot more information about your customers when they make a purchase. And you’d mean, not just simply their transaction information, but you can have depending on what’s in that Blockchain, you can have much deeper information like their age, if they share that with you, like a whole host of very important demographic information that is now connected to the actual purchase of the product, which allows you to know the customer in a much deeper way, in a much easier manner than you’ve ever been able to before.
Me: Now, what are some of the industries that you see adopting this new method of payment?
Jason stated that it’s funny, he thinks any industry that is transacting online, this will be huge for, financial institutions are going to be right on top of this, retailers are going to really care about this. Why? Because it’s an easier way to acquire knowledge, and to know your customer. Now, for example, the types of customers that will care about this, especially customers that are trying to be socially responsible, making sure that their products that they’re purchasing are having minimal negative impact on the environment, or that the company that’s building this product is being inclusive in their hiring practices, all this type of information can be shared across this Blockchain.
And so, at the end of the day, that’s the overall sort of long-term benefit. Now, we’re right in the infancy of this new technology so that’s what’s very exciting. But at the same time, we’re seeing a lot of changes in how customers make decisions, where price used to be the primary drivers, and even for companies, revenue used to be their sort of main goal that they wanted to achieve. Now you’re seeing much wider, sort of the range of metrics that companies measure themselves against for success, including things like environmental impact, social responsibility, and governance, which the short term for that is ESG, which is a really hot topic right now, because customers are very interested in understanding the impact their products are having, both socially and on the environment.
Customers Who Are Adopting Blockchain
Me: Now, in your release, you had mentioned that Blockchain Adoption has highlighted some customers, how some customers are looking for new different offerings, it’s new, and you know for example, as it relates to the different types of buying personas, if that’s the best way to describe it, you will have like the millennials, you have the Gen z’s, who do you see adopting in terms of customer base because for example, I don’t see my mom engaging in this.
Jason stated no, absolutely. This is definitely for the next generation. We know the up-and-coming generation, the young folks, they are much more cognizant of the environment, and of social justice and equality and those are the customers and the ability to have this information will really benefit, not only because they’re more likely to purchase online, but also because they care about these things when they’re making the purchase decision much more so than previous generations have.
Me: Okay, and when you say they care, is that kind of tying back into where you said, emotion will now take an even larger role in decision making all because of the fact that they’re concerned about equality, justice, fairness, those are things that are high on reasons why they buy from a company?
Jason agreed. You better believe it. You’re absolutely right. And what we’re seeing more and more today is that it’s not simply a price comparison, a lot of the products that the younger generation are purchasing, there’s deep reasons behind why they’re purchasing that’s very different than previous generations. And so, absolutely, that’s a huge selling point. And that’s just literally where the marketplace is going in the future, because at the end of the day, why do companies care about ESG? It’s because the customers are demanding that brands be socially responsible, take care of the environment and that has to be taken into account when you’re looking at whether your brand is profitable or not.
Data Transparency and Consumer Loyalty
Me: Now, you also mentioned in your article that there in this whole process, it’s important for the companies to adhere to industry regulation and improve supply chain management and there are three things that you touched on industry regulation, data transparency, and consumer loyalty. Could you expound for us as it relates to data transparency and consumer loyalty?
Jason shared that this is where you intersect a lot of different new trends that we’re seeing. So for example, if you want to be considered environmentally friendly, the government has set up regulations and standards of which to measure your level of environmental impact the company’s having, and in the USA, it’s now starting to roll out and become more adhered to. However, other standards around for example, social responsibilities really haven’t been set. So, how you measure a brand’s level of social responsibility is really up in the air.
And so, right now you’re having different ways of measuring it. But what is going to happen eventually, is that there’s going to be a standardized way of measuring it and this is where it comes back to customer loyalty. Because if these customers care about these things, they’ll be looking at these indicators to understand how the brand they want to purchase from measures up across these very important dimensions.
Me: I get you. So, it’s all connected. And then the general supply chain, how does that tie back in?
Jason stated that supply chain exactly, where are your products coming from? Is it being manufactured in a place that is not setting socially responsible markers for how they treat employees, there’s in terms of wages, in terms of environmental protection in all of these different areas. So in the past, a company could afford to just measure where they’re going to manufacture a product simply on which is the cheapest location – that is going to change as well. Because if that information becomes freely available, customers will be thinking, “You know what, I don’t want to purchase this shirt that’s made in this part of the world where they’re using child labour. I would prefer to pay a few dollars more to purchase it from a brand that’s socially responsible.” Does that make sense?
Me: Yes, it does, it totally does. But it also, I think, will require a lot of research on the part of the consumer or the way how the Blockchain system is set up now, they will be able to delve and capture that information readily when they’re making the purchase.
Jason stated that that’s the future and that’s the sort of Holy Grail is to be able to look at the product, scan this code, be able to understand exactly where all this information about the product and it’s all at your fingertips. So, the customer can make a much more informed decision than they ever have been able to do before.
Me: Over the years I’ve definitely seen customer experience evolve, at one point, if you look back at how customers made decisions before, it was heavily driven by what the organization told them, especially before the age of the internet where you could do your own research. And it’s like the tables have totally turned Jason where I mean, the ball is fully and even more so as you mentioned, this new technology, this new way of decision making, as we go forward, it’s even more in the court of the customer, because the customers are given so much more ammunition now and they should be, because at the end of the day, they’re the ones that are opening their wallets, and spending to create these astronomical profits for these organizations globally. So why not put the decision-making capability in their hands, so they can really make a choice for the product or service that they want to purchase holistically.
Jason agreed, absolutely. And what it’s going to put a lot of pressure on companies to really deliver a bespoke customer experience that’s unique to the needs of every customer, so it will be slightly different. Why? Because that’s what customers are going to demand, “You’re going to care about the things I care about, right? And you’re going to tell me exactly how you are manufacturing these things, and you’re going to deliver a shopping experience the way I want a shopping experience to be delivered.”
And what that allows companies is to actually be able to build a more customized experience, because they will have that information readily available. And so, the transparency that will be possible will benefit both the brands if they take advantage of it. But it’s definitely going to put a lot more power in the hands of the customer especially because it’s exactly you said, knowledge is power and the more knowledge the customer has, the more informed decision they can make.
CX Tips That Will Allow Businesses to Connect and Build Better and Deeper Relationships with Their Customers
Me: Now, Jason, I know the first quarter of the year has passed, but we’re in the beginning of the second quarter. But could you give our listeners maybe one or two CX tips that you think, outside of this new technology, because as you mentioned, it’s still in its infancy stages, but let’s say where they are currently in their business, they’re just not there yet clearly. But they’re looking to ensure that they employ maybe the best, if you could give them maybe one or two tips that you think will allow them to really connect with their customers, build better and stronger and deeper relationships, what would those two tips be for 2022?
Jason stated that 2022 is the year of Listening and here’s why, the marketplace has changed coming out of the pandemic, customers have different expectations for shopping and purchasing experiences and it’s different than ever before. And the customers are really going to dictate how they want to shop and how they’re going to purchase moving forward. There’s a lot of companies out there that are just thinking to themselves, “Oh, I’m just going to hold out until we get back to how things were before.” And the truth of the matter is, things are not going to go back to how they were before, things have changed, and they are different.
And unless you start listening more closely to your customers in every interaction, whether you’re a restaurant listening to your patrons and their feedback in terms of what they want, and how they want it delivered, to major brands who are selling investments in ESG, stocks and ETFs, all and everywhere in between.
If you’re not listening to your customers and understanding how their wants, needs and desires have evolved, you are going to be left behind and that is really his encouragement to companies coming out at the pandemic to start listening to your customers more closely than you ever have before. Because their opinions and their values have changed.
Me: I’ve heard some organizations say that they think customers are way more sensitive, they complain about the least little thing since the pandemic, what are your thoughts on organizations that view their customer feedback as customers being too sensitive and it’s almost like they’re not open to being flexible or being adaptable to take the feedback that the customer is giving them.
Jason shared that it’s funny, the brands that they work with that they hear this from are brands that are stuck in the past. And he often hears, “This is the way we’ve always done it.”
And so, those are the brands that that may have been the way you’ve done it in the past, but if you don’t change your focus from short term financial, quarter over quarter goals, to a longer-term view of what success really means beyond just simply your short-term financial metrics, you’re going to be in big trouble. And this is really the tug of war that’s going on, it’s the old sort of dynamic of, okay, near term profits at any cost versus taking a longer view of your brand, and your brand’s health. And let’s be honest, public corporations are the ones that have been most guilty of that and those are the ones that he believes are going to be at biggest risk if they don’t adapt themselves to the evolving customer.
What Jason is Really Excited About Now!
Jason shared that they’re working to develop a better understanding of the impact ESG is going to have on decision making over the next 12 to 24 months. So, over that time, they’ll be speaking to investors and customers, as well as business leaders to understand who is driving from an organizational point of view, interest in ESG. And what measures companies are taking to implement tactics that address customers ESG concerns? And how important is ESG becoming in the decision making of customers? So, those are the three angles they’re looking at. And so, it’s going to be quite interesting, because he thinks what we’re talking about Blockchain is just one aspect of the bigger evolution that’s taking place.
And so, it’s going to be interesting to see how these things evolve together, because there’s still many that think that this is a fad, it’s going away. They’re betting against that, they’re saying no, these things are here to stay, and these are the changes in evolution and how business is being conducted. So, it’ll be interesting to see what business leaders are thinking in terms of these new and various approaches to thinking about the company’s success.
Where Can We Find Jason Online
LinkedIn – Jason Ten-Pow
Website – www.onrcx.com
Quote or Saying that During Times of Adversity Jason Uses
When asked about a quote or saying that he tends to revert to, Jason stated yes. First thing is, “Stop” whatever you’re doing stop, take a deep breath. If you have a big problem, the first thing you want to do is you want to cut that problem into smaller chunks that are manageable, that are solvable, and then create a pathway ahead, don’t just see a problem and dive in and try to solve it. Because that’s the biggest issue that companies and that’s why they hit the wall, “Oh, I want to improve customer experience. Great. I want this metric up 10%. Let’s throw money at the wall and see what sticks.” No, that’s never the right approach. You have to take a very strategic approach to these types of problems and these types of challenges, and you have to always have a plan. So, make sure you stop and take the time to plan.
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Links
- Unbreakable: A proven process for building unbreakable relationships with customers by Jason Ten-Pow
The ABC’s of a Fantastic Customer Experience
Do you want to pivot your online customer experience and build loyalty – get a copy of “The ABC’s of a Fantastic Customer Experience.”
The ABC’s of a Fantastic Customer Experience provides 26 easy to follow steps and techniques that helps your business to achieve success and build brand loyalty.
This Guide to Limitless, Happy and Loyal Customers will help you to strengthen your service delivery, enhance your knowledge and appreciation of the customer experience and provide tips and practical strategies that you can start implementing immediately!
This book will develop your customer service skills and sharpen your attention to detail when serving others.
Master your customer experience and develop those knock your socks off techniques that will lead to lifetime customers. Your customers will only want to work with your business and it will be your brand differentiator. It will lead to recruiters to seek you out by providing practical examples on how to deliver a winning customer service experience! | https://yaniquegrant.com/episode-167-blockchain-adoption-and-its-impact-on-customer-experience/ |
How have artists resisted the global surge of far-right movements and authoritarian regimes through music and music-making? How has music been mobilized against fascism and fascist tendencies in societies in the 20thand 21st centuries?
The aim of this one-day symposium is to discuss the role of music in antifascist action in response to the recent surge in ultranationalist, xenophobic, discriminatory, and authoritarian regimes throughout the world, which threaten not only democratic governance systems and basic human rights, but also ecological sustainability. Antifascist resistance has long connected people across borders and time periods yet has also manifested in distinctive ways in specific socio-political settings. Through discussions of historical and contemporary examples, participants will initiate dialogue that analyzes a diversity of musical activity related to antifascism, with the aim of illuminating paths for engaging in anti-oppressive politics through music and sound.
We welcome paper proposals and presentations dealing with any genre of music, or any form of musical or sonic expression directly related to antifascist action. We encourage submissions from scholars, artists, activists, curators, journalists, and librarians, among others, and especially proposals from members of underrepresented and marginalized groups.
Paper topics may include but are not limited to the following:
- uses of music and sound in historical antifascist movements
- production, circulation, preservation, and revival of protest music, union and revolutionary songs
- uses of music and sound in contemporary social justice movements (e.g., Black Lives Matter, Idle No More, Arab Spring, Occupy Wall Street, LGBTQ2+ pride events, etc.)
- media promotion or censorship of antifascist music and messaging
- musicians’ involvement in decolonization and liberation movements
- censorship and repression of music by fascist regimes
- co-opting of music and forced compliance of musicians by fascist regimes
- music as agitprop
- weaponization of music and sound; music and sound as defense
- role of music and sound in movement-building, union organizing, and other forms of social solidarity
- commemorative musical works for victims of oppressive regimes
We are pleased to announce that our keynote speaker will be Dr. Federico Spinetti, Professor of Ethnomusicology at the University of Cologne, Germany. Dr. Spinetti’s primary research interests include music and politics, music and memory, auditory cultures and the built environment. He is currently undertaking research into musical memorializations of the WWII antifascist resistance in contemporary Italy. He served as co-editor (with Monika E. Schoop and Ana Hofman) of a 2020 Special Issue of Popular Music and Society, titled “Music and the Politics of Memory: Resounding Antifascism across Borders.” The volume includes his chapter: “Punk Rock on the Gothic Line: Resounding the World War II Antifascist Resistenza in Contemporary Italy.” An active filmmaker, Dr. Spinetti has also directed several documentaries, including Zurkhaneh – The House of Strength: Music and Martial Arts of Iran (2011) and The Enemy – A Partisan Hymnbook (2015).
INSTRUCTIONS FOR SUBMISSION:
Please send the following MS-Word (or PDF) documents to [email protected] by Friday, October 15th, 2021:
- a 250-word abstract and proposed title, without any information identifying the author (followed by specifications of any technical equipment and/or support that may be required).
- a 150-word biographical paragraph, followed by author-contact information and your proposed paper title.
Our program committee will review all submitted proposals by Monday, November 1st, and will reply to all prospective contributors by Monday, November 15th. | https://improvisationinstitute.ca/event/music-and-antifascism-conference-reflections-on-the-past-and-possibilities-in-the-present/ |
Today, Leutkirch is a small town in the Western Allgäu region of Germany, far from big population centers.
Yet recent work by archaeologists and geographers from the ResourceCultures collaborative research center at the University of Tübingen shows that more than three thousand years ago, the Western Allgäu was home to a relatively large population.
The region has a damp climate, long hard winters, and gravelly soil left by retreating glaciers in the last Ice Age. It is higher up than neighboring areas to the north and west, which are warmer and have better soils. To its prehistoric inhabitants however, its location on major trade routes seems to have outweighed the disadvantages. The results of the ResourceCultures excavations have been published in the latest edition of Archäologische Ausgrabungen in Baden-Württemberg, the state heritage authority’s yearbook.
The researchers have been carrying out excavations near Leutkirch since 2017. They revealed a hilltop fortified in the Bronze Age; burial mounds mark a corresponding cemetery, and further Bronze Age settlements were located in the valley below. Soil analyses show high levels of charcoal and widespread erosion in this period, indicating that forests were cut down to grow food for a substantial population around 3,500 BCE.
“A settlement like this did not exist in a vacuum,” says Benjamin Höpfer, a doctoral candidate in the subproject ‘Favor – Disfavor? Development of Resources in Marginal Areas.’ “This indicates a pattern of settlements, and it changes our whole picture of the region at that time. Far from being empty, the prehistoric Allgäu region may have had a village or a farm every five kilometers or so.”
The growing value of long-distance trade
Why did Bronze Age people opt to live in a cold, wet place on stony ground? The advantage lay in the region’s location between the Alps and Lake Constance and the Danube, Iller and Rhine Rivers – all of which were important trade routes. Benjamin Höpfer says the Western Allgäu was a bridge between regions in a wider, pan-European prehistoric economy. “The Alps were not just an obstacle to be overcome – they were a trade hub in themselves,” Höpfer explains.
“Long-distance trade became more and more important; the paths ran along river valleys, and hills were used for orientation,” Höpfer explains. On the rivers and lakes of the regions surrounding the Alps, there are many archaeological sites at which imported wares show that goods such as copper from the eastern Alps and tin from Cornwall were brought here to make bronze alloys. “Copper, tin, amber – these and many other things were traded along routes that pass through here,” Höpfer says. Now it appears that the trade helped attract a permanent population of farmers.
At the same time, the Bronze Age was “an age of huge technical innovation,” Höpfer points out. Metalwork provided new tools. The bronze sickle enabled farmers to harvest not just grain, but also straw and hay. These were fed to animals, which provided milk, meat, hides and wool. Crossbreeding led to new, tougher crop strains, and animals which could adapt to harsher conditions. All this made living in the Western Allgäu more attractive. To be on a major trade route was worth the hard work they invested to make the landscape more livable. “This changes our image of prehistoric people as not just passively accepting what nature provided,” Höpfer says.
Until now, the Western Allgäu has mostly been a blank space on the archaeological map. This was largely because of its remoteness from Universities and Cultural Heritage authorities. This has meant less prospecting, and that building sites – which often reveal archaeological remains – have been subject to less supervision. Especially with the booming construction and development sectors, there is probably a lot still to discover in the region. “We have only scratched the surface,” says Benjamin Höpfer. | https://www.heritagedaily.com/2020/09/the-european-economic-area-dates-back-to-the-bronze-age/134996 |
In the beginning, we were all bushcrafters, we had to be if we wanted to survive. Today, in the modern world, we don’t need all the skills our ancestors had. Still, learning a few bushcraft basics can help us deal with everyday challenges.
The art of bushcraft: Definition
Bushcraft is a practice of survival in the wilderness by using only what you can find in nature with little to none tools. We will call it art because in order to master it you will need a certain amount of creativity. What you will also need is patience since it is no small field.
While your bushcraft knowledge grows, you will also become more self-aware, your confidence will grow and you will easily adapt to changes.
Bushcraft basics: Skills
The whole bushcraft philosophy revolves around the idea of using only what you can find in your environment, so the more bushcraft techniques you learn the less equipment you’ll be needing.
Bushcraft skills can be divided into a few main categories.
- Starting a fire
- Purifying water
- Food gathering
- Building a shelter
Starting a fire
You should begin with learning how to start a fire, this is the essential, probably most important thing you’ll need to know. Start with gathering wood and tinder, and build a small fire. In the beginning, you’ll be needing matches or lighter, but as your knowledge grows you could try building fire-lighting tools such as a fire plough or fire saw. Fire building is a skill that you can always improve, and you should, for it is really in the center of bushcrafting.
For lighting a fire you can also use a sun glass, it can be: magnifying glass, camera lens, glasses, or some similar object. This can be a difficult method since it requires direct sunlight. Some other ways to start a fire are by using a ferro rod, or flint and steel.
READ: How to start a fire in the wild
Purifying water
There are several methods for collecting and purifying water, some of them are boiling, filtering, and distilling. CDC recommends filtering and then boiling water to decontaminate it. For boiling use a metal container or if you don’t have one you can use a wood container and hot stones.
When you start learning, our advice is to practice it with bottled water. There are a lot of diseases you can catch from drinking contaminated water. Always put your safety first! There are iodine or chlorine dioxide tablets/drops for sterilizing water, but you shouldn’t use them for longer than 3 weeks. Later on, as you get more crafty, you can try building a solar still.
READ: How to purify water in the wild
READ: The best backpacking water filters
Food gathering
As a bushcraft novice, you probably won’t be able to hunt animals for food. You can start by learning about local edible plants and mushrooms, but be careful, there are many poisonous species, especially mushrooms.
You can also find great bushcraft cookbooks, and discover some delicious recipes. There is nothing like breakfast made on fire outside your tent!
Building a shelter
When picking a place where you will settle you should consider a few things. The spot should be near water and woods for fire, but you’ll also need to be protected from wind and flooding. Depending on your needs there are different types of shelters you can build. When the nights are warm a “lean-to” shelter will do fine, but when it’s colder or if there’s a storm outside a debris hut or an A-frame is a better choice.
If you decide to spend some time in the wilderness you will need a lot of energy, so a good night’s sleep is essential. Experienced ones would frown on modern tents, but they are great for beginners. Nowadays, with just a little digging, you can get yourself a great one on a budget.
Bushcraft basics: Tools
We already mentioned that as you keep getting better at your bushcraft skills you will need less equipment, but even the most skilled ones need a good knife. So what are some basic things you will need when starting?
Knife
A knife is a part of a standard bushcraft kit. It is used for cutting smaller branches, wood carvings, skinning game, and prepping food. There is a whole range of techniques you can practice that will help you end your tasks easier. Since there are many different types of knives, we recommend starting with something simple like a fixed blade knife.
Conserving your blade is important, keep it sharp and oiled (cooking oil should do the trick). Knives can be made out of stainless steel, but we will recommend using high carbon steel since they stay sharp longer. Always remember, a dull blade is more dangerous than a sharp one.
Axe or hatchet
For preparing firewood and building shelter you will need a more serious tool. Bushcraft axe has a whole range of applications, as you can use it for wood chopping, log splitting or digging. To expand its lifespan, use wedges regularly.
Instead of an axe you can use a saw if you plan on doing a lot of cutting wood. It will prove much more efficient than an axe (especially in winter) and it is also safer for use. Bow saw is great when camping, but when on-trail we recommend foldable saw.
Backpack
Finally, you will need a backpack for carrying all your bushcraft tools and gear. When choosing one, ensure it is comfortable and easy to carry. The material needs to be durable and waterproof, and the backpack should have multiple compartments. This will help you to better organize your stuff.
Get out there!
Now that we covered bushcraft basics you can start your own projects. Begin small, don’t try learning it all at once. Remember to have fun along the way, and in no time you’ll be ready to get out there. | https://zelendom.co/bushcraft-definition-basics/ |
Amanda Filipacchi and Katherine Heiny came to The Center for a discussion on the process of writing, from how to discipline yourself as a writer and methods of routine to where to go for inspiration. They also discussed their latest books, The Unfortunate Importance of Beauty (Filipacchi) and Single, Carefree, Mellow (Heiny).
About The Unfortunate Importance of Beauty
In the heart of New York City, a group of artistic friends struggles with society's standards of beauty. At the center are Barb and Lily, two women at opposite ends of the beauty spectrum, but with the same problem: each fears she will never find a love that can overcome her looks. Barb, a stunningly beautiful costume designer, makes herself ugly in hopes of finding true love. Meanwhile, her friend Lily, a brilliantly talented but plain-looking musician, goes to fantastic lengths to attract the man who has rejected her--with results that are as touching as they are transformative.
To complicate matters, Barb and Lily discover they may have a murderer in their midst, that Barb's calm disposition is more dangerously provocative than her beauty ever was, and that Lily's musical talents are more powerful than anyone could have imagined. Part literary whodunit, part surrealist farce, The Unfortunate Importance of Beauty serves as a smart, modern-day fairy tale. With biting wit and offbeat charm, Filipacchi illuminates the labyrinthine relationship between beauty, desire, and identity, asking at every turn: what does it truly mean to allow oneself to be seen?
Click here to watch a book trialer on T: The New York Times Style Magazine
In “Cranberry Relish” Josie’s ex—a man she met on Facebook—has a new girlfriend he found on Twitter. In “Blue Heron Bridge” Nina is more worried that the Presbyterian minister living in her garage will hear her kids swearing than about his finding out that she’s sleeping with her running partner. And in “The Rhett Butlers” a teenager loses her virginity to her history teacher and then outgrows him.
In snappy, glittering prose that is both utterly hilarious and achingly poignant, Katherine Heiny chronicles the ways in which we are unfaithful to each other, both willfully and unwittingly. Maya, who appears in the title story and again in various states of love, forms the spine of this linked collection, and shows us through her moments of pleasure, loss, deceit, and kindness just how fickle the human heart can be.
Described by The New York Times as a “lovely comic surrealist,” Amanda Filipacchi is the author of three previous novels: Nude Men, Vapor, and Love Creeps. Her writing has appeared in The New York Times, The New Yorker, The Wall Street Journal and The Atlantic, and has been included in Best American Humor and other anthologies. She holds an MFA from Columbia University. Born in France, she has lived in New York City for the past thirty years.
Katherine Heiny's fiction has been published in The New Yorker, Ploughshares, Narrative, Glimmer Train, and many other places. She lives in Washington, D.C., with her husband and children. This is her first book. | http://centerforfiction.org/calendar/in-conversation-amanda-filipacchi-and-katherine-heiny/ |
First published in Coaching World http://www.coachfederation.org International Coaching Federation (ICF) – 18 January 2018
How many times have you found yourself in a situation described by your client that immediately triggers something you may have experienced? And how many times have you been tempted to share your experience or tell them how you felt in that similar situation? Perhaps they may have even asked you, human to human, what your experience has been or how you feel about subject x and whether situation y has happened to you as well.
How can you know whether you are not crossing the very fine line to an extent where you are no longer serving your client? And how can you manage the delicate, sensitive part of our human nature to connect with others, especially when we are supposed to work in an equal partnership with our clients co-creating along this journey?
In coaching people through career transitions, I am often asked how I made my transition. My response is that “I will share my journey; however, let’s have a look first at what it would look like for you.” I do this to avoid unintentionally influencing them, and I use a framework to show on paper what it would look like for them.
Once we feel we have exhausted how they see the issue, only then do I share my journey. I make sure I share the context and circumstances of it. I also make a point of sharing the journey of at least other two people, and then I ask the client how these examples add to what we have discussed about them. So, at this point, I direct the focus back to them and support a process of making sense and adding value to their situation with the objective of expanding on existing perspectives and thinking about aspects that the client may not have previously thought about.
It may be more helpful not to think of self-disclosure in binary terms—you either self-disclose or you don’t—but rather to think of the circumstances, which include the timing. If we looked at the purposes of self-disclosure, it could include modeling of behavior. It’s in a way working with one’s vulnerability, being less guarded, and giving the message that it’s OK. In the example above, this would be sharing things that didn’t work out. As a consequence, you have rapport, and trust may increase in the coaching relationship. In addition, it may support the client in developing new perspectives when they see how others have handled similar situations. In other cases, the mere realization of “I am not the only one” may help them with self-acceptance. In my experience, this last aspect can be very instrumental because they feel empowered to move on.
If not done carefully, self-disclosure may compromise the professional relationship by creating expectations. It may move the focus away from the client, and it may even pressure the client to do something that they may not be ready for. For example, I am very aware that one of my personal biases is to be an optimist, have a can-do attitude and believe that everything is possible. These are my beliefs, but they are not everyone’s beliefs. If I am not careful to be in tune with my client’s state and share my optimism at the wrong time, when they are still processing a negative situation, I may unintentionally alienate them simply because they are not yet ready to move forward.
Another helpful way to look at disclosure is what is referred to as intra-session disclosure, which includes aspects that are happening in the “here and the now,” or during the session. For example, you could offer, “I feel like we are going around in circles today; what is your impression?” Even though this type of sharing is around the coach’s feeling, it’s also about the process. It’s feedback, and it’s direct communication about what is happening. Following the framework provided by Marjorie Shackleton and Marion Gillie, these questions can help keep us, coaches, on track: What’s going on for me? What’s happening between us? What do I observe in my client? What part of my reaction might be useful to my client?
Overall, an effective way to self-disclose would include recognizing and acknowledging the client’s experience and asking for permission to share your own relevant experience, which must be tied back to the client’s story in a relevant way. This could be followed by a powerful question that connects the two stories, which should help the client move forward with their own process. The rationale for telling your story needs to also be explained, and you should ask the client if it created any new awareness about their own story. | https://drmariakatsarou.blog/2019/01/18/self-disclosure-in-coaching/ |
The Centers for Disease Control and Prevention (CDC) released updated recommendations for use of once-weekly isoniazid-rifapentine for 12 weeks (3HP) for treatment of latent tuberculosis (TB) infection. The updated recommendations, published in CDC’s Morbidity and Mortality Weekly Report (MMWR), support expanded use of an effective, shorter treatment regimen to reach even more people with latent TB infection, including people with HIV/AIDS.
In children and adolescents, 2-11 years old.
These updated recommendations are very good news for those with HIV and latent TB infection. Because HIV infection weakens the immune system, people with latent TB infection and HIV infection are at very high risk of developing TB disease, if not treated. Previously, CDC only recommended the 3HP regimen for treatment of latent TB infection in people with HIV who were otherwise healthy and not taking antiretroviral medication. At that time, we did not know enough about the interactions between rifapentine and certain antiviral medications. New data now show an absence of clinically significant drug interactions between once-weekly rifapentine and the antiviral medications efavirenz and raltegravir.
Healthcare providers treating patients with HIV infection and latent TB infection can find information about interactions between rifamycins (including rifapentine) and antiretrovirals in the Department of Health and Human Services’ Guidelines for the Use of Antiretroviral Agents in HIV-1-Infected Adults and Adolescents, as well as useful tables with information on other drug interactions. For those patients on antiretroviral agents, healthcare providers should consult with experts in TB and HIV management to determine the best overall treatment regimen.
The 3HP regimen is the shortest of several regimens available to treat latent TB infection today and can remove barriers to initiate and complete treatment. A shorter treatment timeframe, the option for self-administration, and reduced costs all contribute to increasing the number of people who complete treatment for latent TB infection, helping to prevent future cases of TB disease.
We all play an important role in TB prevention. We encourage clinicians and public health professionals to review and implement the updated recommendations, which include guidance on patient education and monitoring, and to visit the LTBI resource page for additional resources on latent TB infection. | https://www.hiv.gov/blog/latent-tb-infection-and-hiv-cdc-has-updated-recommendations |
According to business intelligence report on Tiotropium Bromide Hydrate market, Covid-19 pandemic will have lasting impact on industry sphere, based on which growth matrix for 2021-2026 is formulated.
The Tiotropium Bromide Hydrate market business intelligence report complies the key trends across the competitive and geographical landscape that are slated to influence the growth trajectory of the industry in the upcoming years. In addition, it addresses the issues plaguing the industry and provides insights into the possibilities that will enable business expansion in both existing and untapped markets. Moreover, it takes into account the changing industry dynamics since the onset of Covid-19 pandemic to help stakeholders make well-informed decisions.
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Key highlights from COVID-19 impact analysis:
- Implications of Covid-19 on the global economy.
- Changes in demand share and supply chain.
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- As per the report, the Tiotropium Bromide Hydrate market size is fragmented into North America, Europe, Asia-Pacific, Southeast Asia, Middle East and Africa, South America.
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Report Answers Following Questions:
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- Jinan Dexinjia Bio&Tech Co.
- Ltd
- BioChemPartner
- Beijing Mediking Biopharm Co.
- Ltd.
- Manus Aktteva Biopharma LLP
- Jinan Dexinjia Bio&Tech Co.
- Ltd.
- Jinan Aery Pharmaceutical Co
- .Ltd
- Vamsi Labs Ltd
- Caming Pharmaceutical Ltd
- Hangzhou Verychem Science And Technology Co. Ltd.
- AFINE CHEMICALS LIMITED
- Pivotal information about the goods manufactured, business profiles, market remuneration, and production trends of the market majors are encompassed in the report.
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- Inhalation Powder
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- Hospital
- Drug Store
- The study assesses the market share and predicts the growth rate during the study period for every application segment.
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- It also boasts of Porter 's five forces analysis and SWOT analysis to ascertain a new project's viability.
Reasons to purchase the research report:
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For More Details On this Report: https://www.marketstudyreport.com/reports/global-tiotropium-bromide-hydrate-market-development-strategy-pre-and-post-covid-19-by-corporate-strategy-analysis-landscape-type-application-and-leading-20-countries
Contact Us:
Corporate Sales, | https://www.altcoinbeacon.com/tiotropium-bromide-hydrate-market-growth-reportc/ |
Southern cuisine has been introduced to the world through foods such as fried chicken and barbecue.
It's a hodegepodge of influences from Native American,African and European cuisine, it's like putting together the best characteristics of influences and make it into one special dish that would really make you crave for more.
What is Southern Cuisine?
As you can see, many states make up the South, which means lots of different influences, opinions, and modern interpretations of its cuisine! In essence, Southern cooking is truly a potluck, with several cultures and backgrounds bringing different cooking styles to the Southern table!
There are dozens of places in the US to try southern cuisine. Some are more traditional than others, but all have their own unique taste and style. Here are some southern cities that are well-known for their southern cuisine.
New Orleans, Louisiana
New Orleans is known for its rich culture and cuisine. It's well known for its authentic southern cuisine that includes dishes like gumbo, jambalaya, crawfish etouffee, and shrimp Creole.
New Orleans has some great restaurants that serve authentic southern cuisine, but it can be difficult to know where to go - especially if you're visiting on your own! Some of the best restaurants in the city include Cochon Restaurant, Ralph’s On The Park, and Clancy’s.
In addition to the city’s rich culture and history, it also has a lot of entertainment options that you can enjoy while visiting. You can visit museums, watch live music performances or visit Mardi Gras celebrations. From jazz to food, this city is one you won’t forget.
Hot Chicken
Knoxville, Tennessee
Southern cuisine is a variety of regional cooking that developed in the Southern United States. It has its roots in the settlement of the Americas and the traditions brought by European immigrants, primarily from England, Ireland, Germany and France.
Knoxville is known for its southern cuisine. The city’s most famous dish is probably hot chicken which is a spicy fried chicken dish that originated in Tennessee and became popular across the South.
The city is also known for its barbecue offering a wide range of meats and sauces in addition to other traditional dishes like sweet potato pie and fried green tomatoes.
But Knoxville doesn’t only have delicious food to offer. It also boasts a rich culture, a sense of community, and plenty of activities and events to partake in. This is why many Knoxville houses for rent are seeing a tremendous increase in interest with all this city has to offer.
Red Beans and Rice
Charleston, South Carolina
The cuisine of Charleston is known for its sweet and savory dishes. The city is also home to many local farms that provide the fresh ingredients needed to make a favorite southern dish.
Charleston is known for its southern dishes like shrimp and grits, red beans and rice, fried green tomatoes, country ham biscuits with butter gravy.
Charleston is a city that has a thriving food scene. Not only does it have some of the best restaurants in the country, but it also has some great local food options.
It is not just about the food, though. It is also a city that is steeped in history and culture. It was once home to many plantations and slave communities, and you can still feel the history all around you when you visit Charleston today.
For those who are looking for an authentic southern experience, Charleston might be just what you're looking for!
These kinds of food makes me want to hop onto the next plane and eat my hearts content but since it's pandemic,traveling is such a chore,so instead of traveling I compiled some websites where you can cook Southern Cuisine at the comfort of your home. | https://www.cleabanal.com/2022/05/best-places-to-try-southern-cuisine.html |
Testimonials:
Leah clarkson, Balham labour party
"As co-facilitators, Camille and Katie provided an insightful, thought provoking and challenging workshop on white privilege. This is naturally a highly sensitive topic and area of work, but with Camille and Katie’s careful and expert guidance in creating a safe and welcoming space, we were able to explore this theme and confront some of our own value and question our own privileges. We were introduced to new concepts and new terminology, which were always fully explained; this ensured that the session was not only thought-provoking but also educational. We are now looking forward to continuing our journey of challenging white privilege and taking further steps to commit to becoming white allies and tackling racial prejudices in our community."
Lyndsay Burtonshaw, Coordinator of Make Change Happen
"Camille Barton co-designed and facilitated a White Allyship workshop for the October 2016 edition of the Make Change Happen activist training program at The University of Sussex. Camille is a highly skilled facilitator, diligent, comprehensive, uncompromising on quality, and meticulously prepared, and deemed a "role model facilitator" in participant feedback. By making an enormous range of intellectual ideas and radical academic content accessible, Camille holds space for a multitude of voices to be acknowledged and heard, creating spaces where participants have "felt uncomfortable – but I think it was done in a good, safe way, and one that has opened my eyes rather than allowed me to sit and feel good about how much I already do", and "Feeling safe and exposed at the same time". The range of resources and activities Camille facilitates reaches all learning styles, a facilitation style which is "very interactive and informative and relevant for everyday life." I am excited to be inviting Camille back for Make Change Happen February 2017 edition and look forward to building on work with her in the future."
Marta Owczarek, Wolf Whistled
"The Introduction to White Allyship, Power and Privilege workshop led by Camille at the Wolf Whistled Day of Doing #7 in January 2017 was a really important experience and we were very pleased to be able to host the session. It was a unique opportunity to confront white privilege and its consequences. What made the session very impactful, and different to a lot of discussions around this subject, was its focus on emotions and processing them. It helped the group go beyond the intellectual, rational political arguments and work on some of the deeper issues that are often obstacles to meaningful work on anti-racism and dismantling white supremacy. Wolf Whistled would recommend this session to everyone."
Clients
Work with us
If you would like to work with us and co-run a workshop, fill out the form below and we will get back to you. | https://www.thecollectiveliberationproject.com/partners/ |
Email us at info(at)facesofgiving.org. Or submit any questions, suggestions, or general feedback in the space below. We appreciate your comments or concerns and will respond to them as soon as possible. | https://www.facesofgiving.org/contact.html |
The utility model relates to an ultra-thin moisture-absorbing quick-drying antibacterial deodorant home textile pillow inner sleeve. The pillow inner sleeve is formed by weaving polyester fibers withfour-leaf-shaped sections. The specification of the four-leaf-shaped polyester fiber is 75D/72F, and the four-leaf-shaped polyester fiber is composed of composite filaments with the monofilament fineness of 1d. The polyester fiber is a polyester fiber containing silver powder and has excellent suction and discharge antibacterial performance. Compared with the prior art, the moisture-absorbing andquick-drying pillow inner sleeve is good in moisture-conducting performance, and the moisture evaporation speed of the moisture-absorbing and quick-drying pillow inner sleeve is more than five times higher than that of a conventional product. The antibacterial rate reaches 95 percent or higher. The pillow inner sleeve is light, thin and breathable, saves product cost, resists bacteria and sweat odor, reduces washing times of the product, and is environment-friendly and energy-saving. The monofilaments are 1d multifilaments and mesh structures of the fabric, so that water evaporation is enhanced, bacteria are resisted, and sweat odor generated by bacteria is reduced. | |
Uranine (sodium fluorescein, UR) has been routinely used in hydrological research to monitor surface and subsurface water flow, transport and mixing processes since the end of nineteenth century. Based on such obtained data, further conclusions can be drawn on the spread and behavior of pollutants (partly on models). Use of UR for qualitative (visual) studies of underground contamination is common, however data available on its environmental behavior (e.g., conversion, degradation or formation and fate of the transformation products, TPs) are incomplete or not readily comparable. UR observations of biodegradation are still speculative. S-metolachlor (SM) is a popular worldwide chloroacetamide herbicide, which highly correspond to the global pesticide use. It is offered on the French market as an effective multicrop herbicide against annual grasses and certain broadleaf weeds under the trade name Mercantor Gold (MG). Photodegradation contributes to the fate of SM in the aquatic environment. TPs were already found in surface and groundwater. However, further fate and assessment of the TPs was not done. Moreover, adjuvants in MG´s formula can affect the solubility, biodegradation, photolysis and sorption properties of the active compound SM. TPs can have different properties (e.g. more mobile, toxic or present at higher concentrations) that enable them to reach the environmental compartments not affected by the parent compound (PC) itself. To assess the ecological impact of pesticides, tracers, and their respective TPs on water organisms, their behavior can be investigated in laboratory screening biodegradation tests. Yet, incomplete data was available on SM, MG and UR transformation or their photo- TPs´ fate in surface and water-sediment systems. The combination of photolysis with aerobic biodegradation in order to identify persistent photo-TPs could provide new insight into the environmental behavior of the selected compounds. Therefore, principle of this thesis was to 1) identify the impact of MG´s adjuvants on the biodegradation, photolysis (Xe lamp) and sorption compared to the SM alone, 2) examine the photolysis and biodegradability of UR 3) monitor the primary elimination (photolysis) of the PCs by HPLC (-UV, -FLD) and measure the degree of mineralization by means of nonpurgeable organic carbon (NPOC) 4) elucidate the photo-TPs of SM, MG and UR by using LCMS/ MS 5) analyze biodegradability of the photo-TPs in order to determine their fate and persistence in aquatic environment 6) conduct in silico toxicity predictions (pesticides) in human (carcinogenicity, genotoxicity and mutagenicity) and eco-toxicity (microtoxicity, bioconcentration factor and toxicity in rainbow trouts). SM, MG and UR were found not readily biodegradable in Closed Bottle test (CBT), Manometric Respiratory test (MRT) and in water-sediment test (WST). Chemical analysis of photolysis samples showed higher elimination of SM in MG compared to SM alone whereas UR displayed high primary elimination rate in general. The overall low degree of mineralization indicated that abundant photo-TPs were formed. Furthermore, the photo-TPs were found not biodegradable in performed biodegradation tests. Only small degradation rates for UR could be observed in the CBT and WST. Additionally, in the MRT and WST new bio-TPs were generated from the photo-TPs of SM and SM in MG. Obtained results suggest that the MG formulation did not significantly affect the biodegradation, however it influenced the diffusion of the active substance (SM) to sediment and potentially affected the photolysis efficiency, which might result in faster formation of photo-TPs in the environment. In silico predictions showed that for many endpoints, biotransformation might lead to an increased toxicity in humans and to water organisms compared with the parent compound SM. No indications were found for UR toxicity. Still, target-oriented investigations on long term impacts of photo-TPs from UR are warranted. The present work demonstrates that a combination of laboratory tests, analytical analysis and in silico tools result in valuable information regarding environmental fate of the TPs from selected compounds. Furthermore, it was shown that photo-TPs formed in the aquatic environment should be taken into account not only the parent compound and its decay. | https://pub-data.leuphana.de/solrsearch/index/search/searchtype/all/rows/50/facetNumber_year/all/facetNumber_subject/all/start/0/subjectfq/Tracer |
As events and openings are canceled, San Diego artists and galleries face an uncertain future
The COVID-19 crisis has hit nearly every major sector of the economy, but for the local arts community, the bad news has come especially fast and swift. A statewide ban on mass gatherings has forced the cancellation or postponement of nearly every arts-based event, from author appearances and theater productions to music concerts and film festivals.
Within the visual arts community, local artists and gallery owners are now being forced to adapt to an entirely unfamiliar circumstance — a world in which art shows, openings and workshops are no longer an option.
“I don’t know what’s going to happen,” says local painter Michelle Guerrero, who recently had to cancel a solo exhibition of her work at Weird Hues, a Chula Vista art space. “I was going to be working with the city of Chula Vista on a project in April and that’s been canceled as well.”
Guerrero, who goes by the name Mr. B Baby, was especially excited for her show at Weird Hues, a gallery known for drawing huge crowds of young adults. Known for her vibrant murals featuring fantastical creatures, Guerrero says she’d been working on the pieces for the show for years. Now, she’s unsure if people will ever see them in person.
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“Pretty much everything I’ve been working on is now ‘until further notice,’ ” says Guerrero. “I’m just trying to be hopeful at this point. I know things are changing with the economy, but for artists like myself, it’s a very scary time.”
Along with galleries such as Good Friday in the East Village and Swish Projects in North Park, Weird Hues has become known for showcasing edgy and contemporary shows from local artists. Weird Hues co-owner Mauro Doñate says the space serves as a source of inspiration for up-and-coming South Bay talent. He now worries those artists will become stifled by not being able to interact with their creative peers.
“None of these artists are thinking that their show is going to get pushed back because of a virus,” Doñate says. “So they put in all this work, and now it’s just paused. We’re frozen in time at the moment. That whole movement, that whole momentum, has come to a pause.”
One of the artists who has benefited from that momentum is Stephanie “Fifi” Martinez. The local cartoonist and musician landed a solo show at Weird Hues in 2019, and the exposure helped her land more appearances and opportunities. Over the past week, she had a reading canceled at the Whistle Stop and is uncertain whether the future shows and conventions she was planning on showcasing will even occur.
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“A lot of my livelihood as an artist comes from these types of interactions,” Martinez says. “Otherwise, it’s just online interactions, and that’s just not as fulfilling. Interacting and being in these spaces is mostly what makes me happy about making art — having that connection.”
Larger events that artists depend on to showcase their work are also being canceled. North Park Main Street recently announced the indefinite postponement of the San Diego County Credit Union Festival of Arts in North Park. The annual three-day festival, known for showcasing hundreds of local artists, was originally scheduled to kick off May 8.
“It was the responsible thing to do to cancel the festival, but the hard part is how are we going to stay afloat without that funding,” says North Park Main Street Executive Director Angela Landsberg, explaining that the Business Improvement District organization depends on sponsorships and funds from the city, all of which are now being either nullified or put on hold.
James Brown is the owner of Bread & Salt, a former bread factory that was converted into a performance and visual arts space in 2013. He’s had to deal with financial troubles over the years, but admits he’s never seen anything like the current COVID-19 crisis.
“It’s all moving so fast,” says Brown, who recently decided to cancel all future events. “Obviously, Bread & Salt will suffer financially as an institution, but the artists are already suffering as well.”
Weird Hues, an art and music collective in Chula Vista, is one of many art spaces across the county that have had to cancel events due to coronavirus concerns. Pictured are Weird Hues owners owners Luis Garcia and Mauro Doñate.
(Courtesy photo )
Larger nonprofit art institutions can remain on solid financial footing thanks to grants, city funding and tax-deductible memberships, but smaller art spaces often don’t have the same benefits. With state quarantine measures likely to intensify, spaces that depend on show openings, foot traffic and even retail sales could suffer major losses. Teros Gallery in City Heights is one such place. Founder Alejandra Frank says she’s trying to “stay positive” after canceling the majority of the space’s upcoming shows and performances. Frank and co-owner Carmela Prudencio have a good relationship with their landlord, but they worry that the COVID-19 crisis could last a long time.
“I think that this is really going to be a time where artists will have to take inventory and understand their resources better,” Frank says. “We don’t have a choice, really. It will have to be community building on a whole new format.”
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Some galleries are doing their best to come up with those new formats and to continue to remain engaged with a public that has no choice but to stay home. Teros Gallery will be hosting an online pop-up show titled “Transitional Meltdown.” Bread & Salt curator Thomas DeMello has begun a series of interactive, sit-down interviews with artists and curators that will be live-streamed on Instagram and later uploaded to YouTube.
The owners of Thumbprint Gallery in La Jolla have decided to close their La Jolla space indefinitely, but are already planning an online exhibition titled “Artists-In-Residence,” which will showcase works from a hundred artists created during the COVID-19 crisis.
“A lot of our income came from our pop-up events at restaurants and bars,” says Thumbprint co-owner Paul Ecdao, referring to the gallery’s curatorial work at places such as Bar Basic and Bluefoot.
“But they’re all closed, and that’s how we’re able to pay rent in La Jolla,” Ecdao says. “We’ll be OK through April, but beyond that, we’re definitely worried.” | |
Recent novels by André Alexis and Suzette Mayr draw on genre fiction conventions in intriguingly plotted and intricately allusive works. The Hidden Keys portrays a Toronto underworld of thieves and thugs, and Dr. Edith Vane and the Hares of Crawley Hall satirizes the more gently cutthroat domain of academia. Race intersects with identity and place in complex and varied ways. Alexis assembles a multiracial and socio-economically diverse cast of characters scattered around Toronto, while in Mayr’s campus novel, white and colonial hegemonic power structures are stubbornly entrenched at the fictional University of Inivea.
The Hidden Keys, Alexis’s follow-up to Fifteen Dogs, is part of his planned series of five connected novels, each taking up a philosophical idea. Tancred Palmieri is a skilled thief with a rigorous sense of personal honour. After several chance encounters of varying degrees of intensity, he develops an affinity with Willow Azarian, an older woman whose drug addiction has alienated her from her siblings and diminished her once-formidable intellectual abilities. Willow is convinced that the objects bequeathed by her father to his five children are clues in an elaborate treasure hunt. She asks Tancred to retrieve them so that she can solve the mystery. In this psychologically astute novel, which Alexis identifies as a loose reading of Treasure Island, Tancred undertakes a quest that changes him more than he had anticipated.
Alexis depicts comically error-ridden heists mounted around the city. Tancred’s investigation precipitates encounters with a motley assortment of underworld tough guys, urbane businessmen, and poised society ladies. In these vividly realized minor characters, Alexis’s novel is reminiscent of Carl Hiaasen’s or Elmore Leonard’s mystery fiction. Below the surface-level pithy dialogue and rapid action is a thoughtful meditation on the lives we choose (or that seem to choose us). Identity, family, home, and belonging are carefully woven in Tancred’s musings and in the reflections of other characters. The humour in the dialogue-heavy work alternates between slapstick and irony, and Alexis’s witty aphorisms resonate beyond the experience of reading the novel.
Mayr also deploys comedy to good effect in her portrait of an anxious English professor. For the title character of Dr. Edith Vane and the Hares of Crawley Hall, “post-tenure Elysium was a rabbit on a greyhound racetrack.” Instead of basking in the anticipated appreciation of her colleagues and students, Edith is beset by a series of woes, from a malfunctioning washing machine to marginalization at work. She hopes that her long-awaited book on “Beulah Crump-Withers, former sporting girl, then housewife, prairie poet, maven memoirist, and all-around African-Canadian literary genius,” will install the author in the canon while elevating her own academic status. Instead, her celebrated former doctoral supervisor joins the department; where she once attempted to quash Edith’s work, she now schemes to usurp it. Edith has few allies, and they are not faring any better than she is.
Mayr notes the divergent fates of academic disciplines under a neo-liberal regime of monetized research and corporate sponsorship. While the Engineering and Business faculties enjoy shiny new facilities, the Arts Faculty is housed in the maggot-infested decay of Crawley Hall, where more than the asbestos needs to be remediated. Edith is reluctant to acknowledge the mounting evidence of “possible paranormal phenomena” because “she doesn’t like having to believe in the supernatural, especially so early in the school year, and so early in the morning.” But as uncanny incidents multiply, it becomes increasingly clear that the building harbours dangerous secrets. Edith’s name conjures up both Shirley Jackson’s insecure Eleanor Vance, from The Haunting of Hill House, and the sturdier Harriet Vane of Dorothy L. Sayers’s celebrated academic mystery, Gaudy Night. Peculiarly menacing jackrabbits that have infiltrated the building and other references to Alice in Wonderland highlight Edith’s disorienting immersion in a world of arbitrary and punitive authority.
Notwithstanding the Gothic trappings, Mayr’s portrait of campus life is disquietingly familiar. This is higher-educational sociology as much as fiction, with Mayr’s protagonist grappling with the issues of overwork, diminished collegiality, and corporatized research agendas outlined in recent critiques, including The Slow Professor, by Maggie Berg and Barbara K. Seeber. Edith entered the profession because she loved books and wanted to share her passion. How did she end up hiding in the bathroom? | https://canlit.ca/article/codes-of-misconduct/ |
The FIFA World Cup began in 1930 and is contested by men’s national teams who are members of the Fédération Internationale de Football (FIFA), and I have the list of countries that have won the World Cup since inception.
1942 and 1946 were the only times the quadrennial tournament did not hold because of World War II.
There has been 21 FIFA World Cups held so far and Brazil is the most successful team.
Also they are the only nation to have participated in every World Cup finals tournament.
READ ALSO: Ex-Super Eagles Stars React To Qatar 2022 W/Cup Qualifying Draw
Italy and Germany have four titles. Current champion France, along with past champions Uruguay and Argentina, have two titles each, while England and Spain have one each.
Cafu is the only player to play in three finals, 1994, 1998 and 2002.
Luis Monti is the only player to have featured in two finals for different national teams. He appeared in the 1930 final as an Argentine, and the 1934 final as an Italian.
Attilio Demaría was also in Argentina’s 1930 squad but switched to Italy’s squad in 1934, but appeared in neither final.
Brazil’s Mário Zagallo became the first to win both as player and manager, winning in 1958 and 1962 as player, and winning in 1970 as the manager.
West Germany’s Franz Beckenbauer is the second. He won as both captain in 1974 and manager in 1990.
Didier Deschamps is the third. Won with France in 1998 as captain, and then in 2018 as manager.
Germany’s Miroslav Klose is the only player to have won four World Cup medals: 2002 (silver), 2006, 2010 (both bronze) and 2014 (gold).
FIFA World Cup Hosts And Winners List
That is the list of countries that have won the World Cup since it began. The question now is when will Africa win the world cup?
Think it’s possible? | https://www.reporterswall.com/see-list-of-countries-that-have-won-the-world-cup-from-1930-to-2018/ |
Hostad is a virtual enterprise focused on offers and services related to hosting plans for websites and web systems. It has in its service roll several packages that focus, such as target audiences, enterprises and micro and small companies.
The construction of a work that could represent a digital enterprise, service provider in the area of hosting of web sites and systems, with unique characteristics, in the base of its infrastructure, being a company possessing a great part of its services allocated in the best servers of the world, was a conceptual challenge that was based on one of the pillars of the company: simplicity in the use of services.
Simplicity is truly a unique feature for any company in any area of the market, both nationally and internationally. This simplicity can mean luxury, sophistication or even power.
In the present project, this attribute was represented very well by the resulting solutions in the resource uses presented and available to subscribers of the Hostad services.
Based on this premise, this apanage, the basic axis in the construction of the graphic elements of the logo, unified the iconographic inspiration of servers, the name of the company with primary color tones referencing security and trust. | https://alexcerqueira.com.br/ac/portfolio/hostad/?lang=en |
Hundreds of women and men gathered in downtown Toronto on Saturday to protest the lack of opportunities for female filmmakers and to call on the entertainment industry to support a wider range of voices in the movies and shows it makes.
The event, dubbed the “Share Her Journey Rally,” is taking place at this year’s Toronto International Film Festival and drew a range of speakers. Actresses Geena Davis and Mia Kirshner; directors Nandita Das and Amma Asante; and academics such as Dr. Stacy L. Smith shared a mixture of personal stories and bleak statistics that demonstrated the barriers that women and particularly women of color face in an industry that is dominated by white men. Only 4% of directors across the 1,100 top films from 2007 to 2017 were female. Those numbers become even starker when they are broken down by race — only four of those films were from black female filmmakers, two were from Asian female filmmakers, and one was from a Latina director.
Carrying signs that read “men of quality don’t fear equality” and wearing buttons that proclaimed “TIFF I Stand With Women,” those gathered expressed optimism that change was achievable.
“We are part of a movement now, and I do believe that each one of us can make a difference,” said Keri Putnam, executive director of the Sundance Institute.
It’s not just a lack of jobs that animated the crowd on Saturday. The fall of Harvey Weinstein, the indie mogul who was accused by dozens of women of sexual harassment, has led to an industry-wide reckoning. In the year since the Weinstein accusations broke, other high-profile media figures such as Matt Lauer, Dustin Hoffman, Russell Simmons, and Brett Ratner have been accused of sexual misconduct. Speakers at the event said there needs to be an easier way for women and men to report instances of harassment and assault and to hold their assailants accountable. Kirshner, for instance, slammed the “indifference of institutions” and said that companies must do more than simply pledge to have “zero tolerance” for that kind of behavior.
“Unless we improve the conditions where our bodies are respected in the workplace, I don’t see how we’ll have our minds respected and achieve leadership,” said Kirshner.
The event also made it clear that as difficult as the situation has been for white women filmmakers and performers, women of color have consistently been denied opportunities for advancement in the entertainment business. They are less likely to have lead roles in major motion pictures, they earn less money for the work they do, and they are not offered as many chances to direct big-budget productions. Race, speakers such as Asante argued, must be a key part of a larger conversation about opportunities for women.
“Every day I dream of a world in which the necessity to talk about this is absent and where my fellow women artists can speak about their art rather than campaign to do it,” said Assante.
Davis, the “Thelma & Louise” star and the founder of the Geena Davis Institute, a research group that explores gender representation in media, said that children’s programming is key. Data suggests that programming geared at younger viewers tends to have a disproportionate number of male protagonists.
“Why are we training [kids] to have unconscious gender bias?” Davis asked.
Earlier in the day, the festival committed to gender parity on its board of directors and within its executive ranks. TIFF also said it will make public the gender and race of all the members of its selection committees and programmers, along with its consultants.
The festival has made a point of promoting female filmmakers and underrepresented directors in its most recent incarnation. Of the 342 films screening this year, about 36% are directed by women. That’s up from 33% in 2017.
On Saturday, many of the people who took the stage said they struggled with being identified as “women directors.” They dreamed of a time when their race or gender would no longer need to be discussed and when they could focus on the films they make. Until that time, they said they were committed to rallying more women to their cause and drawing attention to the need for more representation both in front of and behind the camera.
“Yes, I’m a woman director, and I want more of us out there,” said Das, adding, “we have lots of stories to tell. Please just hear them.”
RELATED: | https://variety.com/2018/film/news/toronto-film-festival-protest-rally-women-underrepresentation-1202933254/ |
This thoroughly updated fourth edition of Clinical Research in Communication Disorders: Principles and Strategies remains an instrumental resource for courses on research methods and design in communication disorders.
The book is separated into three key sections: science and the scientific methods, clinical research designs, and doing, reporting, and evaluating research. Together, these sections provide thorough coverage of both the single-subject and group design strategies along with issues of measurement; philosophy of science; ethics of research; and planning, conducting, and reporting research.
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Instructors and students in communication sciences and disorders will appreciate the text s comprehensive coverage of scientific methods, group and single-subject research designs, report writing, and ethics of research in a single source.
New to the Fourth Edition
- New coauthor, Anthony P. Salvatore, PhD
- A new chapter on statistical analysis of research data, including several statistical techniques for single-subject research data, meta-analysis of both group and single-subject studies
- Updated criteria for visual analysis of single-subject research data
- New sections on translational research, qualitative research, and mixed methods research
- Descriptions of additional research designs not included in the previous edition (e.g., the regression discontinuity design)
- Updated information on research ethics and review of fraudulent biomedical
- Web-based sources that monitor research fraud and recalled studies
- Updated and expanded references throughout
DOWNLOAD THIS BOOK
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Andrea Rosen Gallery is thrilled to announce Back Grounds: Impressions Photographiques II, a historically rooted exhibition organized with Olivier Renaud-Clement that traces a profound lineage of conceptual, process-based photography. Ranging from early experimentations of the early 19th century in France to our contemporary era, this exhibition juxtaposes pioneering historical legacies with divergent contemporary trajectories, as means of building a contextual foundation for the experience and re-experience of such work. The resulting orchestration is an intimately curated dialogue between artists Liz Deschenes, Martin d'Orgeval, Gaylen Gerber, Karl-Heinz Hargesheimer, Sherrie Levine, Baron Adolphe Humbert de Molard, Alfred Stieglitz, and James Welling, which traverses between realms of methodology and intention, and channels attention to the processes of looking.
Built from the foundation of five early paper negatives by Baron Adolphe Humbert de Molard (b. 1800), first exhibited as part of a three-person show originally presented ten years ago at Andrew Kreps Gallery, New York and Galerie Nelson, Paris, this expanded exhibition further investigates a premise that expounds the ways in which our perception of such work, historically bound to context and experience, has shifted over time, and continues to generate a discursivity and evaluation of content. "One thing to remember," states Renaud, "is that in the earliest stages of the medium, the practitioners were not necessarily artists per se or even photographers, but curious experimentalists and chemists who had yet to realize the potential of a new medium that had barely come to life. Today, somehow, in our digital age, it seems we have come full circle and are re-addressing this with the lessons and facts of history."
Beginning in the 1800s, at a time when photography was inherently experimental, such venture into abstraction, as present in de Molard's fragile, golden images of William the Conqueror's Normandy Castle, was an unusual attempt. Attracted to the chemical medium, de Molard experimented with alternative fixing baths and chemical processes, producing what may be perceived definably as both artifacts of a developing technique and the artworks of an amateur pioneer.
Over a century later, Karl-Heinz Hargesheimer's silver gelatin chemigrams introduce early techniques of painting with chemicals on light-sensitive papers, creating liquid, abstract scapes conditionally belonging to a medium understood as representing reality. Re-presenting such early methods, James Welling's works move towards the reanimation of historical approaches; presenting both chemigram surfaces, as well as pristinely isolated autobiographical scenes documented with large format cameras.
Alfred Stieglitz's significant body of work Equivalents (1925-1934), shifts our gaze to consume both expansive and claustrophobic images of the sky, of dissipating and cumulating clouds, provoking, as Stieglitz described, an awareness of the awesome infinity beyond our existence. In an act to re-place historical content before contemporary audiences, Sherrie Levine's reprisal of iconic imagery, such as Stieglitz's Equivalents, provokes questions of authorship, originality, and artistic lineage, and encourages work to be experienced anew.
The natural abstractions of Martin d'Orgeval's wall photographs, developing a specific idea of frame, space and relief through a concern for metaphysical interiority, address subtle and complex issues of perspective. By interacting with the photographed shadows in the image, the real cast shadows of the salient frame create a coalescence of figurative and real spaces, activating the symbolic tension between materiality and immateriality.
Drawing on a realist tradition in the visual arts, Gaylen Gerber's work reinforces understanding as a performative act. In this exhibition Renaud-Clement presents both a "contextual" Backdrop by Gerber that acts as a ground to the exhibition itself, and a series of "discrete" photographic works presented on a constructed partition in the space. This reprise uses two bodies of work that take very different forms, drawing attention to the permeability of the distinction between the contextual and the discrete, and suggesting that everything in the exhibition may alternately be considered both background and subject.
Working to expand the dialogue surrounding photography, Liz Deschenes extends unique viewing experiences that explore the self-reflexive concepts of the medium. As one of the three artists who comprised Renaud's first exhibition in 2002, Deschenes has presented significant and distinct bodies of new work with each incarnation: Blue Screen Process (2002), Black and White (2003), and for this exhibition, a new body of works that stem from the photographs recently exhibited in Bracket (London) at Campoli Presti.
Olivier Renaud-Clement dedicates this exhibition to Philip Nelson, with whom he organized the first exhibition in Paris in 2002.
Liz Deschenes was born in 1966 in Boston, Massachusetts. Her work will be the subject of a solo exhibition at the Walker Art Center in October 2014. Recent exhibitions include Secession, Vienna; National Gallery of Denmark; and Fotomuseum Winterthur, Switzerland. Her work is in the permanent collections of Centre George Pompidou, Paris; MoMA, New York; Whitney Museum of American Art; The Metropolitan Museum of Art; The Walker Art Center; and the Art Institute of Chicago, among others. Deschenes lives and works in New York.
Martin d'Orgeval was born in 1973 in Paris, France. His work has been exhibited internationally at such institutions as Maison Européenne de la Photographie, Paris; Musée de la Chasse et de la Nature, Paris; Villa Oppenheim, Berlin; Museo Archeologico Nazionale, Naples; Galerie Hussenot, Paris; Adamson Gallery, Washington; Pace Gallery, Beijing. D'orgeval lives and works in Paris.
(Chargesheimer) Karl-Heinz Hargesheimer (1924-1971) was born in Cologne, Germany. As photographer, sculptor, stage designer and director, Chargesheimer belongs among the most pioneering and provocative artists of his generation.
Baron Adolphe Humbert de Molard (1800-1874) was born in Paris, France. He took up photography in 1843 using the daguerreotype, and later in the mid 1850s, became one of the first French photographers to use the calotype, a technique on paper developed in England by Fox Talbot, and introducing the principle of positive and negative.
Sherrie Levine was born in 1947 in Hazleton, Pennsylvania. Solo exhibitions of her work have been organized by San Francisco Museum of Modern Art; Philadelphia Museum of Art; Portikus, Frankfurt; Museum of Contemporary Art, Los Angeles; Kunstverein, Hamburg; The Getty Research Institute, Los Angeles; and Hirshhorn Museum, Washington, D.C., among others. Her work has been represented at Documenta 7, the Whitney Biennial, Sydney Biennial, Carnegie International, and São Paulo Bienal. Levine lives and works in New York and Santa Fe.
Alfred Stieglitz (1864-1946) was born in Hoboken, New Jersey. As photographer, critic, dealer and theorist, Stieglitz had a decisive influence on the development of modern art in America during the early 20th century. His photographs are in the permanent collections of the National Gallery of Art, Washington, D.C.; the Metropolitan Museum of Art; MoMA, New York; Philadelphia Museum of Art; and Art Institute of Chicago, among others.
James Welling was born in 1951 in Hartford, Connecticut. Recent exhibitions include a mid-career retrospective at the Cincinnati Art Museum, traveling to the Hammer Museum, Los Angeles and Fotomuseum, Winterthur, as well as solo exhibitions at Palais des Beaux-Arts, Brussels; Museum of Contemporary Art, Los Angeles; and Sprengel Museum, Hannover. His work is in the permanent collections of Centre Georges Pompidou, Paris; Los Angeles County Museum of Art; Metropolitan Museum of Art; MoMA, New York; Solomon R. Guggenheim Museum, New York; Tokyo Metropolitan Museum of Photography; and Whitney Museum of American Art.
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Latitude: 52.6756 / 52°40'32"N
Longitude: -2.969 / 2°58'8"W
OS Eastings: 334577
OS Northings: 309099
OS Grid: SJ345090
Mapcode National: GBR B7.4HQM
Mapcode Global: WH8BX.C0FM
Entry Name: Whitton Hall
Listing Date: 29 January 1952
Grade: II*
Source: Historic England
Source ID: 1055207
English Heritage Legacy ID: 259037
Location: Westbury, Shropshire, SY5
County: Shropshire
Civil Parish: Westbury
Traditional County: Shropshire
Lieutenancy Area (Ceremonial County): Shropshire
Church of England Parish: Westbury
Church of England Diocese: Hereford
SJ 3409 WESTBURY C.P. WHITTON LANE (north-west side)
17/130 Whitton Hall
29.1.52
GV II*
Country house. Circa 1720-30, probably for Alexander Top (II) and later his
son John Topp (I); restored and extended soon after 1920. Red brick
with grey sandstone dressings; slate roofs. Shallow U-plan with
additions to north-east. 2 storeys and attic. Stone bands at ceiling
levels, moulded wooden eaves cornice, and stone coped parapeted gables
with moulded kneelers; large brick ridge stacks off-centre to left and
right and integral brick end stacks at rear of wings, all with stone
caps. South-east (entrance) front: 2:1:1:1:2 bays with projecting
gabled wings; centre bay slightly projecting with open triangular
pediment on flat shaped brackets against wall, and with round-arched
window in tympanum which has moulded architrave with impost blocks and
keystone; glazing bar sashes with exposed boxes and thick bars (except
for 2 late C18 replacements on ground floor to left) stone cills, and
gauged brick heads with triple keystones; central first-floor window
with moulded architrave and single horizontal-sliding glazing bar sashes
in gables; central mid-C18 half-glazed Gothick door with 2 lower raised
and fielded panels and 3 cusped ogee-headed lights, and doorcase with
lugged architrave,frieze and triangular pediment. Early C20 gabled
addition set back to right is of 3 storeys and 2 bays. Left-hand
return front of 3 bays; blind segmental-headed windows except for
central first-floor glazing bar sash. North-west (garden) front:
slightly altered in early C20; balustraded parapet to centre; 3:1:3 bays,
glazing bar sashes with C20 tiled cills and segmental heads; central bay
slightly projecting with wooden balustrade to centre of parapet and door
with 6 raised and fielded panels, moulded architrave, radial fanlight,
flush keystone, and flanking narrow 6-pane windows; early C20 addition
to left has projecting 2-storey semi-circular bay with balustraded
parapet. C18 lead downpipes with moulded rainwater heads. Exception-
ally well preserved early C18 interior; entrance hall: raised and fielded
wainscot panelling, fireplace wall fully panelled with dado rail, moulded
cornice; some reordered C17 panelling with inscribed letters within
lozenges; fireplace consisting of moulded depressed-arched marble
surround with raised and fielded panelled sides, imposts and moulded key,
moulded architrave, and moulded dentil cornice; archway into staircase
hall; door with 6 raised and fielded panels, fluted architrave, radial
fanlight, and surround with fluted Doric pilasters and arch with moulded
architrave and keystone; drawing room: raised and fielded panelling,
moulded cornice, fireplace with shallow-carved frieze to moulded cornice,
raised and and fielded panel above, and flanking fluted Doric pilasters,
each supporting a short section of entablature with triglyphs and guttae;
round-arched buffet to left with keystone and shaped shelves; segmental-
arched recess to right; ground floor front room to left; remodelled
in late C18 (see sashes) with Neo-Classical marble fireplace; dining room
(probable former kitchen): moulded cornice; large open fireplace with
moulded segmental arch and moulded cornice; staircase: 3 flights around
rectangular well with landings, open string with cut brackets, 3 turned
balusters per tread (plain, twisted and fluted), turned newel posts, and
ramped and wreathed moulded handrail with columnular bottom newel post;
back staircase rising to attic: dog-leg with with winders, open string,
2 turned balusters per tread, and moulded ramped handrail; first-floor
corridors: 5 segmental and round archways with panelled piers, moulded
imposts and keystones; one with Y-tracery in fanlight; 3 bedrooms
inspected: one has re-ordered C17 panelling with fluting and inscribed
initials: "INRI", "SPE" and "TSM"; C18 cornice; fireplace with
roll-moulded arch, an moulded cornice; one front bedroom has fireplace
with lugged architrave and moulded dentil cornice; walls with cable-fluted Doric
pilasters, pulvinated frieze and moulded cornice; arched recess; other
front bedroom has fireplace with marble surround, lugged architrave,
central key, and moulded dentil cornice; raised and fielded panel above;
flanking cable-fluted pilasters without entasis, pulvinated frieze and
moulded cornice; panelled window seats. 6-panelled doors (some
with L-shaped hinges), internal panelled window shutters, and fireplaces
with late C18 or early C19 cast iron grates throughout; attic: probable
crucks reused as curved principals, and wall-plates or cill beams reused
as purlins, probably from former house on site. Whitton was the home of
the Lingen family during the C16 and the Topp family during the C17 and
C18, from which time most of the present buildings date. The house
stands within the remains of a moat which can still be discerned
to the south and contains some water to the east, and there is large
fish pond to the south. Several sources record the existence of a
number of reused stones in a garden wall inscribed: "I U" (John and
Ursula Topp),"1727", and "1731". The V.C.H. suggests they probably came
from the south front "where stone dressings have been rebuilt in brick
since c,1830", but this is not proven as the south front still has stone
dressings. The inscribed stones were not located at time of survey
(July 1985). The house forms the centrepiece of a good small country
house group including a former service block (q.v.) and stable block
(q.v.) flanking the forecourt to the south, a dovecote (q.v.) and a
barn (q.v.). A small probably late C18 latticed wooden chinoiserie
summerhouse standing in the garden to the north was dismantled for
repairs at time of survey and an C18 summerhouse on the hill opposite the
house to the south was derelict at the time of survey, neither is
included on this list. Whitton Hall is a complete example of a small
C18 country house, especially notable for its largely unaltered
interior and its outbuildings, V.C.H. Vol. VIII, pp. 313-4; B.O.E.
p. 318; H.E. Forrest, FLS, Some old Shropshire houses and their
owners (1924), pp. 1-7; C. Ryan, The evolution of the peasant house
in Shropshire. Medieval - c.1850. The Parish of Westbury, unpublished
thesis (October 1979), Manchester University, p. 240.
Listing NGR: SJ3457709099
This text is from the original listing, and may not necessarily reflect the current setting of the building.
Source links go to a search for the specified title at Amazon. Availability of the title is dependent on current publication status. You may also want to check AbeBooks, particularly for older titles. | https://britishlistedbuildings.co.uk/101055207-whitton-hall-westbury |
Technology, politics, climate, and the Coronavirus pandemic are four conditions that will influence future trends in the energy industry. In response to these conditions, the trend for the near future will be an increase in technology that enables automation, integration, and consolidation of internal controls. Tightening internal controls will accommodate the increasing layers of regulations, initiatives, and recommendations driven by local, state, federal, and corporate policies.
This brief will examine each condition and potential impacts. In addition, it will provide strategies to manage internal controls effectively and efficiently for long-term success.
Technology, Politics, Climate, and Coronavirus
Technology
Technology continues to advance at an exponential rate, and with that comes an increased demand for cybersecurity. Examples include artificial intelligence, machine learning, virtual reality, augmented reality, blockchain, internet of things, and 5G.
As technology changes, the potential for exploiting that technology changes. Gaps in security are patched as fast as new ones open up. Cybercrime is a constant threat that is forever evolving. Bad actors are getting smarter.
NERC CIP regulations were initiated in 2008 and have continued to evolve to meet the increasing complexity of the cybersecurity landscape. While the next major overhaul to the CIP standards has been pending for a significant amount of time, we are likely to see it come to fruition in the near future.
Currently, NERC Reliability Standards Under Development Project 2016-02 Modifications to CIP Standards encompasses modifications to 11 standards to address some issues identified in earlier versions of the CIP Standards:
- Cyber Asset and BES Cyber Asset Definition
- Network and Externally Accessible Devices
- Transmission Owner (TO) Control Centers Performing Transmission Operator (TOP) Obligations
- Virtualization
These modifications will expand the scope of compliance, and inevitably require changes to NERC CIP internal control and compliance management programs across the industry.
While NERC CIP isn’t the only approach to cybersecurity (there are over 25 different cybersecurity frameworks), each framework has a common theme with variations on implementation, including differing scope, timelines, data required. Entities realize that to be secure, it is not sufficient to stay within the confines of NERC CIP and are implementing additional frameworks and extending cyber controls. Blending the desired cybersecurity frameworks with additional corporate initiatives and applying to the affected IT, OT, & IoT ecosystem is a challenge that all entities will face.
Politics
The political atmosphere in the United States will continue to shape our future. The current administration has made climate control a top-level priority and we will continue to see more intervention and more regulations than in the prior administration.
The Biden-Harris Administration has made it clear that as a nation, we need to invest in our critical infrastructure. This will mean incentives and regulations to drive towards that goal. There are numerous investments outlined in the Bipartisan Infrastructure Law that will help fund modernization efforts. This includes investment in clean energy with a goal of a “zero-emissions future”, an expansion of transmission lines to support the delivery of that new energy, and attention to make our infrastructure resilient against cyber-attacks.
The Cybersecurity and Infrastructure Security Agency (CISA) is tasked with creating a more secure and resilient infrastructure for the future. Their primary goal is to defend against urgent threats and hazards, and the secondary goal is to strengthen critical infrastructure and address long-term risks.
Cybersecurity and zero-emissions initiatives are frequent at the state level and vary state-to-state. Your controls program must support all states that you operate in.
Climate
Extreme weather events have disrupted our lives more frequently in recent years. Regardless of whether severe events are caused by global warming, or just the natural ebb and flow of weather patterns across time, future disruptions will continue to affect the generation, transmission, and distribution of energy.
It is irresponsible to overlook the impact of drought, flood, fire, extreme heat, extreme cold, and strong winds. Communities should strive for a diversity of energy sources with built in redundancy to ensure a reliable system, one that can respond to rapid changes in weather conditions and accommodate extremes. In February 2021, Texas was crippled due to unprecedented low temperatures. California has had years of horrible forest fires. Record numbers of hurricanes attack the South & East.
There are security and compliance risks inherent in reacting to these extreme weather events - unplanned situations cause us to throw out the rule book and react to the moment. When there is no power, computer systems, and/or communications, we are vulnerable to a variety of issues such as loss of critical data, reduction of physical and cyber security, and compromise to health and safety.
Pandemic
The global pandemic due to Covid-19 has changed our lives in so many ways, some of which have resulted in a long-term impact to the energy industry.
Working and learning from home has created a shift in energy demand in the short-term, but the attitudes towards working and learning online have changed permanently.
Step into any grocery store and you can see that the supply chain has been impacted by the pandemic. This impact stretches worldwide and across many types of products. Technology required to maintain the BES may be unavailable, delayed, or cost significantly more than before. This may mean extending the life of existing technology, which warrants additional controls to maintain until the technology is decommissioned.
Additionally, the pandemic has affected the global workforce. Covid-19 has been temporarily or permanently taking workers away from the jobs. Time off to recover from the virus, as well as self-quarantine has kept workers away from their jobs. Furthermore, an increase in early retirement has been hastened by the pandemic and has created a measurable gap in knowledge and experience. It is not sufficient to rely on individuals to execute controls and maintain compliance in a vacuum. Instead, procedures must be well documented, well communicated, and preferably automated to maintain compliance in the event of attrition, illness, or other causes.
Automation, Integration, and Consolidation of Internal Controls
There will be a breaking point. It will no longer be sufficient to have separate departments, groups, or teams for each type of regulation, policy, or initiative. There is so much overlap that they must be coordinated and consolidated enterprise-wide. Manual hands-on tracking of controls and compliance data is inefficient and error-prone and will only become more complicated and time consuming as the regulatory landscape evolves.
Automation
Typical reasons for automation include reducing human error, eliminating repetitive tasks, and ensuring that tasks are assigned and completed quickly and on time.
A solid internal controls program leverages automation for strong controls especially for the following:
- Data collection
- Periodic reviews
- Scheduled activities
- Time-based obligations
When designing controls as part of a cybersecurity program, automation becomes even more critical for monitoring assets and asset baseline, patch management, change control, access management, and more.
Integration
There is no one-size-fits-all software ready to accommodate every requirement. Entities should leverage existing systems and best-in-breed additions to the ecosystem and integrate them together to achieve the best result. For example, consider a NERC CIP Compliance program with a central compliance management software that receives asset and baseline updates from an asset management system, receives patch availability information from a patch discovery system, interacts with a human resources system for user data, and receives training completion information from a learning management system. Automated data feeds can be used to simplify these time intensive tasks, ensure greater accuracy, reduce risk of noncompliance, and serve as cybersecurity controls.
Consolidation
As outlined above, internal controls and the resulting compliance evidence will be required for a myriad of reasons such as federal, state, and local regulations, corporate initiatives, cybersecurity needs, and dynamic workforce. Consolidating the management of these controls in a single software system will ensure visibility and accountability of those controls. Designing controls to meet multiple similar requirements rather than being single-focused will ensure that you minimize duplication of effort.
Conclusion
Strong internal controls are required for protection against situations and bad actors that can cause harm. Reliability and security are the objective of the controls, while provable compliance is a result. Technology, politics, climate, and the pandemic affect the content of your controls program. Enterprise-wide automation, integration, and consolidation of internal controls will be necessary to support an effective program today and into the future.
Invest in a software that can adapt to meet your needs and grow with you as your ecosystem changes.
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Soldiers who suffer more than one mild traumatic brain injury (TBI) face a significantly higher risk of suicide, according to a new study.
Researchers from the National Center for Veterans Studies at the University of Utah also found that the risk for suicidal behaviors and thoughts increased not only in the short term, but during the soldier’s entire life.
“Up to now, no one has been able to say if multiple TBIs, which are common among combat veterans, are associated with higher suicide risk or not,” said the study’s lead author, Craig J. Bryan, Ph.D., assistant professor of psychology at the University of Utah and associate director of the National Center for Veterans Studies.
“This study suggests they are, and it provides valuable information for professionals treating wounded combat servicemen and women to help manage the risk of suicide.”
During a six-month period in 2009, 161 patients who received a suspected brain injury while on duty in Iraq were referred to an outpatient TBI clinic at a combat support hospital.
The researchers found that one in five (21.7 percent) who had sustained more than one TBI reported suicidal ideation, described as thoughts about or preoccupation with suicide.
For those who had received one TBI, 6.9 percent reported having suicidal thoughts. Zero percent of those with no TBIs reported suicidal thoughts.
In evaluating the lifetime risk, researchers asked patients if they had ever experienced suicidal thoughts and behaviors up to the point they were assessed.
The increases were similar for suicidal thoughts during the previous year rather than at any time, according to the researchers. They found that 12 percent of those with multiple TBIs had entertained suicidal ideas during the past year, compared with 3.4 percent with one TBI, and 0 percent for no TBIs.
The researchers explained that they used suicidal ideation as the indicator of suicide risk because too few patients reported a history of making a suicide plan or had made a suicide attempt for statistically valid conclusions to be made.
Researchers also found that multiple TBIs were associated with a significant increase in other psychological symptoms, including depression and post-traumatic stress disorder (PTSD). However, only the increase in depression severity predicted an increased suicide risk, they noted.
“That head injury and resulting psychological effects increase the risk of suicide is not new,” Bryan said. “But knowing that repetitive TBIs may make patients even more vulnerable provides new insight for attending to military personnel over the long-term, particularly when they are experiencing added emotional distress in their lives.”
Because researchers were in Iraq, they were able to compile “a unique data set on active military personnel and head injury,” Bryan said. “We collected data on a large number of service members within two days of impact.”
He noted that researchers assessed only patients with mild or no TBI at the combat hospital. Those with moderate to severe TBI were immediately evacuated from Iraq.
The patients remaining in the study were divided into three groups based the total number of TBIs during their entire lives — zero, one, and two or more. The most recent TBI was typically within the days immediately preceding their evaluation and inclusion in the study.
Each soldier was also surveyed about their symptoms of depression, PTSD and concussions, and their suicidal thoughts and behaviors.
TBI is considered a “signature injury” of the Iraq and Afghanistan conflicts, according to the researchers. They note it is of particular concern because of the frequency of concussive injuries from explosions and other combat-related incidents. Estimated prevalence of TBI for those deployed in these two countries ranges from 8 percent to 20 percent, according to a 2008 study.
Additionally, past studies have found that suicide is the second-leading cause of death among U.S. military personnel, with the rate rising steadily since the conflicts began in Iraq and Afghanistan.
Prevalence of PTSD, depression and substance abuse have risen as well, especially among those in combat, and each has been shown to increase risk for suicidal behaviors, researchers noted.
“Being aware of the number of a patient’s head injuries and the interrelation with depression and other psychological symptoms may help us better understand, and thus moderate, the risk of suicide over time,” Bryan said. “Ultimately, we would like to know why people do not kill themselves. Despite facing similar issues and circumstances, some people recover. Understanding that is the real goal.”
The study was published in JAMA Psychiatry.
Source: University of Utah
Soldier holding his head photo by shutterstock. | https://psychcentral.com/news/2013/05/19/repeated-brain-injuries-up-soldiers-suicide-risk/54981.html |
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Walt Disney Studios’ John Carter opened nationwide on Friday, March 9th, but it wasn’t until last weekend my family and I were able to see it. Three weeks into its theatrical run, John Carter is clearly on life support – mortally wounded not only by Katniss’ arrow from The Hunger Games, but from Walt Disney Studios itself. When my family of 4 entered the theater, we easily doubled the audience for that showing. Disney’s John Carter is about a man transported from Earth to Mars. However, those aren’t the two worlds I’m referring to in this review. No, this discussion is centered on the on-screen world of John Carter vs. what went on off-screen. Continue reading for more about John Carter’s tale of two worlds.
Director Andrew Stanton’s attempt to translate the 100 year old novel by Edgar Rice Burroughs “A Princess of Mars” to the screen was an incredibly ambitious one. By now we all know (or at least we should know) that John Carter is a tale of a post-Civil War Confederate soldier mysteriously transported from Earth to Mars. Once on Mars, John Carter (played by Taylor Kitsch) was adopted by one species of Martians – the 9 foot tall, 4 armed, green skinned Tharks, and Carter becomes embroiled in an age old Civil War between two clans of tattooed, red-skinned Martian Humanoids. The film starts off with a long, expository scene on Mars telling the backstory of the war between the red-skinned Heliumites and Zodangans, and how another race called the Therns (possibly Martians, possibly from another world altogether) tips the balance of the war in the Zodangan’s favor. Notice none of these races are the towering, 4 armed, Martian Tharks, they play another role in the war and John Carter’s development. Confused? You aren’t the only one.
When I saw the opening scene, I flashed back to another ambitious attempt to deliver a cult sci-fi novel to the big screen – Frank Herbert’s “Dune.” When “Dune” was in theaters back in the mid-1980’s, the movie studio distributed handouts explaining the various terminology of the film. I read the flier, but my sister did not. I barely knew what was going on; she had no clue. John Carter’s opening scene was the handout for “Dune” – if you didn’t get it, you weren’t totally lost, but as my 14 year old asked 30 minutes in, “do you know what’s going on?”
To be fair to Andrew Stanton, I blame this confusion on Edgar Rice Burroughs. Stanton did the best he could with a novel full of complicated names and places. It was difficult enough keeping the names straight when reading the book. Hearing everything coming at you, spoken quickly on the screen, sounded like a word salad of uninterpretable name after unpronounceable place. There was nothing for the audience to hang their hat on, and this lack of connection ultimately made the characters feel as distant as Mars.
There were some amazing aspects to what was presented on the screen during John Carter. In particular, Lynn Collins’ portrayal of Dejah Thoris (the Martian princess) was spectacular – she brought life to a strong female lead who was more than a princess and damsel in distress. Collins’ Dejah Thoris was intelligent, scientific, strong, and heroic, and her on screen presence captured the audience’s attention. Taylor Kitsch was acceptable as the strong, silent type, leap-frogging his way across the terrain, but, ultimately, his performance took second place to Collins’.
The visual effects for the film were stunning. Along with John Carter, the audience was transported to a Martian world full of the grit and grime of an ancient, decaying civilization. The animated creatures populating this world including: the green-skinned Tharks, the baby Thark hatchlings, and Woola the Martian, dinosaur-like dog, were seamless and convincing. The movie was epic in its range and scope with different species in far-flung locations. Filmgoers were transported to late 1800’s Victorian New York City, to the wild west of Arizona, to a journey on Mars, and finally back to Earth. However, all those locales and characters ultimately left us with a whole film that was less than the sum of its parts.
That’s where the off-screen world of John Carter comes in – this movie was inherited by the current regime at Walt Disney Studios. The film was green-lighted by then studios chairman Dick Cook who was forced out of the company in 2009. Rich Ross, former President of Disney Channels worldwide, was now responsible for this big-budget baby when he replaced Cook as Chairman of the Studios in October 2009. However, was Ross totally behind the film?
John Carter was clearly Andrew Stanton’s “passion project” – saying as much in numerous interviews. Stanton, the two-time Oscar winning director (Best Animated Feature in 2008 for Wall-E and in 2004 for Finding Nemo), is an extremely talented, creative, and valuable member of the Disney/Pixar team. It would make sense that Ross would want to keep his prized director happy, and with Stanton’s previous successes, he earned the right to his big-budget, risky, passion project. However, as the budget continued to grow, with final estimates between $250-$350 million to create, Disney needed this film to wow everyone. It was now up to the marketing team to get the fannies in the seats, but Disney didn’t quite know how to market the film.
The imagery and trailers for John Carter at best did not give a clear picture of the film and, at worst, were confusing. My wife, a PhD scientist, saw the ads and thought it resembled another “Clash of the Titans” film. Others saw sweeping western vistas, Reconstruction era Union soldiers, alongside Martians, and thought “Cowboys and Aliens.” It certainly didn’t help the film when Marketing President, MT Carney, left Disney in late January 2012 after only 19 months on the job.
Carney’s departure came a little more than 2 weeks ahead of Super Bowl Sunday and the launch of “The John Carter Journey to the Super Bowl Sweepstakes.” The Super Bowl ad involved a slow zoom out showing a montage of images that eventually revealed the words “John Carter” – unfortunately the ad was cut short due to the timing of the game and rushed any mention of the sweepstakes. Compare the two versions of the ad below, and see how quick the mention of the contest is in the 30 second spot that aired during the Super Bowl.
Compared to the final 10 seconds of the longer, 40 second version of the Super Bowl ad.
By all accounts the John Carter Super Bowl ad was underwhelming, and there was very little “buzz” created from the subsequent Super Bowl sweepstakes.
Finally, 10 days after John Carter’s March 9th nationwide release, Walt Disney Studios announced the film was expected to lose $200 million. This statement was picked up and reported worldwide by the press. Dawn Chmielewski quoted Disney in the Los Angeles Times:
“In light of the theatrical performance of John Carter ($184 million global box office), we expect the film to generate an operating loss of approximately $200 million during our second fiscal quarter ending March 31,” Disney said in a statement. “As a result, our current expectation is that the Studio segment will have an operating loss of between $80 and $120 million for the second quarter.”
The timing of this announcement was perplexing. The film was not yet released in all of its international markets. John Carter just had its Tokyo premiere on Sunday, April 1st, and the film won’t open in Japan until April 13th. However, after Disney’s statement, the press have already dubbed John Carter a flop, failure, one of the greatest busts of all time, and thus limiting any success it could have in the US market if not worldwide. Showings of the film are currently extremely limited in US theaters, and John Carter has already been removed from first run theaters in some markets.
So where does this leave us? Disney’s John Carter is certainly not without its problems, the largest of which is a convoluted and confusing story. In retrospect, maybe the story of John Carter was better suited for a television series that could dedicate entire hour long episodes to each locale or scene instead of cramming everything into a 2 hour block. Do the story problems make John Carter the biggest flop of all time? Certainly not. There are some very entertaining parts to the film including Lynn Collins’ performance, the amazing scenery, and impressive special effects. Would I recommend seeing the film? Yes, see if for yourself, and make your own decisions. Although if you haven’t seen it yet, wait until it comes to the bargain theaters. You shouldn’t have to wait too long.
So what do you think? Have you seen John Carter? Share your thoughts below, I’d love to read your impression of the film. For more adventure movie news, be sure to follow Adventures by Daddy on twitter and “like” our facebook page too. | http://www.adventuresbydaddy.com/2012/04/05/review-of-disneys-john-carter-a-tale-of-two-worlds/ |
Housekeeping Attendant (casual)
Posted: 9th October 2017
Flexible hours between 08.00-15.00, including some weekends
We are looking for customer-focused individuals to join our dedicated Housekeeping team. The overall objective is to focus on venue presentation and to clean our guest bedrooms to a high standard. Previous experience in a similar role would be advantageous although training can be provided for the right candidates.
The successful candidate/s will be required to clean our guest bedrooms and public areas to the agreed standard.
Flexibility is essential as are high levels of attention to detail. Experience in a similar role would be advantageous, although training can be provided.
Overall objective
To actively engage with our clients, exceeding their expectations and ensuring all aspects of the customer journey are effectively delivered.
To service and clean the Møller Centre guest bedrooms and other designated areas to the agreed standards.
Main responsibilities
Ensure that the agreed standards of cleanliness and hygiene are maintained in all areas.
Take all necessary action for the security of guests and their belongings, ensuring that guest property is respected at all times.
Ensure linen is dealt with in accordance with the Centre’s procedures.
Ensure all equipment is used correctly and left in good working order at all times.
Ensure beds are made up and rooms serviced to the standards required.
Ensure correct and economical use of equipment and cleaning materials.
Report any maintenance faults or faults on equipment to the Housekeeping Manager/Supervisor immediately.
Ensure lost property is recorded and secured appropriately in line with our procedures.
Carry out spring cleaning as and when requested by the Housekeeping Manager or Front of House Manager.
To use initiative over special/occasional cleaning.
Any other tasks required within the department.
To carry out any reasonable request made by a member of management in a timely and cost effective manner.
Health and Safety
You must ensure that all Health, Safety and Security regulations are adhered to at all times, this includes the fire, manual handling, sharps & syringes procedures and COSHH regulations, all of which you will be made aware. Please ensure that you report any faults or infringements of these procedures or act immediately where appropriate to correct them. You have a duty to carry out work so that you never put yourself or others at risk, creating a safer working environment for everyone.
Training and Development
The Møller Centre has a Company Business Plan which sets out the aims and objectives of the Centre and what we hope to achieve. As part of the process of achieving these objectives, we are committed to train and develop all our staff.
You will be provided with all the relevant statutory training required for you to carry out your role safely and further training requirements would be discussed at your personal development review.
Business Development
As outlined in the Company Business Plan all staff are expected to contribute to the business taking responsibility for the delivery of consistent service excellence to clients thereby exceeding their expectations.
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---
abstract: 'We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely *homotopy invariance*. To do this, we use the result from [@sentai] that any Grothendieck fibration of the kind being considered can automatically be upgraded to a 2-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into 2-dimensional fibrations.'
author:
- Joseph Helfer
bibliography:
- 'fohl.bib'
title: 'First-order homotopical logic'
---
Overview {#sec:overview}
========
The goal of this paper is to introduce a “homotopy-invariant” interpretation of first-order logic with equality, to give a description of this interpretation within the framework of categorical logic, and to give an abstract formulation and proof of the homotopy-invariance property within this framework.
The interpretation can be concisely described by the following commutative diagram. $$\begin{tikzcd}
\text{MLTT}
\ar[r, "\text{Voevodsky-Awodey-Warren-Kapulkin-Lumsdaine}"]&[190pt]\text{Simplicial Sets}\\
\text{IFOL}\ar[u, "\text{Martin-L\"of 1972}"]
\ar[ru, "\text{First-order homotopical logic}"', sloped, near start]
\end{tikzcd}\label{eq:interpretation-diagram}$$
On the bottom left we have intuitionistic first-order logic, on the top-left we have Martin-Löf type theory, and the vertical arrow is the interpretation of IFOL into MLTT which was described in Martin-Löf’s original paper [@ml72]. The long horizontal arrow is the homotopy-theoretic interpretation of type theory [@awodeywarrenid; @kapulkinlumsdaine; @warrenthesis] which initiated the subject of Homotopy Type Theory. Hence, composing these two interpretations, one obtains a homotopy-theoretic interpretation of first-order logic, which it is the purpose of this paper to elaborate.
In fact, one does not need to go through these two interpretations, as the homotopical semantics for first-order logic can be described directly and very simply (in fact, this simplicity, as compared to the interpretation of Martin-Löf type theory, was one of our original motivations for considering this interpretation). Nonetheless, we will give a brief, informal, and somewhat idiosyncratic introduction to Martin-Löf type theory, in order to makes sense of the above commutative diagram.
We consider this paper to be a continuation of [@sentai] and the reader should be prepared to refer to the latter. The expository first half of this paper (Parts \[sec:overview\]-\[sec:fibrational-formulation\]) is basically self-contained, but in the technical second half, we will rely heavily on the definitions and results from [@sentai].
[**Acknowledgments:**]{} We are grateful to McGill’s Logic, Category Theory, and Computation seminar and Carnegie Mellon’s Homotopy Type Theory seminar for allowing us to speak about this project at a very early stage, and to Steve Awodey for encouraging us to write it up, and also for pointing out an interesting connection between our invariance theorem and the Univalence Axiom (which was also independently observed by Ulrik Buchholtz).
Type theory
-----------
###
We begin with the classical notion of type theory – also known as “higher-order logic” – which originated in Russell’s and Whitehead’s *Principia Mathematica* – though we have in mind the later formulation from Lambek’s and Scott’s *Higher-Order Categorical Logic* [@lambek-scott], where it was related to the theory of *elementary toposes*.
To begin with, we put ourselves in the familiar “ZF-style” set theory – i.e., in an axiomatic theory, with two undefined notions, namely “set” and “membership”, and a collection of axioms concerning these notions, from which we derive consequences. We have in mind, of course, the axioms of ZF itself, but we also want to allow the possibility of leaving off some of the axioms, and working with only a fragment of the theory, keeping track as one proceeds of what one is actually using. In particular, we want to allow the possibility of working only within the *intuitionistic* fragment of first-order logic.
Now, we begin by singling out two particular sets, namely the one-element $\tm=\{*\}$, as well as its power set $\{x\mid{}x\subset\tm\}$, which we denote by $\Omega$[^1]. Next, we note that $\Omega$ has the structure of a *Heyting algebra*: it has a partial order $\le$, given by inclusion of subsets, which is a bounded lattice having “relative pseudo-complements” (also known as *exponentials* or *implications*) – i.e., for any elements $q,r\in\Omega$, there is a least element $s$ such that $q\wedge{}s\le{}r$ – namely $\{x\in\tm\mid{}x\in{}p\Rightarrow{}x\in{}q\}$.
Now, the point we want to emphasize is that *every assertion is set theory* is equivalent to one of the form $p=q$ for some $p,q\in\Omega$ – and in fact, one can even take $q=\tm$ – so that we can dispense with the membership relation “$\in$” altogether and restrict ourselves to statements of this form. In a sense, this is obvious: for any proposition $\phi$, we can set $p=\{x\mid{}x\in\tm\wedge{}\phi\}$, and then $\phi$ is equivalent to $p=\tm$. However, we have clearly *not* dispensed with “$\in$”, as we used it in the definition of $p$. The point is that for many propositions $\phi$ of actual mathematical interest (in particular, outside of set theory itself), we can define $p$ directly with the use of a few basic operations (including the Heyting-algebra operations $\top$, $\bot$, $\wedge$, $\vee$, $\To$), staring with a few basic elements of $\Omega$.
In particular, most interesting elements of $\Omega$ will arise as specific values of some *predicates*, i.e. functions $X\to\Omega$ from some set $X$. In order to specify particular predicates, let us introduce the “function abstraction” notation; given sets $X$ and $Y$, we have the usual set of functions $$Y^X=\{f\mid{}f\subseteq{}X\times{}Y\wedge{}\forall{}x\,\exists!y(\br{x,y}\in{}f)\},$$ and we write $\lambda(x:X)t$ for $\{z\mid{}\exists{}x(x\in{}X\wedge{}z=\br{x,t})\}\in{}Y^X$ (where $t$ is some expression involving $x$ and denoting an element of $Y$). Next, for a predicate $P:X\to{}\Omega$, let us write $\forall{}P$ for $\bigcap_{x\in{}X}Px$ and $\exists{}P$ for $\bigcup_{x\in{}X}Px$. Let us also write $(-)=_X(-)$ for the predicate $X\times{}X\to{}\Omega$ taking $a,b\in{}X$ to $\{x\in\tm\mid{}a=b\}$.
Then, for example, we can express the predicate “n is prime” as $$\lambda(n:\N)
\Bigg(\forall
\Big(\lambda(d:\N)
\big(
\exists(\lambda(e:\N)(d\cdot{}e=_\N{}n))\To(d=_\N1\vee{}d=_\N{}n)
\big)
\Big)
\Bigg)$$ and (now abbreviating $\forall(\lambda{}x:X.t)$ as $\forall_{x:X}t$), the principle of induction can be stated as $$\label{eq:ind-princ-omega}
\forall_{P:\Omega^N}
\Bigg(
\Big(
P(0)\wedge\forall_{n:\N}
\big(
P(n)\To{}P(n+1)
\big)
\Big)
\To\forall{}P
\Bigg)$$ – or rather, the principle of induction states $p=\tm$, where $p\in\Omega$ is the element (\[eq:ind-princ-omega\]).
The upshot is that we obtain a very elegant and concise language for stating mathematical facts, which consists entirely of operations for forming *terms* of various *types* (i.e., elements of various sets) – in particular, of the set $\Omega$ – together with the single binary relation “$(-)=(-)$”, taking two arguments of type $\Omega$, which we use to make assertions.
Importantly, we can use this language not only to *express*, but also to *prove* mathematical facts – thus making it in a sense self-contained – using an appropriate system of rules. For example, from $p\To{}q=\tm$ and $p=\tm$, we can conclude $q=\tm$. Thus, we end up being able to do much of mathematics in what is essentially a very small fragment of set theory; in particular one in which every statement is *atomic* – i.e. consists of a single equation $p=q$ with no further logical connectives.
We now observe that, when something is proven within this system, one has actually proven something *stronger* than the corresponding fact in set theory, since the assumptions are weaker, but the conclusion is the same. For example, the assumptions made in the type theory about the sets $\N$, or about the operations $X\times{}Y$ and $Y^X$ are satisfied not only by the usual set of natural numbers, or the usual product and function-set operations, but, for example, by any sets isomorphic to these.
It would seem that this only involves an obvious and trivial generalization, since when we proved our statement in ZF, we had no doubt that it would also hold, say, with $\N$ replaced by any other system $(N,0,S)$ satisfying the Peano axioms. But there are also more surprising generalizations. In particular, if we drop the assumption that the notion of “set” (or rather, “type”) in the theory be interpreted as meaning an actual set, we find that there are interesting “non-standard” models[^2]. In particular, we can interpret the types as denoting objects of an arbitrary *elementary topos* – for example, the category of sheaves on any topological space – and the terms as denoting certain morphisms.
### {#subsubsect:mltt}
We now move on to “Martin-Löf type theory” (or “dependent type theory”). The main point here will be that, by a variation on the above scheme, we can see that every sentence in set theory is equivalent, not only to one of the form “$p=\tm$” with $p\in\Omega$, but also to one of the form $\exists{}x\colon{}x\in{}X$[^3], so that we can dispense with the set of truth values $\Omega$. In fact, this is again in a sense obvious, since for $p\in\Omega$, the statement $p=\tm$ is equivalent to $\exists{}x\colon{}x\in{}p$ – however, again, this is not what we want, since we are still making reference to (elements of) the set $\Omega$.
The key observation is that, for any element $p\in\Omega$ defined using the operations on $\Omega$ included in the type theory sketched above, one can, using just the set-forming operations $\tm$, $\times$ and $(-)^{(-)}$ – as well as the empty set $\emptyset$, disjoint union $+$ and, most importantly, the indexed product $\prod_{x\in{}X}$ and sum $\sum_{x\in{}X}$ – define a set $X$ whose inhabitedness is equivalent to that of $p$. A concise way to explain this is to say that, under the correspondence of sets with elements of $\Omega$ taking each set $P$ to $\{z\in\tm\mid{}\exists{}y\colon{}y\in{}P\}\in\Omega$ – and more generally, taking any family of sets $\{P_x\}_{x\in{}X}$ to the function $X\to\Omega$; $x\mapsto\{z\in\tm\mid{}\exists{}y\colon{}y\in{}P_x\}$ – each of the following operations in the top row is taken the adjacent operation in the bottom row. $$\def\arraystretch{1.5}
\begin{array}{c||c||c||c||c||c||c}
\tm&\emptyset&+&\times&(-)^{(-)}&\prod_{x\in{}X}&\sum_{x\in{}X}\\
\hline
\tm&\emptyset&\vee&\wedge&\To&\forall_{x\in{}X}&\exists_{x\in{}X}\\
\end{array}\label{eq:props-as-types-chart}$$ That this is so is seen directly by inspection. For example, given two sets $X$ and $Y$, we have that $\exists{}z(z\in{}X)\wedge\exists{}z(z\in{}Y)$ is equivalent to $\exists{}z(z\in{}X\times{}Y)$.
Actually, what we just said is only true *assuming the law of the excluded middle*; intuitionistically, the statement that $Q^P$ or $\prod_{x\in{}X}P_x$ is inhabited is *stronger* than the statement $\exists{}x(x\in{}P)\To\exists{}x(x\in{}Q)$ or $\forall{}x{\in}X\ \exists{}y(y\in{}P_x)$. However, this is not necessarily a disadvantage; it simply means that one is forced in dependent type theory to prove stronger statements than one normally would.
This correspondence between set-building operations and logical operations has a complicated history (involving Kleene’s notion of *realizability*, Gödel’s “Dialectica interpretation”, and the “Curry-Howard isomorphism” – see [@troelstrahist] for a detailed account), but originates in the “BHK” (Brouwer-Heyting-Kolmogorov) interpretation of intuitionistic logic, and was given the above, simple, set-theoretic formulation by Läuchli [@lauchlisemantics].
Hence, by using a type theory – that is, a formal system which allows the construction of types and of elements of these types – which includes the above type-forming operations, and which has in addition the single unary relation $\exists{}x\colon{}x\in{}X$ (with $X$ a type), one can express everything that was expressible in the above type theory using the type $\Omega$ of truth values[^4]. Actually, in dependent type theory, one does not have the unary relation $\exists{}x\colon{}x\in{}X$, but only the binary relation $x\in{}X$ (with $x$ an element and $X$ a type), usually written[^5] $x:X$ – thus, instead of proving the statement, $\exists{}x\colon{}x\in{}X$, one just *constructs* a particular term $t$ for which $t:X$ holds.
This may seem to undermine what we set out to do – namely, dispense with the membership relation “$\in$” – since now we have reintroduced it. In fact, in dependent type theory, we also need an equality relation between terms and between types, so that we seem not to have eliminated anything! The point is that, as with the type theory from the previous section, the important thing is that everything is now expressed by *atomic* formulas $x:y$ or $x=y$, with no need for further logical connectives.
In order to generate sets representing propositions of mathematical interest, we need sets corresponding to *atomic formulas* – i.e., equalities. Thus, for each type $A$ in the type theory, and each pair of terms $s,t:A$, we introduce a type $s=_At$ denoting the $(X\times{}X)$-indexed family of sets $(\{z\in\tm\mid{}x=y\})_{x,y\in{}X}$, where $X$ is the set denoted by $A$.
We are now in a position to describe the vertical arrow in (\[eq:interpretation-diagram\]) – i.e., the interpretation of first-order logic into dependent type theory; here, we consider the case of first-order logic with equality, but in which there are no other relation symbols. Given a (possibly multi-sorted) first-order signature $\sigma$ – i.e., a set $\Ob\sigma$ of “sorts”, and a (possibly empty) set $\sigma(\vec{A},B)$ of “function symbols” with “arity” $\vec{A}$ and “codomain sort” $B$, where $\vec{A}$ is a finite sequence of sorts, and $B$ is a sort – we consider dependent type theory augmented with a type constant $[A]$ for each sort $A\in\Ob\sigma$, and a term-constant $f$ of type $[A_1]\times\cdots\times{}[A_n]\to{}[B]$ for each $f\in\sigma(\vec{A},B)$.
There is then an obvious, recursively defined assignment, taking each term $t$ of $\sigma$ of sort $A\in\Ob\sigma$ to a term $[t]$ in dependent type theory of type $[A]$. Then, we can recursively assign to each *formula* $\phi$ of $\sigma$ a *type* $[\phi]$ of dependent type theory. In particular, an atomic formula $s=t$ (with $s$ and $t$ terms of some sort $A$) is interpreted as the equality type $[s]=_{[A]}[t]$; and to interpret formulas built up using the logical connectives, we use the correspondence displayed in (\[eq:props-as-types-chart\]) – for example, $\phi\wedge\psi$ is interpreted as $[\phi]\times[\psi]$.
Now, the fundamental fact about this interpretation, which was proven in [@ml72], is that it is *sound*, in the sense that for any formula $\phi$ of $L$ which is intuitionistically valid, there exists a term $t$ in dependent type theory with $t:[\phi]$.
### {#subsubsec:intro-hott}
Now, as was the case with the first type theory we described, there are interesting interpretations of dependent type theory besides the obvious set-theoretic one that we used to motivate it. Namely, again, it can be interpreted into any elementary topos – and more generally, any so-called *locally cartesian closed category with finite coproducts* [@seely-loc-cart-closed; @hofmann-loc-cart-closed] – $\C$ by interpreting the types and terms as certain objects and morphisms of $\C$ (to be more precise, of the various *slice categories* $\C/X$).
But with dependent type theory, there is yet another new and very interesting interpretation whose discovery initiated the subject of Homotopy Type Theory; namely, when interpreting it into certain categories (for example, the category of simplicial sets, which is in fact a topos), one can interpret the *equality types* in a non-standard way.
Now, we have not actually said anything about the *rules* of dependent type theory, which are quite interesting, and to some extent surprising, in their own right. In particular, the rules governing the use of the equality type are such that they bear a certain similarity to properties of path-spaces in homotopy theory, and it is this similarity which is exploited in the homotopy-theoretic interpretation.
However, even without going into the details of the rules of dependent type theory, this re-interpretation of equality can be motivated as follows. In the obvious, set-theoretic interpretation described above, we took the equality type $s=_At$ to be a set containing at most one element. However, it is natural to ask whether there are also interpretations in which the sets $s=_At$ can have more than one element (i.e., in which $s$ and $t$ can be equal “in different ways”).
In fact, this question has a counterpart within the type theory itself – known as the UIP (“uniqueness of identity proofs”) principle: is it the case that for any two terms $e,e':s=_At$, there exists a term of type $e=_{s=_At}e'$? In [@hofmannstreicher], this question is given a negative answer by defining an interpretation of dependent type theory in *groupoids*, so that a type $A$ is interpreted as a groupoid $X$, terms $s,t:A$ as objects $x,y\in\Ob{}X$, and the equality type $s=_At$ as the discrete groupoid (i.e., set) $\Hom_{X}(x,y)$. At the end of that paper, they also remark that, under this interpretation, although the equality types $s=_At$ can have more than one element, the “iterated” equality types $e=_{s=_At}e'$ can still only have at most one. They therefore suggest looking for interpretations in which *all* the “iterated” equality types can have more than one element by considering *higher-dimensional* groupoids.
We do not want to delve into the complicated and fascinating story of higher-dimensional categories here. We mention only that one of the driving ideas in this subject has been the connection, known as the “homotopy hypothesis”, drawn by Grothendieck (in [@pursuingstacks]) between higher-category theory and homotopy theory. Specifically, the idea is that the notion of “infinite-dimensional groupoid” (whatever that may mean) should in some sense be equivalent to that of “homotopy type” (i.e., topological space up to homotopy-equivalence). The analogy begins with the construction of the *fundamental groupoid* $\Pi(X)$ of a topological space, whose objects are the points of $X$ and whose morphisms $p\to{}q$ are homotopy classes of paths from $p$ to $q$, composition being given by concatenation of paths. But now, in an infinite-dimensional groupoid, the morphisms between two objects should form not a *set*, but *another infinite-dimensional groupoid*. Similarly, while the *homotopy-classes* of paths between two points in a space naturally form a *set*, the collection of all paths naturally form *another topological space*. Thus the suggestion is that, by iteratively considering points, paths, paths between paths, and so on, one should be able to build a infinite-dimensional groupoid from each topological space, and that every infinite-dimensional groupoid should arise this way.
Now, without worrying about whether such a correspondence can actually be established, one can just go ahead and define an interpretation of dependent type theory into topological spaces (or rather – since the category of topological spaces is not locally cartesian closed – into simplicial sets) in such a way that the equality types are interpreted as path-spaces. This is precisely what was carried out in the horizontal arrow in (\[eq:interpretation-diagram\]).
This ends our description of the vertical and horizontal arrows in (\[eq:interpretation-diagram\]). From this, it is easy to see what the diagonal arrow should do. Namely, it should interpret each *sort* of the given first-order language as a *space*, and then interpret each *formula* as another space, with equality being interpreted as a path space, and with other formulas being interpreted according to the analogues in the category of simplicial sets of the set-theoretic operations in (\[eq:props-as-types-chart\]). However, we will now “start over”, and explain the homotopical semantics for first-order logic directly.
Before we do that, we mention an important caveat, namely that in composing with the vertical arrow in (\[eq:interpretation-diagram\]), *much expressivity is lost*. That is, there are many interesting homotopy-theoretic properties of spaces which can be expressed in dependent type theory (to give one example: that of being $n$-connected for each $n$), but which cannot be expressed in first-order logic. In fact, first-order logic is very limited in this respect (we discuss this further in §\[subsec:questions\]), though some important properties are still expressible, such as contractibility or path-connectedness (see §\[subsec:examples\]).
Propositions as sets {#subsec:props-as-sets}
--------------------
We now explain directly how to define the homotopical semantics for first-order logic. To do this, we will first explain an analogous, but simpler, semantics, in which formulas are interpreted not as *spaces* but rater as *sets*. This corresponds, in the type-theoretic formulation of the semantics given above, to replacing the horizontal arrow in (\[eq:interpretation-diagram\]) with the obvious, set-theoretic interpretation of type theory.
As we mentioned in §\[subsubsect:mltt\], our starting point for the homotopical semantics is the BHK interpretation of intuitionistic logic, the general idea of which is that the meaning of a proposition $\phi$ is given by explaining what it takes to prove $\phi$ – or, put another way, by describing the *set of proofs* of $\phi$. Each logical connective then corresponds to an operation on sets which, when applied to the set of proofs of the constituent propositions, yields the set of proofs of the resulting proposition.
The easiest way to make this into a concrete mathematical definition, essentially due to Läuchli [@lauchlisemantics], is as follows. Let us again, as in §\[subsubsect:mltt\], fix some signature $\sigma$, and let us also fix some $\sigma$-structure $M$ – i.e., a set $M(A)$ for each $A\in\Ob\sigma$ and functions $M(f):M(A_1)\times\cdots\times{}M(A_n)\to{}M(B)$ for each $f\in\sigma(\vec{A},B)$. To each closed formula $\phi$ over $\sigma$, we wish to assign a certain *set* $\tau(\phi)$ (the set of “abstract proofs” of the formula). More generally, to a formula $\phi$ with free variables $x_1,\ldots,x_n$ of sorts $A_1,\ldots{},A_n$, we wish to assign a family of sets $\tau(\phi)[a_1,\ldots,a_n]$, indexed by elements $a_1\in{}M(A_1),\ldots,a_n\in{}M(A_n)$. The definition is given by induction on the complexity of $\phi$ as follows (where we omit the indices $[a_1,\ldots,a_n]$ when they are the same in all instance of $\tau$ appearing on one line): $$\label{eq:propositions-as-sets-clauses}
\begin{split}
\tau(\top)&=\tm\\
\tau(\bot)&=\emptyset\\
\tau(\phi\wedge \psi)&=\tau(\phi)\times\tau(\psi)\\
\tau(\phi\vee \psi)&=\tau(\phi)+\tau(\psi)\\
\tau(\phi\To \psi)&=\tau(\psi)^{\tau(\phi)}\\
\tau((\forall x_n)\phi)[a_1,\ldots,a_{n-1}]&=\prod_{a_n\in{}M(A_n)}\tau(\phi)[a_1,\ldots,a_n]\\
\tau((\exists x_n)\phi)[a_1,\ldots,a_{n-1}]&=\coprod_{a_n\in{}M(A_n)}\tau(\phi)[a_1,\ldots,a_n]\\
\end{split}$$ Note that we are including the atomic formulas $\top,\bot$ (“true” and “false”) as part of first-order logic, and omitting negation, which we define as $\phi\To\bot$. As in §\[subsubsect:mltt\], $\tm$ denotes some fixed one-element set, $+$ denotes disjoint union, $Y^X$ denotes the set of functions from $X$ to $Y$, and $\prod$ and $\coprod$ denote indexed product and indexed disjoint union.
It remains to define the set $\tau(s=t)[a_1,\ldots,a_n]$, where $s$ and $t$ are terms over the signature of the same sort $A$. Here, we note that, in the usual way, $s$ and $t$ can be interpreted as some elements $M(s)[a_1,\ldots,a_n],M(t)[a_1,\ldots,a_n]\in{}M(A)$, and we then define $\tau(s=t)[a_1,\ldots,a_n]$ to be $\tm$ if $M(s)[a_1,\ldots,a_n]=M(t)[a_1,\ldots,a_n]$, and $\emptyset$ otherwise.
Now, the main observation (which is proven easily by induction) is that a formula is satisfied in the structure $M$ (with respect to an interpretation of its free variables) in the usual (Tarskian) sense if and only if the corresponding set of “abstract proofs” is non-empty.
It may come somewhat as a surprise that this interpretation is equivalent to the classical one – in particular, that it satisfies the law of the excluded middle – given that it is supposed to implement an interpretation for intuitionistic logic. In fact, what Läuchli defined was a variation on what we have described, and is in fact sound and complete for intuitionistic predicate logic.
We would also like to emphasize that, though this interpretation is equivalent to the classical one in terms of which formulas it designates as *true*, it is still interesting. We give an example. Let $\phi$ be the sentence $$\forall m,n\ \exists{}d\ \exists k,l\ (k\cdot d=m\wedge l\cdot d=n)$$ in the usual language of arithmetic (“every pair of numbers has a greatest common divisor”), and consider its interpretation $\tau(\phi)$ in the standard model. By carrying out the definitions, one easily sees that the set $\tau(\phi)$ is isomorphic to $$\set{f\in\N^{\N\times\N} \mid \forall{}m,n(f(m,n) \text{ divides } m,n)}.$$ Now, we know this set is inhabited, since $\phi$ is *true*; indeed, given $m$ and $n$ we could take $d$ to be their greatest common divisor – or, alternatively, take $d=1$. Note that, each of these *proofs* of $\phi$ gives an *element* of $\tau(\phi)$ – namely, the functions $f(m,n)=\gcd(m,n)$ and $f(m,n)=1$. Indeed, one can refine this interpretation so as to give for each proof (in some formal system) of a given formula $\phi$, an element of the set $\tau(\phi)$. In fact, such a refinement is an automatic byproduct of the fibrational formulation of the semantics which we give later on (see Part \[sec:fibrational-formulation\]).
Propositions as objects of $\C$ {#subsec:props-as-obs-of-c-direct}
-------------------------------
The idea for the homotopical semantics is to repeat the above definition but replace “set” with “space”. Here, we will take *space* to mean *simplicial set*; in order to handle topological structures, we can first apply the singular simplicial set functor $\Sing:\Top\to\sSet$, and then proceed as below.
In fact, at this stage, the choice of simplicial sets is (almost) immaterial; objects in any sufficiently nice category will do.
There is a well-known way, due to Lawvere, of interpreting a first-order signature $\sigma$ into any category $\C$ with finite products (see Definition \[defn:signature\] below). Given such an interpretation $M$ with associated objects $M(A)$ for $A\in\Ob\sigma$, we then want to associate to each closed formula over this signature, not a *set*, but an object of $\C$. And more generally, to each formula with free variables $x_1,\ldots,x_n$ of sorts $A_1,\ldots,A_n$, we want, rather than a *family of sets indexed by $M(A_1)\times\cdots{}\times{}M(A_n)$*, an arrow in $\C$ with codomain $M(A_1)\times\cdots{}\times{}M(A_n)$.
We now proceed to define the interpretation of formulas in the same way as we did before, by generalizing each of the operations in (\[eq:propositions-as-sets-clauses\]) from sets to objects in an arbitrary (sufficiently nice) category.
In the clauses for “$\wedge$” and “$\vee$”, the operations are “indexwise product or disjoint sum of two indexed families of sets”. Here, we recall that a family of sets indexed by $I$ is the same as an object of the slice category $\Set/I$; and it is then seen that these operations are just given by the categorical product and coproduct in $\Set/I$. Similarly, the operation for “$\To$”; namely, “indexwise set of functions” is given by the exponential objects (Definition \[defn:exponential-obs\]) in this category.
As for the quantifiers, the associated operations take a family indexed by a product of sets $I\times{}J$, and give the product or disjoint sum over (say) the second factor, returning a family indexed by $I$. Hence, these should be described by functors $\Set/(I\times{}J)\to\Set/I$. In fact, we have natural functors $\Set/I\to\Set(I\times{}J)$, taking $p:X\to{}I$ to $p\times{}\id_J:X\times{}J\to{}I\times{}J$, and the functors in question turn out to be the right and left adjoints to this functor.
We conclude that we can carry out our definition of “Propositions-as-objects-of-$\C$” whenever: $\C$ has finite products; each slice $\C/I$ has finite products, coproducts, and exponentials; and each of the functors $(-\times{}\id_J):\C/I\to\C/(I\times{}J)$ has a left and right adjoint[^6]. These are precisely the so-called “locally cartesian closed categories with finite coproducts” ($\sSet$ is such a category; $\Top$ is not).
Actually, there is an important part of the “Propositions-as-sets” definition that we have yet to treat for a general category $\C$ – namely, the interpretation of equality. The categorical description of this is as follows. For a given set $X$, the $(X\times{}X)$-indexed family of sets $$\operatorname{Id}^X_{ab}=
\begin{cases}
\tm&a=b\\
\emptyset&\text{otherwise}
\end{cases}$$ is given by the diagonal map $\Delta_X:X\to{}X\times{}X$; hence, taking $X=M(A)$, this is the interpretation of the formula $x=y$ (with $x,y$ variables of sort $A$). In general, if the terms $s$ and $t$ have free variables $x_1,\ldots,x_n$, then $s=t$ should be interpreted as a family $X\to{}M(\vec{A})$ (where we write $M(\vec{A})$ for $M(A_1)\times\cdots{}M(A_n)$, with $n$ the length of $\vec{A}$); it is given by the following pullback. $$\label{eq:eq-as-pullback-def}
\begin{tikzcd}
M(\vec{A})\times_{M(A)\times{}M(A)}M(A)
\ar[r]\ar[d]\ar[rd, phantom, "\lrcorner", pos=0]&M(A)\ar[d, "\Delta_{M(A)}"]\\
M(\vec{A})\ar[r, "\br{M(s),M(t)}"]&M(A)\times{}M(A)
\end{tikzcd}$$
Since any locally cartesian closed category $\C$ has diagonal maps and pullbacks, we see that we can interpret equality in $\C$ as well.
Homotopical semantics {#subsec:htpical-semantics-direct}
---------------------
*However, for the homotopical semantics, this is not how we want to interpret equality!* Indeed, the whole point of the homotopical semantics is that we want to interpret equality as the *space of paths*.
Hence, for the homotopical semantics in $\sSet$, we interpret all the logical connectives as above, but we change the interpretation of equality; this is the first place where we really need to use simplicial sets (or some similar category of “spaces”) and not an arbitrary locally cartesian closed category with finite coproducts.
The one additional fact that we need about simplicial sets is that for each simplicial set $A$, there is another one $A^I$ – the “path-space of $X$” – which comes with a natural map $A^I\to{}A\times{}A$. Hence, to interpret equality, we repeat the construction (\[eq:eq-as-pullback-def\]), but replace $\Delta:A\to{}A\times{}A$ with this map $A^I\to{}A\times{}A$.
This completes the definition of the homotopical semantics.
In analogy to the set-based version, we say that a closed formula is *true* under the homotopical semantics if the associated simplicial set is non-empty. We also reiterate that, given a structure for our signature in the category of topological spaces, we can apply the singular simplicial set functor $\Sing:\Top\to\sSet$ and, since this functor preserves finite products, obtain a structure in $\sSet$. The interpretation of formulas with respect to this structure in $\Top$ is then defined to be the interpretation with respect to the resulting structure in $\sSet$.
Basic properties; invariance {#subsec:properties-and-invariance}
----------------------------
Having defined the homotopical semantics, we would now like to be able to say something about it.
The first, most obvious question is whether it is *sound*; i.e., is it the case that every closed formula which is deducible by the usual rules of first-order logic with equality is true under the homotopical semantics in every structure? The answer to this is *yes*, provided we restrict ourselves to *intuitionistic* first-order logic. This will follow more or less automatically from the fibrational formulation which we introduce below. That the semantics are *not* sound with respect to classical logic (i.e., the law of the excluded middle) will be shown in §\[subsec:examples\].
We next want to consider the question of *homotopy invariance*, which will be our chief occupation for most of the remainder of this paper. We first consider the analogous question – namely *isomorphism-invariance* – in the classical case.
An easy and well-known property of the classical semantics for first-order logic is that any two isomorphic structures satisfy the same closed formulas; more generally, if a formula has free-variables $x_1,\ldots,x_n$, and these are interpreted as elements $a_1,\ldots,a_n$ in one structure and as corresponding (under the isomorphism) elements $b_1,\ldots,b_n$ in the second structure, then the resulting truth values are the same. This is one justification of the intuitive idea that isomorphic structures have “all the same properties”. One proves this by an induction on the complexity of the formula.
We next consider the “propositions-as-sets” semantics defined above in §\[subsec:props-as-sets\]. Here, we can ask for a slightly stronger property; namely that, given two isomorphic structures, the two sets associated to any closed formula are *isomorphic* (and an obvious analogous property for general formulas). Again, this is easily proved by induction on the complexity of the formula.
Now for the homotopical semantics, we consider *homotopy-equivalence* of structures – i.e., a homotopy-equivalence between the underlying spaces of the structures (say, $h_A:M(A)\rightleftarrows{}N(A):k_A$) such that, for the interpretations $M(f):M(\vec{A})\to{}M(B)$ and $N(f):N(\vec{A})\to{}N(B)$ of each operation in the signature, the squares $$\begin{tikzcd}
M(\vec{A})\ar[r, "h_{\vec{A}}"]\ar[d, "M(f)"']&
N(\vec{A})\ar[d, "N(f)"]\\
M(B)\ar[r, "h"]&N(B)
\end{tikzcd}
\hspace{50pt}
\begin{tikzcd}
N(\vec{A})\ar[from=r, "k_{\vec{A}}"']\ar[d, "N(f)"']&
M(\vec{A})\ar[d, "M(f)"]\\
N(B)\ar[from=r, "k"']&M(B)
\end{tikzcd}$$ commute up to homotopy (where $h_{\vec{A}}$ is short for $h_{A_1}\times\cdots\times{}h_{A_n}$ and similarly with $k_{\vec{A}}$). In case these are topological structures, “homotopy” and “homotopy-equivalence” are to be understood in the usual way. In the case of simplicial sets, there is again a notion of homotopy and (hence) of homotopy-equivalence, and this is what is meant.
Now, the property we would like to hold is that, in this situation, every closed formula has the same truth value with respect to the two structures[^7]. Next, the natural statement corresponding to the above invariance property for the propositions-as-sets semantics is that the two simplicial sets associated to any closed formula with respect to the two structures are *homotopy-equivalent* (rather than isomorphic). The statement for general formulas is as follows. Such a formula is interpreted with respect to the two structures as morphisms $x:X\to{}M(\vec{A})$ and $y:Y\to{}N(\vec{A})$, respectively; we then demand that there be a homotopy-equivalence $X\simeq{}Y$ *over* the homotopy-equivalence $h:M(\vec{A})\rightleftarrows{}N(\vec{A}):k$ – i.e., that there be a homotopy-equivalence $p:X\rightleftarrows{}Y:q$, with associated homotopies $X\times{}I\to{}X$ and $Y\times{}I\to{}Y$ making the following squares (strictly) commute. $$\begin{tikzcd}
X\ar[r, "p"]\ar[d, "x"']&Y\ar[d, "y"]\\
M(\vec{A})\ar[r, "f"]&N(\vec{A})
\end{tikzcd}
\hspace{10pt}
\begin{tikzcd}
Y\ar[r, "q"]\ar[d, "r"']&X\ar[d, "x"]\\
N(\vec{A})\ar[r, "g"]&M(\vec{A})
\end{tikzcd}
\hspace{10pt}
\begin{tikzcd}
X\times{}I\ar[r, "h"]\ar[d, "x\times\id_I"']&X\ar[d, "x"]\\
M(\vec{A})\times{}I\ar[r]&M(\vec{A})
\end{tikzcd}
\hspace{10pt}
\begin{tikzcd}
Y\times{}I\ar[r, "k"]\ar[d, "y\times\id_I"']&Y\ar[d, "y"]\\
N(\vec{A})\times{}I\ar[r]&N(\vec{A})
\end{tikzcd}$$ (where the bottom vertical maps in the last two diagrams are the homotopies associated to the homotopy-equivalence $h:M(\vec{A})\rightleftarrows{}N(\vec{A})$).
We note that this homotopy-invariance property is a generalization of certain familiar facts from algebraic topology, such as that any continuous binary operation on a space which is homotopy-equivalent to a topological group is (not necessarily associative but) homotopy associative. This follows from the above general homotopy-invariance property since a binary operation on a space is homotopy-associative if and only if it satisfies (under the homotopical semantics) the sentence $$\forall{}x,y,z\ ((x\cdot{}y)\cdot{}z)=(x\cdot{}(y\cdot{}z)$$ (see §\[subsec:examples\]). In fact, this notion of “homotopy-invariant algebraic structures” is the beginning of a large and interesting story which plays a central central in modern homotopy theory (see, for example, [@boardmanvogt; @vogtbrave; @lurieha]).
In §\[subsec:examples\], we will give more examples of sentences and their interpretations under the homotopical semantics.
Fibrational formulation of the homotopical semantics {#sec:fibrational-formulation}
====================================================
We now introduce the “algebraic” or “functorial” (or “fibrational”) presentation of the homotopical semantics. Let us first review the idea of functorial semantics in general (see [@marquis-reyes-history] for a thorough history of these and related ideas).
Schematically, we might represent the general idea of semantics as follows: $$\begin{tikzpicture}
\path[draw,arrows=->,
shorten >=5pt,shorten <=5pt]
(0, 0) node[rectangle,draw,anchor=east] {Language} --
++(2.5,0) node[rectangle,draw,anchor=west] {The Universe}
node[midway,above] {Semantics};
\end{tikzpicture}$$ That is, we assign to each linguistic entity some other (not necessarily linguistic) entity, in some consistent manner. Now, Lawvere’s idea of “functorial semantics” [@lawverethesis] consists roughly in replacing each element of the above scheme as follows: $$\label{eq:cat-logic-schema}
\begin{tikzpicture}[baseline]
\path[draw,arrows=->,
shorten >=5pt,shorten <=5pt]
(0,0) node[rectangle,draw,anchor=east,align=center] {Some\\category} --
++(3,0) node[rectangle,draw,anchor=west,
align=center] {Some other category\\(probably $\Set$)}
node[midway,above] {Some functor};
\end{tikzpicture}$$ That is, the linguistic entities are gathered together into some categorical structure, the universe is gathered together into some other structure (somewhat miraculously, of the same kind), and then the semantics is mediated by a structure preserving map from the first structure to the second.
Propositional logic
-------------------
The simplest instance of this setup – which long predated Lawvere – arises in the case of propositional logic, in which very special kinds of categories come into play, namely Boolean (or – in the case of intuitionistic logic – Heyting) algebras. Here, given some set $\Sigma$ of propositional atoms, we take the category on the left of (\[eq:cat-logic-schema\]) to be the “Lindenbaum-Tarski algebra” $\mathrm{B}_\Sigma$ associated to $\Sigma$: the elements are equivalence classes propositions built (using $\top,\bot,\wedge,\vee,\To$) from the elements of $\Sigma$, in which $P\le{}Q$ if and only if $Q$ is derivable from $P$ by the rules of propositional logic.
The category on the right of (\[eq:cat-logic-schema\]) is then traditionally taken to be the 2-element Boolean algebra $\mathbf{2}$, and the functor is taken to be an arbitrary homomorphism of Boolean algebras.
### {#subsec:prop-logic-freeness}
What we want to emphasize here is that the Boolean (or Heyting) algebra $\mathrm{B}_\Sigma$ is *free* on $\Sigma$: there is a natural inclusion $\Sigma\hookrightarrow\mathrm{B}_\Sigma$ from $\Sigma$ into the underlying set of $\mathrm{B}_\Sigma$, such that any map $\Sigma\to{}B$ from $\Sigma$ to the underlying set of some Boolean (or Heyting) algebra admits a unique extension to a homomorphism $\mathrm{B}_\Sigma\to{}B$ of Boolean (or Heyting) algebras. In particular, any map $\Sigma\to\mathbf{2}$ – i.e., an “assignment of truth values” – extends uniquely to a homomorphism $\mathrm{B}_\Sigma$ – this is “evaluation by truth tables”.
### {#subsubsec:prop-logic-as-sets}
Above, we used that a Heyting algebra – being by definition a *poset* satisfying certain conditions – is in particular a *category*. We now observe that the conditions that a poset needs to satisfy to be a Heyting algebra are all such that they make sense for arbitrary categories and not just for posets (for example, the existence of finite meets and joins is the same as the existence of finite products and coproducts). Hence, we can consider the notion of a *non-posetal Heyting algebra*, defined so that a Heyting algebra is precisely a non-posetal Heyting algebra which is also a poset.
In particular, instead of the *free Heyting algebra* $\mathrm{B}_\Sigma$ considered above, we can consider the *free non-posetal Heyting algebra* $\C_\Sigma$ – i.e., a non-posetal Heyting algebra admitting a map $\Sigma\to\Ob\Sigma$ having the same universal property as above, but with respect to morphisms of free non-posetal Heyting algebras. This is Lambek’s “category of proofs” (see [@makkai-harnik-lauchli]), so called because the morphisms $P\to{}Q$ can be interpreted as (equivalence classes) of proofs of $Q$ from $P$.
If we now take $\mathbf{2}$ on the right side of (\[eq:cat-logic-schema\]) as before, we obtain the same notion of semantics, since every morphism $\C_\Sigma\to\mathbf{2}$ of non-posetal Heyting algebras factors through $\B_\Sigma$ (which is the “posetal reflection” of $\C_\Sigma$ – the poset obtained by identifying any two parallel morphisms). However, we can now put other non-posetal Heyting algebras on the right of (\[eq:cat-logic-schema\]) – for instance $\Set$. Doing this, we recover the “propositions-as-sets” semantics for propositional logic: a map $\Sigma\to\Ob\Set$ assigns to each atomic proposition $P$ a set – to be thought of as the set of “primitive proofs” of $P$ – and then the induced functor $\C_\Sigma\to\Set$ associates to each proposition a set according to the rules in (\[eq:propositions-as-sets-clauses\]) on p. .
### {#subsubsec:invariance}
We can now obtain a “baby” version of the *isomorphism invariance* property (see §\[subsec:properties-and-invariance\]), but one whose proof will serve as a template for proofs of similar properties below. Namely, one can show that $\C_\Sigma$ actually satisfies a stronger (“2-categorical”) universal property than the one which defines it.
Namely, the defining universal property says that the map $\Ob\operatorname{HAFun}(\C_\Sigma,\D)\to\Ob\operatorname{Fun}(\Sigma,\D)$ obtained by composing with the functor $\Sigma\hookrightarrow\C_\Sigma$ is a bijection, where $\mathrm{Fun}$ denotes the usual functor category, $\mathrm{HAFun}$ denotes the full subcategory thereof on the morphisms of non-posetal Heyting algebras, and where the set $\Sigma$ is considered as a discrete category. But what is also true is that the functor $\operatorname{HAFun}(\C_\sigma,\D)^{\mathrm{iso}}\to\operatorname{Fun(\Sigma,\D)}^{\mathrm{iso}}$ is an isomorphism of categories, where $(-)^\mathrm{iso}$ denotes the “maximal subgroupoid” – the subcategory containing only the isomorphisms.
In particular, any (pointwise) isomorphism between assignments $M,M':\Sigma\to\Set$ of sets to atomic propositions induces a natural isomorphism between the induced functors $M,M':\C_\Sigma\to\Set$ – thus the objects $M(P)$ and $M'(P)$ are isomorphic for every proposition $P$.
Of course, this isomorphism-invariance is easy to prove directly by induction, but the point is that we now have a more conceptual proof, which we can generalize.
Predicate logic {#subsec:intro-predicate-logic}
---------------
We now ask: if Heyting algebras were the appropriate kinds of structures to use in the scheme (\[eq:cat-logic-schema\]) in order to describe the semantics for propositional logic, what are the structures appropriate to the semantics for predicate logic?
There are different possible answers to this question (again, we refer the interested reader to [@marquis-reyes-history]), but the one which will be relevant for us is given by Lawvere’s notion of *hyperdoctrine*[^8] [@lawvereadj; @lawvereequality].
This involves *Grothendieck fibrations* (see [@sentai §2]). Given a (possibly multi-sorted) algebraic signature $\sigma$, we associate to it a fibration $\fibr{Form_\sigma}{Form_\sigma}{Tm_\sigma}$ (roughly) as follows (we describe this in detail in the appendix). The base category $\TM_\sigma$ is the finite product category associated by Lawvere to (the empty theory over) this signature: the objects are “contexts” – i.e., finite sequences of sorts of $\sigma$ – and the morphisms are given by sequences of terms of $\sigma$. The objects of $\cat{Form}_\sigma$ are first-order formulas over $\sigma$, and in particular, the *fiber* $\fib{Form_\sigma}^{\vec{A}}$ over a context $\vec{A}$ is a Heyting algebra, whose objects are formulas with free variables in the context $\vec{A}$, and with the ordering given by intuitionistically provable implication of formulas.
### {#subsubsec:props-as-sets-fib}
The “universe” in (\[eq:cat-logic-schema\]) is now given by the “subobject fibration” $\fibr{P(\Set)}{P(\Set)}{\Set}$ (see [@makkai-lauchli1 p. 349]), in which the fiber $\fib{P(\Set)}^A$ over a set $A$ is the power set $\mathcal{P}(A)$. This fibration has the same kind of structure as $\fib{Form}_\sigma$ – namely, it is an *$h^=$-fibration* (Definition \[defn:h-fibration\]) – and the arrow in (\[eq:cat-logic-schema\]) is now given by a morphism of $h^=$-fibrations $\fib{Form_\sigma}\to\fib{P(\Set)}$ (Definition \[defn:morphism-of-fibs\]). In particular, such a morphism associates to each sort of $\sigma$ a set, and to each formula $\phi$ with free variables $x_1,\ldots,x_n$ a subset of the set $A_1\times\cdots\times{}A_n$ (the “set of elements satisfying $\phi$”), where $A_i$ is the set associated to the sort of $x_i$. Moreover, this gives the correct interpretation of the formulas, since each of the logical connectives is given by a certain operation in $\fib{Form_\sigma}$ (or better, is characterized by a certain universal property), the corresponding operations in $\fib{P(\Set)}$ are given by the usual ones used to interpret the logical connectives (intersection, union, etc.), and a morphism of $h^=$-fibrations by definition preserves these operations.
As in §\[subsubsec:prop-logic-as-sets\], we again have a “non-posetal” version $\fibr{Pf_\sigma}{Pf_\sigma}{Tm_\sigma}$, in which the fibers of $\fib{Pf_\sigma}$ have the same objects as before, but are now non-posetal Heyting algebras (the morphisms between two formulas $\phi$ and $\psi$ being equivalence classes of proofs of $\psi$ from $\phi$). Again as in §\[subsubsec:prop-logic-as-sets\], if we keep the same fibration $\fib{P(\Set)}$ on the right-hand side of (\[eq:cat-logic-schema\]), we obtain the same semantics. But if we instead take the “codomain fibration” $\fibr{F(\Set)}{\Set^\to}{\Set}$ (see [@sentai §12]), we now recover the “proposition-as-sets” semantics from §\[subsec:props-as-sets\]. Indeed, in this case, a morphism $\fib{Pf_\sigma}\to\fib{F(\Set)}$ assigns to each formula over $\sigma$ with free variables $x_1,\ldots,x_n$ a map with codomain the corresponding product $A_1\times\cdots\times{}A_n$ – i.e., a family of sets over $A\times\cdots\times{}A_n$ – and the formulas are interpreted correctly for the same reason as above.
### {#section-1}
Again, the important feature of the fibration $\fib{Pf_\sigma}$ is that it is *free*, in fact in two senses. Firstly, the f.p. (finite product) category $\TM_\sigma$ admits an interpretation of the algebraic signature $\sigma$ (in the sense of Definition \[defn:signature\]), and it is the initial such f.p. category, in the sense that any other interpretation of $\sigma$ into an f.p. category $\C$ is obtained by composing the interpretation of $\sigma$ into $\TM_\sigma$ with a unique (up to isomorphism) f.p. functor from $\TM_\sigma\to\C$. Secondly, $\fib{Pf_\sigma}$ is the *free $h^=$-fibration over $\TM_\sigma$* (Definition \[defn:free-hfib\]), i.e. given any $h^=$-fibration $\fibr{C}CB$ and an f.p. functor $M:\TM_\sigma\to\B$ there is a unique (up to isomorphism) functor $\widehat{M}:\Pf_\sigma\to\C$ such that $(M,\widehat{M})$ is a morphism of $h^=$-fibrations $\fib{Pf}_\sigma\to\fib{C}$. Combining these two freeness properties, we have that any interpretation of $\sigma$ into $\Set$ gives rise to a unique (up to isomorphism) morphism $\fib{Pf_\sigma}\to\fib{F(\Set)}$ of $h^=$ fibrations.
### {#section-2}
We now come again to the question of *isomorphism invariance*. This will again result from our free structure having an additional, stronger universal property. Firstly, the interpretations of $\sigma$ into a finite product category $\D$ are the objects of a category $\fpints(\sigma,\D)$, in which the morphisms are homomorphisms of $\sigma$-structures in $\D$. The freeness property of $\TM_\sigma$ mentioned above is that the functor $\FPFun(\TM_\sigma,\D)\to\fpints(\sigma,\D)$ (where $\FPFun$ is the category of f.p. functors) induced by composing with the canonical interpretation of $\sigma$ into $\TM_\sigma$ is surjective on objects and “essentially injective” – i.e., any two objects with the same image are isomorphic. However, it is in fact an equivalence of categories.
The second freeness property of $\fib{Pf_\sigma}$ mentioned above amounts to saying that the restriction functor $h^=\operatorname{Mor}(\fib{Pf_\sigma,\fib{D})}\to\FPFun(\TM_\sigma,\B)$ (where $h^=\operatorname{Mor}$ denotes the category of morphisms of $h^=$-fibrations) is surjective on objects and “essentially injective”. Again, we have the stronger property that the induced functor $h^=\operatorname{Mor}(\fib{Pf_\sigma,\fib{D})}^{\operatorname{iso}}
\to\FPFun(\TM_\sigma,\B)^{\operatorname{iso}}$ is in fact an equivalence of categories[^9].
Combining these two freeness properties immediately yields the isomorphism-invariance property for the proposition-as-sets semantics. Indeed, they imply that any isomorphism of $\sigma$-structures – i.e., of objects in $\fpints(\sigma,\Set)$ – yields an isomorphism between the induced functors $\TM_\sigma\to\Set$, and an induced isomorphism between the induced functors $\Pf_\sigma\to\Set^\to$ lying over this. Unwinding the definitions, this amounts precisely to the desired isomorphism-invariance property.
We note that, in the above proof of isomorphism-invariance, the *only* thing we needed to know about the fibration $\fib{Pf}_\sigma$ was this (strengthened) freeness property. In this sense – and since moreover the universal property (even the weaker one) determines $\fib{Pf}_\sigma$ up to equivalence – the precise description of $\fib{Pf}_\sigma$ is not really important. However, in order to know that such a fibration in fact exists – and moreover, that the above “abstract” isomorphism-invariance proof really implies the actual isomorphism-invariance property for the propositions-as-sets semantics – we must explicitly construct the fibration $\fib{Pf}_\sigma$ along the lines sketched above, and show that it has the desired universal property (this we do in the appendix).
Homotopical semantics {#subsec:htpical-semantics-fib}
---------------------
Having seen the general setup of the “fibrational semantics”, the path toward adapting it to our homotopical semantics should be clear: simply replace the “target” fibration $\fib{F(\Set)}$ with one that is suitable to our purposes.
The most obvious choice would be $\fib{F(\Top)}$ – or rather, $\fib{F(\sSet)}$, since the former is not an $h^=$-fibration, while the latter is (since $\sSet$ is a locally cartesian closed category with finite coproducts – see Proposition \[prop:lccc-iff-hfib\]). As was the case in §\[subsec:htpical-semantics-direct\], this “almost” does the right thing, but not quite: the interpretation of equality is wrong.
In an $h^=$-fibration $\fibr{C}CB$, the *equality object* $\Eq_B\in\fib{C}^{B\times{}B}$ associated to an object $B\in\Ob\B$ is described by a certain universal property (see [@sentai §5]). In a codomain fibration $\fib{F(\C)}$, this is satisfied by the diagonal morphism $(\Delta:B\to{}B\times{}B)\in\Ob\fib{F(\C)}^{B\times{}B}$. As in §\[subsec:htpical-semantics-direct\], the consequence of this is that, in the resulting semantics, equality is interpreted as “identity”, and not “existence of a path”. Hence it seems, at first sight, that the framework of fibrational semantics, as it stands, will not serve our purposes. Fortunately, however, we can fix the situation by suitably modifying the fibration $\fib{F(\sSet)}$.
We recall that our desired equality object $\Eq_B$ is the path space $B^I\to{}B\times{}B$. Since this is not (in general) isomorphic to the diagonal $\Delta_B$, it cannot itself be an equality object. However, as we explain in [@sentai p.32] – and this is, in a sense, the fact at the heart of this work, and of Homotopy Type Theory – it is “almost” an equality object. The “almost” is because the universal property is only satisfied, in some sense, “up to homotopy”. We might therefore try to coerce the path space into having the desired universal property by “modding out by homotopy”.
Indeed, in [@sentai], we show that for any *right-proper model category* $\C$ (of which $\Top$ and $\sSet$ are both examples), the fibration $\fibr{HoF(\C_\fb)}{Ho(\C_\fb)^\to}{\C_\fb}$, obtained by passing to the homotopy category of each fiber in $\fib{F(\C)}$ and restricting to the fibrant objects, is a “$\wedgeq$-fibration”, and in §§\[subsec:kan-hfib\] and \[subsec:top-spaces\] below, we show that it is in fact an $h^=$-fibration for appropriate $\C$, including $\sSet$ and $\Top$. Moreover, in the case of $\C=\sSet$, the localization morphism $\fib{F(\C_\fb)}\to\fib{HoF(\C_\fb)}$ and the inclusion $\fib{F(\C_\fb)}\hookrightarrow\fib{F(\C)}$ are both morphisms of $h$-fibrations, which means that the “operations” in $\fib{HoF(\C_\fb)}$ corresponding to the logical connectives are computed as in $\fib{F(\C)}$ – i.e., according to the prescription in §\[subsec:htpical-semantics-direct\][^10]. In the case of $\Top$, we have that the morphism $\fib{HoF(\Top)}\to\fib{HoF(\Kan)}$ (where $\Kan:=\sSet_\fb$) induced by the singular set functor $\Top\to\Kan$ is a morphism of $h^=$-fibrations and induces an equivalence on each fiber, hence the interpretation in $\Top$ can be computed as in §\[subsec:htpical-semantics-direct\] by first passing to $\sSet$.
### {#subsubsec:fibrational-htpy-inv}
With this fibration in place, we can now deploy the above argument to obtain the isomorphism-invariance property for the homotopical semantics. But of course, what we are interested in is *homotopy*-invariance. It is not even clear how to *express*, let alone prove, homotopy invariance with this setup. In particular, the notion of *isomorphism* of $\sigma$-structures (in some category $\C$) was readily expressible by *natural isomorphism* of f.p. functors $\TM_\sigma\to\C$. But what of the notion of *homotopy-equivalence*?
One way to express homotopy-equivalence of $\sigma$ structures in $\Top$ or $\sSet$ is to use the *2-categorical* structure on these categories, in which the 2-cells are given by (homotopy-classes of) homotopies between maps. In fact, the notion of homotopy-equivalence of $\sigma$-structures in either of these categories is captured precisely by the notion of *pseudonatural equivalence* of functors $\TM_\sigma\to\Top$ (Definition \[defn:psnt\]).
Next, we use the alternative description of fibrations over a category $\C$ in terms of *pseudo-functors* $\C^\op\to\Cat$ (see [@sentai §9]), and in particular the fact that the pseudo-functor $\Top^\op\to\Cat$ (we can also use $\sSet$ here) associated to the fibration $\fib{HoF(\Top)}$ is compatible with the 2-categorical structure on $\Top$ (we prove this in [@sentai]). It follows from this that the fibration $\fib{HoF(\Top)}$ can be upgraded to a so-called *1-discrete 2-fibration* (Definition \[defn:1d2f\]) – and in particular, that the total category $\Ho(\Top)^\to$ can be give a 2-categorical structure. We can thus also consider pseudonatural equivalences between functors from $\Pf_\sigma\to\Ho(\Top)^\to$ lying over a given pseudonatural equivalence between functors $\TM_\sigma\to\Top$; and it turns out that (once we restrict to the category $\Top_\cf$ of spaces homotopy equivalent to a cell complex) these are given precisely by “fiberwise homotopy-equivalences” as in §\[subsec:properties-and-invariance\].
Hence, the proof – which is given in §\[sec:abstract-invariance-theorem\] – is finished by showing that the freeness property of $\fib{Pf_\sigma}$ extends to pseudonatural equivalence, i.e., that for any 1-discrete 2-fibration $\fibr{C}CB$ and any pseudonatural equivalence between functors $\TM_\sigma\to\B$, there is a pseudonatural equivalence lying over it between the induced functors $\Pf_\sigma\to\C$. This is done using a device which reduces it to the original freeness property.
There is one lingering blemish on this proof of homotopy-invariance. Namely, in the above proof of *isomorphism*-invariance, nothing at all was used about the fibration $\fib{F(\Set)}$ – the argument would have gone through with any other $h^=$-fibration in its place. However, when proving the homotopy-invariance, we have made use of the special 2-categorical structure occurring on $\Top$ (or $\sSet$). Hence, in order to set this argument in its proper context, we should (ideally) show that in *any* $h^=$-fibration $\fibr{C}CB$, the base category $\B$ carries a 2-categorical structure satisfying the necessary properties, and such that in the case of $\Top$ or $\sSet$, this recovers the usual 2-categorical structure. This is precisely what was done in [@sentai].
This ends our overview of the fibrational presentation of the homotopical semantics. We now must tie up the many loose ends in the above account. We must construct $\fib{Pf_\sigma}$ and show that it has the desired universal property (this we do in the appendix), we must prove that $\fib{HoF(\Kan)}$ and $\fib{HoF(\Top)}$ are really $h^=$-fibrations (this we do in Part \[sec:h-fib-of-spaces\]), and we must fill in the details of the above homotopy-invariance argument (this we do in Part \[sec:1d2f-and-inv-thm\]).
The $h^=$-fibration of spaces {#sec:h-fib-of-spaces}
=============================
We now formally introduce the concept of $h^=$-fibration which was discussed in Part \[sec:fibrational-formulation\].
As was mentioned there, there are two (classes of) of $h^=$-fibrations which are of interest to us: the “syntactic” – or “free” – ones, and the “semantic” ones, morphisms into which give rise to the various notions of semantics; in particular, we are interested in fibrations built out of some kind of spaces, thus giving rise to the homotopical semantics. Here in Part \[sec:h-fib-of-spaces\], we will only be defining the “semantic” fibrations; as for the former, we describe the relevant universal property in Part \[sec:1d2f-and-inv-thm\], and prove the existence in the appendix.
In fact, much of the work involved in defining the “semantic” fibration was already carried out in [@sentai]. There, we defined the fibration $\fib{HoF(\C_\fb)}$ for any right-proper model category $\C$, and showed that it was a $\wedgeq$-fibration, which was all we needed at that point. Hence, to see that it is an $h^=$-fibration, it remains to show that this fibration has the necessary extra structure – namely, that it supports the “logical” operations $\vee,\To,\forall,\exists$. In order to do this, we need to impose further restrictions on the model category $\C$ (see Definition \[defn:suitable\]). The category $\Top$ will not satisfy these restrictions, but we will still be able to show that $\fib{HoF(\Top)}$ is an $h^=$-fibration; the reason, essentially, is that $\Top$ is *Quillen-equivalent* to $\sSet$, which does satisfy the requirements.
$h^=$-fibrations
----------------
We will use the definitions and notation concerning fibrations from [@sentai]. In particular, we take everything from [@sentai Part II] for granted, and our present discussion of fibrations will, so to speak, continue from there. We will also use material from other parts of [@sentai], but we will indicate when we do so.
### {#section-3}
A fibration $\fib{C}$ has *fiberwise finite coproducts* if every fiber of $\fib{C}$ has finite coproducts. We have the notion of a coproduct diagram in a fiber of $\fib{C}$ being *stable* (under pullbacks), analogous the corresponding notion for products (see [@sentai Definition 4.2]).
We follow the conventions concerning coproducts from [@sentai §13.2], except that we use the symbols $\vee$ and $\bot$ instead of $+$ and $\init$ when the category under consideration is the fiber of some fibration. Also, we denote by $\inn_1$ and $\inn_2$ the coprojections into a coproduct.
### {#defn:exponential-obs}
We recall the definition of exponential objects.
Given objects $B,C$ in a category $\C$, an *exponential diagram based on $B$ and $C$* is a diagram $$\begin{tikzcd}[column sep=6pt, row sep=10pt]
&C^B\times{}B\ar[dl, "\pi_1"]\ar[dr, "\pi_2"']\ar[rr, "\expev"]&&C\\
C^B&&B&
\end{tikzcd}$$ in which $C^B\xot{\pi_1}C^B\times{}B\tox{\pi_2}B$ is a product diagram, and $\expev$ (the *evaluation morphism* of the *exponential object* $C^B$) has the following universal property: given any product $A\xot{\pi_1}A\times{}B\tox{\pi_2}B$ in $\C$ and any morphism $f:A\times{}B\to{}C$, there is a unique $\expind{f}:A\to{}C^B$ such that, with the induced morphism $\expind{f}\times\id_B:A\times{}B\to{}C^B\times{}B$, the following diagram commutes. $$\begin{tikzcd}
C^B\times{}B\ar[r, "\expev"]&C\\
A\times{}B\ar[u, "\tilde{f}\times\id_B"]\ar[ru, "f"']
\end{tikzcd}$$ In other words, the following composite must be a bijection. $$\Hom(A,C^B)\tox{(-)\times{}\id_B}
\Hom(A\times{}B,C^B\times{}B)\tox{\expev\circ}
\Hom(A\times{}B,C)$$ A category is *cartesian closed* if it has finite products, and there is an exponential object based on each pair of objects. It is *bicartesian closed* if it is cartesian closed and has finite coproducts. A functor between cartesian closed categories is *cartesian closed* if it preserves finite products and takes exponential diagrams to exponential diagrams, and a *bicartesian closed* functor is defined similarly.
We will generally use the above notation for exponential objects, except when the category in question is the fiber of some fibration, in which case we will write $B\To{}C$ instead of $C^B$.
A $\wedge$-fibration $\fib{C}$ has *fiberwise exponentials* if each fiber of $\fib{C}$ is cartesian closed. We have the notion of *stability (under pullbacks)* of exponential diagrams in fibers, analogous to that of product and coproduct diagrams.
### {#section-4}
Let $\fibr{C}CB$ be a fibration, $f:A\to{}B$ a morphism in $\B$, and $P\in\Ob\fib{C}^A$. A *$\prod$-diagram over $f$ based on $P$* is a diagram $$\begin{tikzcd}
f^*\prod_fP\ar[d, "\prodev"']\ar[r, "\ct"]&\prod_fP\\
P\\[-10pt]
A\ar[r, "f"]&B
\end{tikzcd}$$ in $\C$ in which $\ct$ is cartesian over $f$, and $\prodev$ lies over $A$ and has the following universal property: given any cartesian morphism $\ct:f^*Q\to{}Q$ over $f$ and any morphism $p:f^*Q\to{}P$, there is a unique morphism $\tilde{p}:Q\to\prod_fP$ over $B$ such that, with the induced morphism $f^*\tilde{p}:f^*Q\to{}f^*\prod_fP$, the following diagram commutes (actually, the trapezoid commutes by definition of $f^*\tilde{p}$, so the condition is just that the triangle commutes). $$\begin{tikzcd}
f^*Q\ar[ddr, "p"']\ar[dr, "f^*\tilde{p}"]\ar[rrr, "\ct"]&&&Q\ar[dl, "\tilde{p}"]\\
&f^*\prod_fP\ar[d, "\prodev"]\ar[r, "\ct"]&\prod_fP\\
&P\\[-10pt]
&A\ar[r, "f"]&B
\end{tikzcd}$$ In other words, the following composite must be a bijection. $$\Hom_{\fib{C}^B}(Q,\tprod_fP)\tox{f^*}
\Hom_{\fib{C}^A}(f^*Q,f^*\tprod_fP)\tox{\prodev\circ}
\Hom_{\fib{C}^A}(f^*Q,P)$$
We will usually use the above notation when dealing with $\prod$-diagrams.
### {#section-5}
Given a fibration $\fibr{C}CB$ and morphisms $g:C\to{}D$ and $k:B\to{}D$ in $\B$, we say that a $\prod$-diagram $$\begin{tikzcd}
g^*\prod_gP\ar[d, "\prodev"]\ar[r, "\ct"]&\prod_gP\\
P
\end{tikzcd}$$ over $g$ is *stable along $k$* if for every pullback diagram $$\begin{tikzcd}
A\ar[r, "f"]\ar[d, "h"']\ar[rd, phantom, "\lrcorner", pos=0]&B\ar[d, "k"]\\
C\ar[r, "g"]&D
\end{tikzcd}$$ in $\B$ and every commutative diagram $$\begin{tikzcd}
h^*g^*\prod_gP\ar[d, "h^*\varepsilon"']\ar[rr, "q"]\ar[dr, "\ct"']
&&[-50pt]k^*\prod_gP\ar[dr, "\ct"]&\\
h^*P\ar[rd, "\ct"']&g^*\prod_gP\ar[d, "\varepsilon"]\ar[rr, "\ct"]&&\prod_gP\\
&P&\\[-10pt]
A\ar[rr, "f"]\ar[rd, "h"']&&B\ar[rd, "k"]&\\
&C\ar[rr, "g"']&&D
\end{tikzcd}$$ in which the morphisms $\ct$ are cartesian and each morphism in $\C$ lies over the corresponding morphism in $\B$ as shown, the diagram $$\begin{tikzcd}
h^*g^*\prod_gP\ar[d, "h^*\varepsilon"']\ar[r, "q"]
&k^*\prod_gP&\\
h^*P&\\[-10pt]
A\ar[r, "f"]&B
\end{tikzcd}$$ is also a $\prod$-diagram. This condition is also known as the *Beck-Chevalley condition*.
### {#defn:h-fibration}
A fibration $\fibr{C}CB$ is a *$h$-fibration* if it satisfies the following four conditions.
1. $\fib{C}$ has stable fiberwise finite products and coproducts and exponentials.
2. $\B$ has finite products.
3. \[item:hfib-def-lift-cond\] For any product projection $\pi_2:A\times{}B\to{}B$ in $\B$ and any $P\in\Ob\fib{C}^{A\times{}B}$, there is a cocartesian lift of $\pi_2$ with domain $P$ and a $\prod$-diagram over $g$ based on $P$.
4. All cocartesian lifts of product projections and $\prod$-diagrams over product projections are stable along all morphisms.
$\fib{C}$ is an $h^=$-fibration if, in addition
1. Each diagonal $\Delta_B:B\to{}B\times{}B$ has a cocartesian lift with domain any terminal object $\top_{B}\in\fib{C}^B$.
### {#defn:morphism-of-fibs}
In [@sentai Definition 15.4], we give the definition of *morphism* of fibrations, and of $\wedge$-fibrations, over a given category $\B$. This is a special case of the following definition.
Given prefibrations $\fibr{C}CB$ and $\fibr{C'}{C'}{B'}$, a *morphism of prefibrations* $\fib{C}\to\fib{C}'$ is a pair $(\phi,\Phi)$, where $\phi:\B\to\B'$ and $\Phi:\C\to\C'$ are functors such that the square $$\begin{tikzcd}
\C\ar[r, "\Phi"]\ar[d, "\fib{C}"']&\C'\ar[d, "\fib{C'}"]\\
\B\ar[r, "\phi"]&\B'
\end{tikzcd}$$ commutes (strictly). We say that $(\phi,\Phi)$ is a morphism of prefibrations *over* $\phi$. If $\B=\B'$ and $\phi=\id_\B$, we may just write $\Phi$ instead of $(\id_B,\Phi)$, and we say in this case that $\Phi$ is *over $\B$*. If $\fib{C}$ and $\fib{C'}$ are fibrations, then $(\phi,\Phi)$ is a *morphism of fibrations* if $\Phi$ takes cartesian morphisms to cartesian morphisms.
Note that for each $A\in\Ob\B$, $(\phi,\Phi)$ induces a functor $\Phi:\fib{C}^A\to(\fib{C'})^{\phi{}A}$.
If $\fib{C}$ and $\fib{C'}$ are $*$-fibrations (where $*$ is one of $\wedge$, $h$, $\wedgeq$, $h^=$), we say that $(\phi,\Phi)$ is a *morphism of $*$-fibrations* it preserves the relevant structure: (i) in all cases, the induced functors on fibers should be f.p. (ii) if $*$ is $h$ or $h^=$, the induced functors on fibers should moreover be bicartesian closed, and $\Phi:\C\to\C'$ should preserve $\prod$-diagrams, and cocartesian morphisms, over product projections (iii) if $*$ is $\wedgeq$, $h$, or $h^=$, $\phi:\B\to\B'$ should preserve finite products (iv) if $*$ is $h^=$ or $\wedgeq$, $\Phi$ should preserve cocartesian lifts of diagonal morphisms with domain a terminal object.
### {#section-6}
Every $h^=$-fibration is a $\wedgeq$-fibration.
Referring to the definition of $\wedgeq$-fibration from [@sentai Definition 5.6], we see that we are only missing the Frobenius reciprocity and stability conditions. These follow from Propositions \[prop:frobexp\] and \[prop:beck-chev-swap\] below.
### {#prop:frobexp}
In a fibration $\fibr{C}CB$ with stable fiberwise products and exponentials, every cocartesian morphism satisfies Frobenius reciprocity[^11]. Given a commutative diagram $$\begin{tikzcd}
P\rac[r, "\tc"]&\sum_fP\\
P\wedge{}f^*Q\ar[r, "\tc\wwdge\ct"]\ar[u, "\pi_1"]\ar[d, "\pi_2"']&
\sum_fP\wedge{}Q\ar[u, "\pi_1"']\ar[d, "\pi_2"]\\
f^*Q\car[r, "\ct"]&Q\\[-10pt]
A\ar[r, "f"]&B
\end{tikzcd}$$ in $\C$ with $\ct$ cartesian, $\tc$ cocartesian, and the two sides product diagrams, we need to show that $\tc\!\wwdge\!\ct$ is cocartesian. By [@sentai Proposition 5.2] it suffices to show that for each $R\in\Ob\fib{C}^B$, the map $\circ(\tc\!\wwdge\!\ct):\Hom_{\fib{C}^B}(\sum_fP\wedge{}Q,R)\to\Hom_f(P\wedge{}f^*Q,R)$ is a bijection.
Choose an exponential object $Q\To{}R$, pullbacks $f^*R$ and $f^*(Q\To{}R)$, and a product $f^*(Q\To{}R)\wedge{}f^*R$. By Proposition \[prop:longprod\] below, the induced morphism $\ct\wwdge\ct:f^*(Q\To{}R)\wedge{}f^*Q\to{}(Q\To{}R)\wedge{}Q$ is cartesian. We also have an induced morphism $\cind{\varepsilon(\ct\wwdge\ct)}:f^*(Q\To{}R)\wedge{}f^*Q\to{}f^*R$ – i.e., the unique morphism over $A$ making the following diagram commute. $$\begin{tikzcd}[column sep=40pt]
f^*(Q\To{}R)\wedge{}f^*Q
\ar[r, "\ct\wwdge\ct"]
\ar[d, "\cind{\varepsilon(\ct\wwdge\ct)}"', dashed]
&(Q\To{}R)\wedge{}Q\ar[d, "\varepsilon"]\\
f^*R\car[r, "\ct"]&R.
\end{tikzcd}$$ It then follows from the stability of exponentials in $\fib{C}$ that the following is an exponential diagram. $$\begin{tikzcd}
&[-35pt]f^*(Q\To{}R)\wedge{}f^*Q\ar[rr, "\cind{\varepsilon(\ct\wwdge\ct)}"]
\ar[dl, "\pi_1"']\ar[dr, "\pi_2"]&[-35pt]&
f^*R\\
f^*(Q\To{}R)&&f^*Q
\end{tikzcd}$$
The claim now follows from the commutativity of the following diagram, and the fact that the vertical composites on the left and right are isomorphisms. Here, we abbreviate $\Hom_{\fib{C}^X}(\cdots)$ by $[\cdots]_X$ and $\Hom_f(\cdots)$ by $[\cdots]_f$. $$\begin{tikzcd}[column sep=35pt]
{[P\wedge{}f^*Q,f^*R]}_A\ar[r, "\sim"', "\ct\circ"]&
{[P\wedge{}f^*Q,R]}_f&
{[\sum_fP\wedge{}Q,R]}_B\ar[l, "\circ(\tc\wwdge\ct)"']\\
{[P\wedge{}f^*Q,f^*(Q\To{}R)\wedge{}f^*Q]}_A
\ar[r, "(\ct\wwdge\ct)\circ"]
\ar[u, "(\cind{\varepsilon(\ct\wwdge\ct)})\circ"]&
{[P\wedge{}f^*Q,(Q\To{}R)\wedge{}Q]}_f
\ar[u, "\varepsilon\circ"]&
{[\sum_fP\wedge{}Q,(Q\To{}R)\wedge{}Q]}_B
\ar[l, "\circ(\tc\wwdge\ct)"']
\ar[u, "\varepsilon\circ"']\\
{[P,f^*(Q\To{}R)]}_A
\ar[r, "\ct\circ", "\sim"']
\ar[u, "(-)\wedge{}\id_{f^*Q}"]&
{[P,Q\To{}R]}_f
\ar[u, "(-)\wwdge\ct"]&
{[\sum_fP,Q\To{}R]}_B
\ar[l, "\circ\tc"', "\sim"]
\ar[u, "(-)\wedge\id_Q"']
\end{tikzcd}
\tag*{\qed}$$
### {#prop:frobexp-converse}
The following converse of \[prop:frobexp\] holds: if $\fib{C}$ has fiberwise exponentials and stable fiberwise products, then for any morphism $f:A\to{}B$ in $\B$, if $f$ admits a cocartesian lift with domain $P$ satisfying Frobenius reciprocity for each $P\in\fib{C}^A$, then the exponential diagrams in $\fib{C}^B$ are stable along $f$. The proof is essentially the same as that of Proposition \[prop:frobexp\].
It suffices to show that for each exponential diagram (say, with exponential object $Q\To{}R$) in $\fib{C}^B$, *some* pullback along $f$ is an exponential diagram in $\fib{C}^A$. We choose one as in the proof of Proposition \[prop:frobexp\]. We then need to show the universal property for each $P\in\fib{C}^A$. We then choose a cocartesian morphism $\tc:P\to\sum_fP$, conclude by Frobenius reciprocity that the induced morphism $\tc\wwdge\ct$ is cocartesian, and now the commutativity of the last diagram in the proof of Proposition \[prop:frobexp\] gives us the desired universal property.
### {#prop:beck-chev-swap}
Let $\fibr{C}CB$ be a fibration, and let $g:C\to{}D$ and $k:B\to{}D$ be morphisms in $\B$. If there is a $\prod$-diagram over $k$ based on $Q$ that is stable along $g$ for each $Q\in\Ob\fib{C}^B$, then every cocartesian lift of $g$ is stable along $k$[^12]. Suppose we have a pullback diagram in $\B$ and a diagram lying over it, as shown below, with cartesian and cocartesian morphisms as indicated. $$\begin{tikzcd}
A\ar[r, "f"]\ar[d, "h"']\ar[rd, phantom, "\lrcorner", pos=0]&B\ar[d, "k"]\\
C\ar[r, "g"]&D
\end{tikzcd}
\hspace{50pt}
\begin{tikzcd}
h^*P\ar[r, "\cind{\tc\ct}"]\car[d, "\ct"']&
k^*\sum_gP\car[d, "\ct"]\\
P\rac[r, "\tc"]&\sum_gP
\end{tikzcd}$$ We need to show that $\cind{\tc\ct}$ is cocartesian. By [@sentai Proposition 5.2] it suffices to show that for each $Q\in\fib{C}^B$, the map $\circ\cind{\tc\ct}:\Hom_{\fib{C}^B}(k^*\sum_gP,Q)\to\Hom_f(h^*P,Q)$ is a bijection.
Choose a $\prod$-diagram over $k$ based on $Q$, as shown below, and choose pullbacks $f^*Q$, $f^*k^*\prod_kQ$ and $g^*\prod_kQ$, so that we have, by stability, an induced $\prod$-diagram over $h$. $$\begin{tikzcd}
k^*\prod_kQ\ar[r, "\ct"]\ar[d, "\varepsilon"']&\prod_kQ\\
Q\\[-10pt]
B\ar[r, "k"]&D
\end{tikzcd}
\hspace{50pt}
\begin{tikzcd}
f^*k^*\prod_kQ\ar[r, "\cind{\ct\ct}"]\ar[d, "f^*\varepsilon"']&g^*\prod_kQ\\
f^*Q\\[-10pt]
A\ar[r, "h"]&C
\end{tikzcd}$$ The claim now follows from the commutativity of the following diagram since the vertical composites on the left and right are isomorphisms. We use the same abbreviations as in the proof of \[prop:frobexp\]. $$\label{eq:swap-proof-diagram}
\begin{tikzcd}[column sep=60pt]
{[k^*\sum_gP,Q]}_B\ar[r, "\circ\cind{\tc\ct}"]&
{[h^*P,Q]}_f&
{[h^*P,f^*Q]}_A\ar[l, "\ct\circ"', "\sim"]\\
{[k^*\sum_gP, k^*\prod_kQ]}\ar[u, "\varepsilon\circ"]\ar[r, "\circ\cind{\tc\ct}"]&
{[h^*P,k^*\prod_kQ]}_f\ar[u, "\varepsilon\circ"]&
{[h^*P,f^*k^*\prod_kQ]}_A\ar[u, "f^*\varepsilon\circ"']
\ar[l, "\ct\circ"', "\sim"]\\
{[\sum_gP,\prod_kQ]}_D\ar[u, "k^*"]\ar[r, "\circ\tc", "\sim"']&
{[P,\prod_kQ]}_g\ar[u, "\cind{(-)\ct}"]&
{[P,g^*\prod_kQ]}_C\ar[l, "\ct\circ"', "\sim"]\ar[u, "\cind{\cind{\ct(-)\ct}}"']
\end{tikzcd}$$ The commutativity of the individual squares follows from [@sentai Propositions 3.4, 3.5, 3.6] and their duals (those propositions were stated in the context of a fixed cleavage, but they still hold in this context). Finally, we note that the strange-looking map $\cind{\cind{\ct(-)\ct}}$ is just the “pullback functor” – i.e., for $p:P\to{}g^*\prod_kQ$ over $C$, $\cind{\cind{\ct{}p\ct}}$ is the unique morphism over $A$ making the following diagram commute (this again follows from loc. cit. and the fact $\cind{\ct{}p\ct}=p\ct:h^*P\to{}g^*\prod_kQ$). $$\begin{tikzcd}[baseline=(QED.base)]
h^*P\ar[r, "\ct"]\ar[d, dashed, "\cind{\cind{\ct{}p\ct}}"']&P\ar[d, "p"]\\
f^*k^*\prod_kQ\car[r, "\cind{\ct\ct}"]&g^*\prod_kQ\\[-10pt]
A\ar[r, "h"]& |[alias=QED]| C
\end{tikzcd}
\tag*{\qed}$$
### {#prop:beck-chev-otherswap}
The converse of Proposition \[prop:beck-chev-swap\] holds: if there is a cocartesian lift of $g$ with domain $P$ that is stable along $k$ for every $P\in\Ob\fib{C}^C$, then every $\prod$-diagram over $k$ is stable along $g$. The proof is the same as that of Proposition \[prop:beck-chev-swap\]. There, we fixed a $P\in\Ob\fib{C}^A$ and a cocartesian lift of $g$ with domain $P$, and then for each $Q\in\Ob\fib{C}^B$, chose a $\prod$-diagram over $k$ based on $Q$. Now, we do the opposite, first fixing a $Q\in\Ob\fib{C}^B$ and $\prod$-diagram, and then, for each $P\in\Ob\fib{C}^A$, choosing a cocartesian lift. In the commutative diagram , we then have that the morphism $\cind{\tc\ct}$ is an isomorphism, and it follows that the vertical composite on the right is an isomorphism as desired.
### {#prop:prodf-is-sections}
Let $\C$ be a cartesian closed category. We recall the well-known way to compute the objects $\prod_{!_A}(X,A,x)\in\Ob(\C/\tm_{\C})\cong\C$ in the codomain fibration $\fib{F(\C)}$ as the “object of sections” of the morphism $x:X\to{}A$ (see [@elephant1 Lemma 1.5.2] and Remark \[subsubsec:adjoints\] below).
Let $A,Y\in\Ob\C$ and $(X,x)\in\Ob(\C/A)$. Also, fix a terminal object $\tm_\C$, a cartesian morphism $(\pi_1,!_A):(Y\times{}A,A,\pi_2)\to(Y,\tm_\C,!_Y)$ over $!_A$, and exponential objects $X^A$ and $A^A$.
Note that since $Y\xot{\pi_1}Y\times{}A\tox{\pi_2}A$ is a product, for any $p:Y\times{}A\to{}X$, we have an induced $\expind{p}:Y\to{}X^A$.
The diagram $$\begin{tikzcd}
(Y\times{}A,\pi_2)\ar[d, "p"']\ar[r, "\pi_1"]
&(Y,!_Y)&\\
(X,x)&\\[-10pt]
A\ar[r, "!_A"]&\tm_\C
\end{tikzcd}$$ is a $\prod_{!_A}$-diagram if and only if the following is a pullback square, where $\qq{\id_A}$ denotes the morphism induced by $\pi_2:\tm_\C\times{}A\to{}A$. $$\begin{tikzcd}
Y\ar[r, "\expind{p}"]\ar[d, "!_Y"']&X^A\ar[d, "x^A"]\\
\tm_\C\ar[r, "\qq{\id_A}"]&A^A
\end{tikzcd}$$
The first condition amounts to the composite $$\label{eq:sections-proof-map1}
\Hom_{\C/\tm_\C}((Z,!_Z),(Y,!_Y))\tox{\times{}A}
\Hom_{\C/A}((Z\times{}A,\pi_2),(Y\times{}A,\pi_2))\tox{p\circ}
\Hom_{\C/A}((Z\times{}A,\pi_2),(X,x))$$ being a bijection for each $Z\in\Ob\C$, while the second condition amounts to the map $$\label{eq:sections-proof-map2}
\Hom_\C(Z,Y)\tox{\br{\expind{p}\circ,!_Y\circ}}
\Hom_\C(Z,X^A)\times_{\Hom_\C(Z,A^A)}\Hom_\C(Z,\tm_\C)$$ being a bijection for each $Z\in\Ob\C$.
Clearly, the domains of (\[eq:sections-proof-map1\]) and (\[eq:sections-proof-map2\]) are in bijection with one another. The codomains are isomorphic, respectively, to the following sets: $$\set{p\in\Hom_\C(Z\times{}A,X)\mid{}xp=\pi_2}
\quad\text{and}\quad
\set{p\in\Hom_\C(Z,X^A)\mid{}x^A\cdot{}p=\qq{\id_A}\cdot!_Z}.$$ The canonical bijection $\Hom_\C(Z\times{}A,X)\cong\Hom_C(Z,X^A)$ restricts to a bijection between these sets.
The morphisms (\[eq:sections-proof-map1\]) and (\[eq:sections-proof-map2\]), together with the above two bijections, form a commuting square, and hence (\[eq:sections-proof-map1\]) is a bijection if and only if (\[eq:sections-proof-map2\]) is.
### {#subsubsec:adjoints}
It is easy to see (and well-known – see [@makkai-lauchli1]) that, given a fibration $\fibr{C}CB$ and a morphism $f:A\to{}B$ in $\B$, “the” associated pullback functor $f^*:\fib{C}^B\to\fib{C}^A$ has a left or right adjoint if and only there exists a choice, for each $P\in\Ob\fib{C}^A$ of a cocartesian lift of $f$ with domain $P$ or $\prod$-diagram over $f$ based on $P$, respectively.
To formulate this in a way which does not refer to any choices (and is hence more general in the absence of the axiom of choice) is to say there is always a canonical *anafunctor* (see [@avoiding-choice]) $f^*:\fib{C}^B\to\fib{C}^A$ and that this anafunctor has a left or right adjoint anafunctor (which is then also canonically defined) if and only if there exists, for each $P\in\Ob\fib{C}^A$, a cocartesian lift of $f$ with domain $P$ or $\prod$-diagram over $f$ based on $P$, respectively.
### {#defn:lccc}
A category $\C$ is *locally cartesian closed* if each slice category $\C/X$ is cartesian closed and $\C$ has a terminal object (so in particular, $\C\cong{}\C/\tm$ is itself cartesian closed).
### {#prop:lccc-iff-hfib}
$\fib{F(\C)}$ is an $h$-fibration (in fact, an $h^=$-fibration) if and only if $\C$ is locally cartesian closed and has finite coproducts. This amounts to various easy or well-known facts, all of which can be found in [@elephant1].
By [@sentai Proposition 12.5], we know that $\fib{F(\C)}$ is a fibration if and only if $\C$ has finite limits. If $\fib{F(\C)}$ also has fiberwise exponentials, then $\C$ is clearly locally cartesian closed (hence, in this direction, we actually need much less than $\fib{F(\C)}$ being an $h$-fibration).
In the other direction, if $\C$ is locally cartesian closed, then $\fib{F(\C)}$ clearly has fiberwise exponentials, and moreover these are stable by Proposition \[prop:frobexp-converse\], since, as noted in the proof of [@sentai Proposition 12.5], every morphism in $\C$ admits a cocartesian lift with any domain, and these satisfy Frobenius reciprocity. Since (as was also noted in loc. cit) all cocartesian morphisms in $\C^\to$ are stable, this also gives us the required stable cocartesian lifts of product projection.
To see that for every product projection (in fact, every morphism) $f:A\to{}B$, there is a $\prod$-diagram based on any $P\in\Ob\C/A$ – i.e., by Remark \[subsubsec:adjoints\], that $f^*$ has a right-adjoint – we first reduce to the case in which $B$ is terminal, by noting that there are canonical isomorphisms $(\C/A)\cong(\C/B)/(A,f)$ and $(\C/B)\cong(\C/B)/(B,\id_B)$ under which the functor $\sum_f:\C/A\to\C/B$ corresponds to the functor $\sum_f:(\C/B)/(A,f)\to(\C/B)/(B,\id_B)$, and hence their right adjoints $f^*$ correspond as well. When $B$ is terminal, we can construct a $\prod_f$-diagram as the “object of sections” of $f$ as in Proposition \[prop:prodf-is-sections\].
The stability of the $\prod_f$-diagrams in $\C^\to$ follows by Proposition \[prop:beck-chev-otherswap\] from the stability of the cocartesian morphisms.
Finally, coproducts in the fibers are given by $(A,a)+(B,b)=(A+B,[a,b])$, and these are preserved by the pullback functors since – by Remark \[subsubsec:adjoints\] – the latter have right adjoints.
$\fib{F_\fb(\Kan)}$ is an $h$-fibration
---------------------------------------
In [@sentai Proposition 16.3], we showed that for any model category $\C$, the fibration $\fib{F_\fb(\C)}$ – and hence also $\fib{F_\fb(\C_\fb)}$ (see Definition \[defn:fib-restrictions\]) – is a $\wedge$-fibration. We now want to show in the particular case $\C=\sSet$, that $\fib{F_\fb(\sSet_\fb)}=\fib{F_\fb(\Kan)}$ is in fact an $h$-fibration. To begin with, we isolate, in Definition \[defn:suitable\], the exact properties of $\sSet$ which allow the proof to go through.
### {#defn:suitable}
We call a model category $\C$ *suitable* if the following four conditions are satisfied.
1. \[item:suitable-rp\] $\C$ is *right-proper*, i.e. weak equivalences are closed under pullbacks along fibrations.
2. \[item:suitable-cof-monos\] The cofibrations are precisely the monomorphisms.
3. \[item:suitable-fib-coprods\] $[f,g]:A+B\to{}C$ is a fibration whenever $f:A\to{}C$ and $g:B\to{}C$ are, and the unique morphism $0\to{}A$ is always a fibration.
4. \[item:suitable-lcc\] $\C$ is locally cartesian closed (as a category).
We note that the standard model structure on simplicial sets (see [@quillen-ha II.3.14]) is suitable. Condition \[item:suitable-cof-monos\] holds by definition. Condition \[item:suitable-rp\] is non-trivial, but well-known, and follows from the existence of finite-limit and pullback preserving fibrant replacement functors (see, e.g. [@may-ponto p. 370]). As for \[item:suitable-lcc\], it is well-known that any presheaf category is locally cartesian closed (see [@elephant1 p. 48]).
To see \[item:suitable-fib-coprods\], note that the “horns” $\Lambda^n_k$ are all *connected*, in the sense that any two vertices are connected by a path of edges (in fact, for $n>2$, by a single edge). It follows that any morphism $\Lambda^n_k\to{}A+B$ to a coproduct must factor through one of the summands $A,B$. Now, for $[f,g]:A+B\to{}C$ to be fibrant, it must lift against each horn inclusion $\Lambda^n_k\to{}\Delta^n$. But if $f$ and $g$ are both fibrant, this follows immediately from the fact that any given morphism $\Lambda^n_k\to{}A+B$ factors through $A$ or $B$. The corresponding lifting problem for morphisms $0\to{}A$ is trivial, since there are no morphisms $\Lambda^n_k\to0$.
We suspect there are other interesting suitable model structures (we have in mind the so-called *Cisinski* model structures, which always satisfy (2) and (4)).
### {#prop:slice-suitable}
If $\C$ is a suitable model category, then any slice category $\C/A$ of $\C$, with its induced model structure, is also suitable. It is well-known (and easy to see) that each slice of a locally cartesian closed category is locally cartesian closed; this follows from the existence of the canonical isomorphisms $(\C/A)/B\cong\C/B$. Condition \[item:suitable-cof-monos\] is immediate since a morphism $(p,\id_A)$ in $\C/A$ is a monomorphism or a cofibration if and only if $p$ is. Condition \[item:suitable-rp\] is immediate since a square in a slice category $\C/A$ is a pullback square if and only if its image under the forgetful functor $\C/A\to\C$ is. Condition \[item:suitable-fib-coprods\] follows similarly since the forgetful functor preserves and creates coproducts.
### {#section-7}
In a suitable model category $\C$, every object is cofibrant. Hence $\C_\cfb=\C_\fb$, and $\Ho(\C_\fb)=\pi(\C_\cfb)$.
Since the cofibrations are the monomorphisms, this amount to checking that each morphism from the initial object is a monomorphism. It is well-known that this holds in any cartesian closed category (see [@elemelem p. 61]).
### {#prop:suitable-fib-exp}
Let $\C$ be a suitable model category, let $p:X\to{}Y$ be a fibration, and let $C$ be any object. We then have an induced map $p^C:X^C\to{}Y^C$. $p^C$ is a fibration. We must show that for any and solid commutative diagram $$\begin{tikzcd}
A\ar[d, "i"']\ar[r]&X^C\ar[d, "p^C"]\\
B\ar[r]\ar[ru, dashed]&Y^C
\end{tikzcd}$$ with $i$ a trivial cofibration, there exists a dashed morphism making the diagram commute.
Using the adjunction $(-\times{}C)\dashv(-)^C$, this is seen to be equivalent to the existence of a dashed morphism making the corresponding diagram $$\begin{tikzcd}
A\times{}C\ar[d, "i\times\id_C"']\ar[r]&X\ar[d, "p"]\\
B\times{}C\ar[r]\ar[ru, dashed]&Y
\end{tikzcd}$$ commute. For this, it suffices that $i\times\id_C$ be an trivial cofibration. Noting that it is the pullback of $i$ along the projection $B\times{}C\to{}B$ (which is a fibration since $C$ is fibrant), we have that $i\times\id_\C$ is a cofibration since monomorphisms are stable under pullback, and is a weak-equivalence, since $\C$ is right-proper.
### {#prop:suitable-fb-bcc}
Let $\C$ be a suitable model category. We know from [@sentai Proposition 16.2] that $\C_\fb$ is an f.p. category and that the inclusion $\C_\fb\hookrightarrow\C$ is an f.p. functor. $\C_{\fb}$ and the inclusion $\C_{\fb}\hookrightarrow\C$ are bicartesian closed. Since $\C_\fb$ is a full subcategory of $\C$, it suffices to show that the fibrant objects in $\C$ are closed under exponentials and coproducts.
Condition \[item:suitable-fib-coprods\] of “suitable” (Definition \[defn:suitable\]) implies that the fibrant objects are closed under coproducts. That they objects are closed under exponentials follows from Proposition \[prop:suitable-fib-exp\].
### {#defn:fib-restrictions}
In [@sentai Definition 17.7], for a model category $\C$ and a full subcategory $\D\subseteq\C$, we defined $\fib{HoF_*(\D)}$ to be the restriction of $\fib{HoF_*(\C)}$ to $\D$. We similarly define $\fib{F_*(\D)}$ to be the restriction of $\fib{F_*(\C)}$ to $\D$, and we denote the total categories of the fibrations $\fib{F_*(\D)}$ and $\fib{HoF_*(\D)}$ by $(\D^\to)_*$ and $\Ho(\D^\to)_*$, respectively[^13].
We note that, in general, the restriction of a $\wedgeq$-fibration, $h$-fibration or $h^=$-fibration to any full subcategory having finite products is again an $\wedgeq$-fibration, $h$-fibration or $h^=$-fibration.
### {#prop:suitable-mfib-hfib}
Let $\C$ be a suitable model category. Since $\C$ is locally cartesian closed, we know by Proposition \[prop:lccc-iff-hfib\] that $\fib{F(\C)}$, and hence $\fib{F(\C_\fb)}$, is an $h$-fibration, and by [@sentai Proposition 16.3], we know that $\fib{F_\fb(\C_\fb)}$ is a $\wedge$-fibration, and that the inclusion $\fib{F_\fb(\C_\fb)}\hookrightarrow\fib{F(\C_\fb)}$ is a morphism of $\wedge$-fibrations. $\fib{F_\fb(\C_\fb)}$ is an $h$-fibration and the inclusion $\fib{F_\fb(\C_\fb)}\hookrightarrow\fib{F(\C_\fb)}$ is a morphism of $h$-fibrations.
It follows from Propositions \[prop:slice-suitable\] and \[prop:suitable-fb-bcc\] that the fibers of $\fib{F_\fb(\C_\fb)}$, and the functors on fibers induced by the inclusion, are bicartesian closed.
Next, we need to check that, given a product projection $\pi_2:A\times{}B\to{}B$ in $\C_{\fb}$ and a cocartesian morphism $(p,\pi_2):(X,A\times{}B,x)\to{}(Y,B,y)$ in $\C^\to$ lying over $\pi_2$, if $(X,x)$ is in $\fib{F_\fb(\C_\fb)}^{A\times{}B}$, then $(Y,y)$ is in $\fib{F_\fb(\C_\fb)}^B$ – i.e., if $x$ is a fibration, then so is $y$. But the product projection $\pi_2$ is a fibration since $A$ is fibrant, and by [@sentai Proposition 12.3], $p$ is an isomorphism, hence $y=\pi_2xp\I$ is a fibration as well.
Similarly, we need to check that if $(X,x)$ is a fibration, then $\prod_{\pi_2}(X,x)$ is a fibration (actually, for this, we don’t need $\pi_2$ to be a product projection). In the case where $B\cong\tm_\C$, this follows from Propositions \[prop:prodf-is-sections\] and \[prop:suitable-fib-exp\] since fibrations are stable under pullback. The general case is reduced to this case, as in the proof of Proposition \[prop:lccc-iff-hfib\], by using the isomorphism $(\C/A\times{}B)\cong(\C/B)/(A\times{}B,\pi_2)$ and Proposition \[prop:slice-suitable\].
It remains to see that all the operations are “stable”, i.e., that the pullback functors are bicartesian closed, and that the cocartesian morphisms and $\prod_f$-diagrams over projections are stable. In each case, this follows immediately from the corresponding fact in $\fib{F(\C_\fb)}$.
$\fib{HoF_\fb(\Kan)}$ is an $h^=$-fibration {#subsec:kan-hfib}
-------------------------------------------
### {#prop:suitable-hofb-bcc}
Let $\C$ be a suitable model category. We know from Proposition \[prop:suitable-fb-bcc\] that $\C_{\fb}$ is bicartesian closed, and we know from [@sentai Proposition 16.4] that $\Ho(\C_\fb)$ is an f.p. category and that the functor $\gamma:\C_\fb\to\Ho(\C_\fb)$ is an f.p. functor. The category $\C_{\fb}$ and the functor $\gamma:\C_\fb\to\Ho(\C_\fb)$ are bicartesian closed. That $\Ho(\C_\fb)$ has, and $\gamma$ preserves, finite coproducts, follows from an argument dual to the one given in [@sentai Proposition 16.4].
We next turn to exponentials. Consider an exponential diagram $$\begin{tikzcd}[column sep=6pt, row sep=10pt]
&C^B\times{}B\ar[dl, "\pi_1"]\ar[dr, "\pi_2"']\ar[rr, "\expev"]&&C\\
C^B&&B.&
\end{tikzcd}$$ We already know that $C^B\times{}B$ is still a product in $\Ho(\C_\fb)$, so it remains to see that for any $A\in\Ob\C_\fb$ and product $A\xot{\pi_1}A\times{}B\tox{\pi_2}B$, the composite $$\pi(A,C^B)\tox{(-)\times\id_B}\pi(A\times{}B,C^B\times{}B)\tox{\expev\circ}\pi(A\times{}B,C)$$ is a bijection. That this map is surjective is clear, since it is already surjective before passing to homotopy classes. To show injectivity, we need to show that if two morphisms $f_1,f_2:A\times{}B\to{}C$ are homotopic, then the corresponding morphisms $\expind{f_1},\expind{f_2}:A\to{}C^B$ are.
Let $A+A\tox{[\partial_1,\partial_2]}A\times{}I\tox{\sigma}A$ be a cylinder object for $A$ with $[\partial_1,\partial_2]$ a cofibration. Because the functor $(-\times{}B)$ is a left-adjoint, it preserves coproducts, and hence the canonical morphism $[\br{\inn_1\pi_1,\pi_2},\br{\inn_2\pi_1,\pi_2}]:A\times{}B+A\times{}B\to(A+A)\times{}B$ is an isomorphism. Applying $(-\times{}B)$ to our cylinder object for $A$, we have a sequence of morphisms $$A\times{}B+A\times{}B\tox{\sim}
(A+A)\times{}B\tox{[\partial_1,\partial_2]\times\id_B}
(A\times{}I)\times{}B\tox{\sigma\times\id_B}
A\times{}B$$ and we claim that this exhibits $(A\times{}I)\times{}B$ as a cylinder object for $A\times{}B$. Indeed, the composite is clearly equal to $\nabla_{A\times{}B}$, and $\sigma\times\id_B$ is a weak equivalence by the right-properness of $\C$, since it the pullback of a weak equivalence along the projection $(A\times{}I)\times{}B\to{}A\times{}I$, which is a fibration since $B$ is fibrant. Moreover, the first two morphisms are cofibrations (the first being an isomorphism and the second being the pullback of a monomorphism and hence a monomorphism).
Hence, by [@sentai Proposition 13.5 (ii)], given two homotopic maps $f_1,f_2:A\times{}B\to{}C$, we obtain a left-homotopy $h:(A\times{}I)\times{}B\to{}C$ between them, and hence a morphism $\expind{h}:A\times{}I\to{}C^B$. It remains to see that this is a homotopy between $\expind{f_1}$ and $\expind{f_2}$, i.e. that $\expind{h}\partial_i=\expind{f_i}:A\to{}C^B$. It suffices to see that $\expev\cdot((\expind{h}\partial_i)\times\id_B)=f_i$, which follows from the definition of $h$.
### {#thm:suitable-hfib-hfib}
Let $\C$ be a suitable model category. By Proposition \[prop:suitable-mfib-hfib\], we know that $\fib{F_\fb(\C_\fb)}$ is an $h$-fibration, and by [@sentai Propositions 16.5 and 17.8], we know that $\fib{HoF_\fb(\C_\fb)}$ is a $\wedge=$-fibration and that $\gamma:\fib{F_\fb(\C_\fb)}\to\fib{HoF_\fb(\C_\fb)}$ is a morphism of $\wedge$-fibrations.
$\fib{HoF_\fb(\C_\fb)}$ is in fact an $h^=$-fibration, and the localization morphism $\gamma:\fib{F_\fb(\C_\fb)}\to\fib{HoF_\fb(\C_\fb)}$ is a morphism of $h$-fibrations. By Proposition \[prop:suitable-hofb-bcc\], we know that the fibers of $\fib{HoF_\fb(\C_\fb)}$ and the functors on the fibers induced by $\gamma:\fib{F_\fb(\C_\fb)}\to\fib{HoF_\fb(\C_\fb)}$ are bicartesian closed.
Next, we consider sum objects $\sum_{\pi_2}P$. That is, we need to show that the image under $\gamma$ of any cocartesian morphism in $(\C_\fb^\to)_\fb$ over a product projection is cocartesian in $\Ho(\C_\fb^\to)_\fb$. This follows from [@sentai Propositions 12.3 and 17.2] and the fact that every isomorphism is a weak equivalence.
We next consider product objects $\prod_{\pi_2}P$. Let $\pi_2:A\times{}B\to{}B$ be a product projection in $\C_\fb$, and let $$\begin{tikzcd}
\pi_2^*\prod_{\pi_2}P\ar[r, "\ct"]\ar[d, "\expev"']&\prod_{\pi_2}P\\
P\\[-10pt]
A\times{}B\ar[r, "\pi_2"]&B
\end{tikzcd}$$ be a $\prod$-diagram in $(\C_\fb^\to)_\fb$. We need to see that its image in $\Ho(\C^\to)_\fb$ is a also a $\prod_{\pi_2}$-diagram.
We know already that the image of $\ct$ is cartesian. Hence, it remains to show that for each $Q\in(\C/B)_\fb$ and cartesian $\ct:f^*Q\to{}Q$ over $\pi_2$, the composite $$\pi(Q,\tprod_{\pi_2}P)
\tox{\pi_2^*}
\pi(\pi_2^*Q,\pi_2^*\tprod_{\pi_2}P)
\tox{\expev\circ}
\pi(\pi_2^*Q,P)$$ is a bijection. As in the proof of Proposition \[prop:suitable-fb-bcc\], it is immediate that it is surjective, and injectivity follows by a similar argument to the one there.
It remains to check the various “stability” conditions for $\fib{HoF_\fb(\C)}$. These are proven in the same way as the stability of products in [@sentai Proposition 16.5]. Namely, in each case, we reduce to showing the stability of *some* (rather than every) diagram of the appropriate kind, and then we choose the diagram coming from $(\C_\fb^\to)_\fb$, where we already know that stability holds.
Topological spaces {#subsec:top-spaces}
------------------
We now explain how to extend the above considerations in order to include the category $\Top$ of topological spaces – which, as it stands, is excluded, since $\Top$ is not a suitable model category (Definition \[defn:suitable\]), as it is not locally cartesian closed. Here, we are considering $\Top$ to be endowed with the mixed model structure (see [@sentai §19.1]).
In particular, we cannot directly apply Theorem \[thm:suitable-hfib-hfib\] in the case of topological spaces. However, it turns out that the fibration $\fib{HoF_\fb(\Top)}$ *is*, after all, an $h^=$-fibration.
The reason is that the singular simplicial set functor $\Sing:\Top\to\sSet$ gives us a morphism of fibrations $\fib{F_\fb(\Top)}\to\fib{F_\fb(\sSet)}$ and hence, upon passage to homotopy categories, a morphism of fibrations $\fib{HoF_\fb(\Top)}\to\fib{HoF_\fb(\sSet)}$. This turns out be a fiberwise equivalence, which then implies that $\fib{F(\Top_\cf)}$ is an $h^=$-fibration and that that the morphism is a morphism of $h^=$-fibrations.
In practice, we will want to consider the cofibrant objects in $\Top$ (i.e., the spaces homotopy equivalent to a cell complex), so we will use the fibration $\fib{HoF_{\cfb}(\Top_\cf)}$; this is equivalent to the fibration $\fib{Ho_\fb(\Top_\cf)}$, which is an $h^=$ fibration since $\Top_\cf\subseteq\Top$ has finite products.
### {#prop:induced-slice-adjunction}
Let $\C$ and $\D$ be categories, $F:\C\rightleftarrows\D:U$ functors, and $(\eta,\varepsilon):F\dashv{}U$ an adjunction. Recall from [@sentai §14.4] that, for each morphism $f:A\to{}B$ in $\D$, we have a functor $\sum_f:\D/A\to\D/B$. Note also that, for each $A\in\Ob\D$ the functor $U$ induces a functor $U:\D/A\to\C/UA$ sending $(X,x)$ to $(UX,Ux)$ and $p:(X,x)\to(Y,y)$ to $Up:(UX,Ux)\to(UY,Uy)$, and similarly $F$ induces a functor $F:\C/UA\to\D/FUA$.
For each $A\in\Ob\D$, the functor $U:\D/A\to\C/UA$ is right adjoint to the composite functor $\C/UA\tox{F}\D/FUA\tox{\sum_{{\scriptstyle\varepsilon}_A}}D/A$. Let us show that, for $(X,x)\in\Ob\D/A$ and $(Z,z)\in\Ob\C/UA$, we have a natural isomorphism $$\label{eq:slice-bijection}
\Hom_{\C/UA}((Z,z),U(X,x))
\cong
\Hom_{\D/A}(\tsum_{\varepsilon_a}F(Z,z),(X,x)).$$ These two sets are in bijection with certain subsets of $\Hom_\C(Z,UX)$ and $\Hom_\D(FZ,X)$, respectively. Now, the adjunction $F\dashv{}U$ gives us a bijection between the two latter sets, and it is easily seen (using the naturality of this bijection) that it restricts to give a bijection (\[eq:slice-bijection\]). It remains to see that this bijection is natural. This follows from the fact that it is equal to the following composite of natural transformations: $$\begin{split}
\Hom_{\C/UA}((Z,z),U(X,x))
&\tox{F}
\Hom_{\D/FUA}(F(Z,z),FU(X,x))\\
&\tox{\tsum_{{\scriptstyle\varepsilon}_A}}
\Hom_{\D/FUA}(\tsum_{\varepsilon_A}F(Z,z),\tsum_{\varepsilon_A}FU(X,x))\\
&\tox{\varepsilon_X\circ}
\Hom_{\D/A}(\tsum_{\varepsilon_a}F(Z,z),(X,x)).
\end{split}$$
### {#section-8}
Given model categories $\C$ and $\D$, a functor $U:\D\to\C$ is a *right Quillen functor* if it is a right adjoint and preserves fibrations and trivial fibrations. $U$ is a *right Quillen equivalence* if it is a right Quillen functor, and admits a left adjoint $F:\C\to\D$ for which the associated bijections $\Hom_{\D}(FA,B)\cong\Hom_{\C}(A,UB)$ preserve weak equivalences (i.e., an element on one side of this bijection is a weak equivalence if and only if the corresponding element on the other side is) whenever $A$ is cofibrant and $B$ is fibrant.
The important fact about Quillen equivalences, for our purposes, is that, if $U$ is a right Quillen equivalence and preserves weak equivalences, then the induced functor $U:\Ho(\D)\to\Ho(\C)$ is an equivalence of categories (see [@hovey]).
### {#prop:induced-slice-quillen-equiv}
For any right Quillen equivalence $U:\D\to\C$ and any fibrant $A\in\Ob\D$, the induced functor $U:\D/A\to\C/UA$ is a right Quillen equivalence with respect to the induced model structures. By Proposition \[prop:induced-slice-adjunction\], we know that $U:\D/A\to\C/UA$ is a right adjoint. It clearly preserves fibrations and trivial fibrations, since a morphism $p:(X,x)\to(Y,y)$ is a fibration or trivial fibration if and only if $p:X\to{}Y$ is, in which case $Up:UX\to{}UY$, and hence $Up:U(X,x)\to{}U(Y,y)$, is also a fibration or trivial fibration, since $U$ is a right Quillen functor. Hence $U:\C/A\to\D/UA$ is a right Quillen functor.
To see that $U$ is a right Quillen equivalence, note that in the proof of Proposition \[prop:induced-slice-adjunction\], the bijection $\Hom_{\C/UA}(F(Z,z),(X,x))\cong\Hom_{\D/A}((Z,z),U(X,x))$ establishing $U:\C/A\to\D/UA$ as a right adjoint is a restriction of the bijection $\Hom_\C(FZ,X)\cong\Hom_\D(Z,UX)$. Now, if $(Z,z)$ is cofibrant in $\D/A$, then $Z$ is cofibrant in $\D$, and if $(X,x)$ is fibrant in $\C/A$, then $X$ is fibrant in $\C$ since $A$ is fibrant. Hence, since $U$ is a right Quillen equivalence, the latter bijection preserves weak equivalences, and hence the former bijection preserves weak equivalences as well.
### {#section-9}
A morphism $(\phi,\Phi):\fibr{C}CB\to\fibr{C'}{C'}{B'}$ of prefibrations is a *fiberwise equivalence* if the induced functor $\fib{C}^A\to\fib{C'}^{\phi{}A}$ is an equivalence for each $A\in\Ob\B$. If $\fib{C}$ and $\fib{C'}$ are fibrations, $\B=\B'$, $\phi=\id_\B$, and $\Phi$ is a morphism of fibrations over $\B$, this is the same as $\Phi:\C\to\C'$ being an equivalence of categories.
### {#prop:fib-equiv-is-hfib-mor}
If $(\phi,\Phi):\fibr{C}CB\to\fibr{C'}{C'}{B'}$ is a morphism of fibrations which is a fiberwise equivalence, $\B$ and $\B'$ are f.p. categories and $\phi$ an f.p. functor, and $\fib{C'}$ is an $h^=$-fibration, then $\fib{C}$ is an $h^=$-fibration and $(\phi,\Phi)$ is a morphism of $h^=$-fibrations. This follows from the considerations in Definition \[defn:pullback-fib\] below, since if $(\phi,\Phi)$ is a fiberwise equivalence, then the induced morphism $\cind{\Phi}:\fib{C}\to{}F^*\fib{C'}$ of fibrations over $\B$ is an equivalence. It then remains to see that a fibration equivalent to an $h^=$-fibration is again one (and the equivalence is a morphism of $h^=$-fibrations).
### {#section-10}
The singular simplicial set functor $\Sing:\Top\to\sSet$ is a right Quillen equivalence and preserves weak equivalences (in fact, we will not define $\Sing$, as this is all we will need about it). That $\Sing$ is a right Quillen equivalence when $\Top$ has the Quillen model structure is well-known, and is in fact the archetypical example of a Quillen equivalence (see, e.g., [@may-ponto Theorem 17.5.2]); for the case of the mixed model structure, see [@may-ponto Theorem 17.4.2]. That $\Sing$ preserves weak equivalences follows from [@may-ponto Corollary 17.5.11] and the definition of weak equivalences in $\sSet$.
### {#thm:top-sset-comparison}
Since $\Sing:\Top\to\sSet$ preserves fibrations (being a right Quillen functor), it induces a morphism $\fib{F_\fb(\Top)}\to\fib{F_\fb(\sSet)}$ of prefibrations, which is a morphism of fibrations since $\Sing$ is right adjoint and hence preserves pullback squares. We can then restrict to obtain a morphism of fibrations $\fib{F_\cfb(\Top_\cf)}\to\fib{F_\fb(\sSet)}$. The functor $\Sing:(\Top_\cf^\to)_\cfb\to(\sSet^\to)_\fb$ preserves vertical weak equivalences, and hence descends to localizations, giving a morphism $\fib{HoF_\cfb(\Top_\cf)}\to\fib{HoF_\fb(\sSet)}$ of prefibrations.
$\fib{HoF_\cfb(\Top_\cf)}$ is an $h^=$-fibration, and the morphism $\fib{HoF_\cfb(\Top_\cf)}\to\fib{HoF_\fb(\sSet)}$ induced by $\Sing$ is a morphism of $h^=$-fibrations. This will follow from Proposition \[prop:fib-equiv-is-hfib-mor\] once we show that the morphism in question is a morphism of fibrations and a fiberwise equivalence.
The inclusion $\fib{HoF_\cfb(\Top_\cf)}\hookrightarrow\fib{HoF_\cfb(\Top)}$ is obviously a morphism of fibrations and fiberwise equivalence, and the inclusion $\fib{HoF_\cfb(\Top)}\hookrightarrow\fib{HoF_\fb(\Top)}$ is a morphism of fibrations and fiberwise equivalence by [@sentai §§13.6,15.7]. Hence, it remains to consider $\fib{HoF_\fb(\Top)}\to\fib{HoF_\fb(\sSet)}$.
To see that it is a morphism of fibrations, note that any cartesian morphism in $\fib{HoF_\fb(\Top)}$ is isomorphic to one which is in the image of the localization morphism $\fib{F_\fb(\Top)}\to\fib{HoF_\fb(\Top)}$, hence its image in $\fib{F_\fb(\sSet)}$ will be cartesian, since we have a commuting square $$\begin{tikzcd}
\fib{F_\fb(\Top)}\ar[r]\ar[d]&\fib{F_\fb(\sSet)}\ar[d]\\
\fib{HoF_\fb(\Top)}\ar[r]&\fib{HoF_\fb(\sSet)}
\end{tikzcd}$$ where the top and right morphisms are morphisms of fibrations.
That it is a fiberwise equivalence follows Proposition \[prop:induced-slice-quillen-equiv\].
1-discrete 2-fibrations and the invariance theorem {#sec:1d2f-and-inv-thm}
==================================================
We now formulate and prove the “abstract” version of the homotopy-invariance property of the homotopical semantics, and deduce from it the concrete version.
The main new ingredient which will be involved in this is that of *1-discrete 2-fibrations* (Definition \[defn:1d2f\]). Let us say something about the significance of this notion.
We have already seen, in [@sentai §9.2], part of the correspondence between Grothendieck fibrations and pseudo-functors – namely, that a pseudo-functor $\fpsf{C}:\C^\op\to\Cat$ can be associated to a cloven fibration over $\C$. One can also go the other way, and construct a fibration from a given pseudo-functor – this is known as the *Grothendieck construction*. These operations are inverse to each other, in the sense that they establish an equivalence between naturally defined 2-categories of pseudo-functors $\C^\op\to\Cat$ and of fibrations over $\C$.
By applying the Grothendieck construction to the identity functor $\Cat\to\Cat$, one obtains a *universal fibration* over $\Cat^\op$ (in which the fiber over a category $\D$ is $\D$ itself), so that the Grothendieck construction can be described as simply pulling back the universal fibration along a given pseudo-functor $\C\to\Cat^\op$ (however, note that this is not entirely trivial to make precise, as one must clarify the notion of pulling back a fibration along a pseudo-functor).
There is a simpler and well-known version of the Grothendieck construction, in the case that the given pseudo-functor $\C^\op\to\Cat$ is valued in *discrete* categories – i.e., it is just a functor $\C^\op\to\Set$. In this case, the Grothendieck construction is the usual “category of elements”. The fibrations obtained in this way are precisely the *discrete fibrations* (these are prefibrations for which every morphism has a unique lift with a given codomain). In this case, the universal fibration is simply the forgetful functor $(\Set_*)^\op\to\Set^\op$ from the category of *pointed sets*.
Now, we note that, in the case of discrete fibrations, we are considering prefibrations of $$\text{
\emph{1-categories} with \emph{0-categories} as fibers, and a particular \emph{1-category} $\Set$
}$$ with a universal such prefibration over it. By contrast, in the case of general fibrations, we are considering prefibrations of $$\text{
\emph{1-categories} with \emph{1-categories} as fibers, and a particular \emph{2-category} $\Cat$
}$$ with a universal such prefibration over it. Hence, we see that in generalizing discrete fibrations, we should consider an arbitrary *2-category* as a base but (still) 1-categories as fibers – this is the notion of *1-discrete 2-fibration* which, as we see, is in a sense *more* natural than the notion of fibration.
Now, just as the notion of discrete fibration can be defined without reference to general fibrations, the notion of 1-discrete 2-fibration can be defined without reference to general 2-fibrations, and this is what we will do. More generally, when it comes to such higher-categorical generalities, there tends to be a lot of natural elegant extra structure lurking close at hand, as well as the ever-present possibility of “weakening” the notions being considered (pseudo-functors instead of strict 2-functors, etc.), but the elaboration of this extra structure tends to increase very quickly in complexity. Thus, below, we try to be as economical as possible, and define only those notions that we need, foregoing certain natural more general formulations.
2-categorical preliminaries {#sec:2-cat-prelims}
---------------------------
We begin with some generalities about 2-categories. The totality of 2-categories forms a *3-dimensional structure* (i.e., the collection of morphisms between 2-categories is 2-dimensional), and this comes in various flavours according to the weakness of the notions being considered. Here, we elaborate a very small part of this 3-dimensional structure.
### {#defn:psnt}
Given a category $\C$, a 2-category $\D$ and functors $F,G:\C\to\D$, a *pseudonatural transformation* $\alpha:F\to{}G$ consists of the following data (i)-(ii), subject to the conditions (iii)-(iv):
1. A 1-cell $\alpha_A:FA\to{}GA$ for each $A\in\Ob\C$
2. An isomorphism 2-cell $$\begin{tikzcd}[column sep=60pt]
FA
\ar[r, "Ff"]\ar[d, "\alpha_A"']&
FB
\ar[d, "\alpha_B"]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\alpha_{f}", pos=0.4, "\sim"' {sloped, pos=0.35, inner sep=3pt},
start anchor={[xshift=-10pt]}, end anchor={[xshift=10pt]}]
\\
GA\ar[r, "Gf"']
&
GB
\end{tikzcd}$$ for each morphism $f:A\to{}B$ in $\C$
3. For each pair $A\tox{f}B\tox{g}C$ of composable morphisms in $\C$, we have $(\id_{Gg}\circ\alpha_f)(\alpha_g\circ\id_{Ff})=\alpha_{gf}$
4. For each $A\in\Ob\C$, we have $\alpha_{\id_A}=\id_{\alpha_A}$
Given pseudonatural transformations $F\tox{\alpha}G\tox{\beta}H$, their *composite* $\beta\circ\alpha$ is defined by the prescriptions $(\beta\circ\alpha)_A=\beta_A\circ\alpha_A$ and $(\beta\circ\alpha)_f=(\beta_f\circ\alpha_A)(\beta_A\circ\alpha_f)$. We leave it to the reader to verify that this is again a pseudonatural transformation.
A 1-cell $f:A\to{}B$ in a 2-category is an *equivalence* if there exists a 1-cell $g:B\to{}A$ and isomorphism 2-cells $g\circ{}f\cong{}\id_A$ and $f\circ{}g\cong{}\id_B$ – such a $g$ is called a *quasi-inverse* to $f$. The pseudonatural transformation $\alpha:F\to{}G$ is a *pseudonatural equivalence* if $\alpha_A$ is an equivalence in $\D$ for each $A\in\Ob\C$.
### {#defn:psnt-whisker}
Given categories $\B$ and $\C$, 2-categories $\D$ and $\E$, functors $F:\B\to\C$ and $G,H:\C\to\D$, a 2-functor $K:\D\to\E$ and a pseudonatural transformation $\alpha:G\to{}H$, as in $$\begin{tikzcd}[row sep=0pt, column sep=20pt]
&&
\ar[dd, shorten >=0pt, shorten <=-5pt, Rightarrow, "\alpha"' inner sep=5pt, pos=0.3]
&
&\\[-5pt]
\B\ar[r, "F"]&
\C\ar[rr, "G", bend left]\ar[rr, "H"', bend right]
&&
\D\ar[r, "K"]&\E,\\
&&{}&&{}&
\end{tikzcd}$$ we define (i) the *whiskering of $\alpha$ by $K$*, which we denote by $K\circ\alpha$, to be the pseudonatural transformation $KG\to{}KH$ defined by $(K\circ\alpha)_A=K(\alpha_A)$ and $(K\circ\alpha)_f=K(\alpha_f)$ for $A\in\Ob\C$ and $f\in\Ar\C$; and (ii) the *whiskering of $\alpha$ by $F$*, denoted by $\alpha\circ{}F$ to be the pseudonatural transformation $GF\to{}HF$ defined by $(\alpha\circ{}F)_A=\alpha_{FA}$ and $(\alpha\circ{}F)_f=\alpha_{Ff}$.
We leave to the reader the easy proof that these are in fact pseudonatural transformations as claimed.
We note that the whiskering of a pseudonatural equivalence (on either side) is again a pseudonatural equivalence.
1-discrete 2-fibrations
-----------------------
We now introduce 1-discrete 2-fibrations (1D2Fs). We then prove some analogues for 1D2Fs of basic properties of fibrations. We also describe (a part of) the “Grothendieck construction” for 1D2Fs.
### {#defn:2-functor}
Given 2-categories $\C$ and $\D$, we define a *2-functor* $F:\C\to{}\D$ to be a pseudo-functor such that $F_{f,g}$ and $F_A$ are identity 2-cells for every $A\in\Ob\C$ and $A\tox{f}B\tox{g}C$ in $\C$.
More concretely, a 2-functor is a “homomorphism” in the obvious sense from $\C$ to $\D$; it takes $i$-cells to $i$-cells (for $i=0,1,2$) and preserves all compositions and identities. Note that a 2-functor induces a functor on underlying 1-categories.
### {#defn:1d2f}
A *pre-2-fibration* is simply a 2-functor $\fibr{C}CB$. We use similar terminology for pre-2-fibrations as we do for prefibrations: $\B$ is the *base* 2-category; $\C$ is the *total* 2-category; a 0-, 1-, or 2-cell in $\C$ *lies over* its image in $\B$; and so on. The *fiber* $\fib{C}^A$ of $\fib{C}$ over $A\in\Ob\B$ is the sub-2-category consisting of 0-cells, 1-cells, and 2-cells lying over $A$, $\id_A$, and $\id_{\id_A}$, respectively. The *underlying prefibration* of a pre-2-fibration is just the induced functor on the underlying 1-categories.
The pre-2-fibration $\fib{C}$ is a *1-discrete 2-fibration*[^14] (or 1D2F) if (i) the underlying prefibration is a fibration, and (ii) for every 2-cell $\alpha:f\to{}g$ in $\B$ and every 1-cell $p$ over $f$, there is a unique 2-cell over $\alpha$ with domain $p$, as depicted below (this says that the functor $\HOM_\C(P,Q)\to\HOM_\B(A,B)$ is a *discrete op-fibration*). $$\begin{tikzcd}[column sep=30pt]
P
\ar[r, bend left, "p"{name=p}]
\ar[r, bend right, ""'{name=q}, dashed]
&Q
\ar[Rightarrow, from=p, to=q, dashed, shorten <=3pt, shorten >=3pt]
\\[10pt]
A
\ar[r, bend left, "f"{name=f}]
\ar[r, bend right, "g"'{name=g}]
&B
\ar[Rightarrow, from=f, to=g, "\alpha", shorten <=3pt, shorten >=3pt]
\end{tikzcd}$$
Note that the fibers of a 1D2F are 1-categories and that, if $\B$ is a 1-category, then a 1D2F over $\B$ (seen as a 2-category with only trivial 2-cells) is the same thing as a fibration over $\B$.
### {#prop:1d2f-cart-gen}
In a 1D2F, we have the following generalization of the universal property for cartesian morphisms.
Let $\fibr{C}CB$ be a 1D2F, and suppose we have a solid diagram $$\begin{tikzcd}[row sep=5pt]
&|[alias=Q]|Q\car[rd, "q"]&\\
P\ar[ru, dashed, "p"]\ar[rr, "r"'{name=rar}]
\ar[from=rar, to=Q, Rightarrow, dashed, shorten <=2pt, shorten >=1pt]&&
R\\[5pt]
&|[alias=B]|B\ar[rd, "g"]&\\
A\ar[ru, "f"]\ar[rr, "h"'{name=h}]
\ar[from=h, to=B, Rightarrow, "\alpha"' inner sep=4pt, shorten <=2pt, shorten >=1pt]&&
C\\
\end{tikzcd}$$ with $q$ cartesian over $g$, $r$ lying over $h$, and $\alpha$ a 2-cell $h\to{}gf$.
There exists a unique 1-cell $p:P\to{}Q$ over $f$ for which there exists a (necessarily unique) 2-cell $r\to{}qp$ over $\alpha$. We know that there is a unique 2-cell $\sigma:r\to{}r'$ over $\alpha$ with domain $r$, and we can (and must) then take $p$ to be unique morphism $P\to{}Q$ over $f$ such that $qp=r'$: $$\begin{tikzcd}[row sep=5pt, baseline=(C.base)]
&Q\car[rd, "q"]&\\[10pt]
P\ar[ru, "p", dashed]
\ar[rr, "r"'{name=rar}]
\ar[rr, "r'"{name=rarr}, bend left]
\ar[from=rar, to=rarr, Rightarrow, "\sigma"' inner sep=4pt, shorten <=2pt, shorten >=2pt]&&
|[alias=C]|C\\
\end{tikzcd}
\tag*{\qed}$$
### {#prop:1d2f-cartuniq}
Next, we have a generalization in 1D2Fs of the uniqueness up to isomorphism of cartesian morphisms in fibrations [@sentai Proposition 2.9].
Given a 1D2F $\fibr{C}CB$ and a diagram $$\begin{tikzcd}[row sep=5pt]
P\car[rd, "p" name=rar]\ar[dd, "r"']&\\
&R&\\
|[alias=Q]|Q\car[ru, "q"']&
\ar[from=rar, to=Q, Rightarrow, shorten <=6pt, shorten >=6pt,
"\sigma" pos=0.4, "\sim"' {sloped, pos=0.25}]
\ar[Rightarrow, from=f, to=g, "\alpha" inner sep= 3pt, shorten <=3pt, shorten >=3pt,
"\vsim"' {inner sep=3pt, pos=0.4}]
\end{tikzcd}$$ in which $r$, $p$, $q$, $\sigma$ lie over $\id_A$, $f$, $g$, $\alpha$, respectively, and $q$ and $r$ are cartesian: if $\alpha$ (and hence $\sigma$) is an isomorphism 2-cell, then $r$ is an isomorphism 1-cell. By Proposition \[prop:1d2f-cart-gen\], there is a unique morphism $r':Q\to{}P$ over $A$ for which there exists a (necessarily unique) 2-cell $\sigma':q\to{}pr'$ over $\alpha\I$: $$\begin{tikzcd}[row sep=16pt, column sep=40pt]
P\ar[rd, "p" name=rar]\ar[d, "r"']&\\
|[alias=Q]|Q\ar[r, "q"' {name=q, pos=0.4}]\ar[d, "r'"']&R
\ar[from=rar, to=Q, Rightarrow, shorten <=6pt, shorten >=6pt,
"\sigma" pos=0.4, "\sim"' {sloped, pos=0.25}]\\
|[alias=P]|P\ar[ru, "p"']&
\ar[from=q, to=P, Rightarrow, shorten <=-2pt, shorten >=9pt,
"\sigma'" {pos=0.0, inner sep=1pt}, "\sim"' {sloped, pos=-0.15},
start anchor={[xshift=-3pt]}]\\
\\[-9pt]
A
\ar[r, bend left=60pt, "f"{name=f}]
\ar[r, "g"' {description, name=g}]
\ar[r, bend right=60pt, "f"'{name=ff}]
&B.
\ar[Rightarrow, from=f, to=g, "\alpha" inner sep= 3pt, shorten <=4pt, shorten >=2pt,
"\vsim"' {inner sep=3pt, pos=0.5}]
\ar[Rightarrow, from=g, to=ff, "\alpha\I" {inner sep= 3pt, pos=0.3}, shorten <=3pt,
shorten >=3pt, "\vsim"' {inner sep=3pt, pos=0.3}]
\end{tikzcd}$$ Then $r'r$ and $\id_P$ are both the unique morphism $t:P\to{}P'$ over $\id_A$ for which there exists a 2-cell $p\to{}pt$; hence $r'r=\id_P$. Similarly, $rr'=\id_Q$.
### {#prop:1d2f-2-of-3}
We also have a generalization in 1D2Fs of the “2-of-3” property [@sentai Proposition 2.5] of cartesian morphisms in fibrations.
Given a 1D2F $\fibr{C}CB$ and a diagram $$\begin{tikzcd}[row sep=5pt]
&|[alias=Q]|Q\car[rd, "q"]&\\
P\ar[ru, "p"]\car[rr, "r"'{name=rar}]
\ar[from=rar, to=Q, Rightarrow, shorten <=4pt, shorten >=1pt, "\sigma"' inner sep=3pt,
"\vsim" inner sep=3pt]&&
"\vsim" inner sep=3pt]&&
C\\
\end{tikzcd}$$ with $p$, $q$, $r$, $\sigma$ lying over $f$, $g$, $h$, $\alpha$, respectively: if $\alpha$ (and hence $\sigma$) is invertible, and $q$ and $r$ are cartesian, then so is $p$. Choosing a cartesian lift $\ct:f^*Q\to{}Q$ of $f$, we obtain a factorization $p=\ \ct\cind{p}$ as in $$\begin{tikzcd}
f^*Q\car[r, "\ct"]&|[alias=Q]|Q\car[rd, "q"]&\\
P\ar[ru, "p"]\car[rr, "r"'{name=rar}]\ar[u, "\cind{p}"]
\ar[from=rar, to=Q, Rightarrow, shorten <=4pt, shorten >=1pt, "\sigma"' inner sep=3pt,
"\vsim" {inner sep=3pt}, start anchor={[xshift=2pt]}]&&
R.
\end{tikzcd}$$ It then follows from Proposition \[prop:1d2f-cartuniq\] that $\cind{p}$ is an isomorphism, and hence (by [@sentai Propositions 2.6 and 2.7]) that $p$ is cartesian.
### {#constr:rev-groth-constr-ext}
In [@sentai § 9.2], we reviewed the well-known construction of a pseudo-functor $\B^\op\to\Cat$ from a cloven fibration $\fibr{C}CB$.
We now show how, if $\fib{C}$ is a 1D2F, the resulting pseudo-functor can be extended to a pseudo-functor on the 2-category $\B^\op$.
To do this, we need to (i) extend the map $\fpsf{C}:\Hom_\B(A,B)\to\Hom_\Cat(\fib{C}^B,\fib{C}^A)$ to a functor $\HOM_\B(A,B)\to\HOM_\Cat(\fib{C}^B,\fib{C}^A)$ for each $A,B\in\Ob\B$ – and in particular, define a natural transformation $\alpha^*:f^*\to{}g^*$ for each 2-cell $\alpha:f\to{}g$ in $\B$ – and (ii) prove that the 2-cells $\fpsf{C}_{fg}:f^*g^*\to(gf)^*$ are then natural in $f$ an $g$.
Given 1-cells $f,g:A\to{}B$ and a 2-cell $\alpha:f\to{}g$ in $\B$, we define $\alpha^*_Q:f^*Q\to{}g^*Q$ for each $Q\in\Ob\fib{C}^B$ to be the (by Proposition \[prop:1d2f-cart-gen\]) unique morphism $f^*Q\to{}g^*Q$ over $A$ for which there exists a 2-cell $\crt{f}Q\to\crt{g}Q\cdot\alpha^*_Q$ over $\alpha$: $$\begin{tikzcd}[column sep=45pt]
f^*Q\ar[rd, "\crt{f}Q"{name=fQ}, pos=0.3]
\ar[dd, "\alpha^*_Q"', dashed]&\\[-15pt]
&Q\\[-15pt]
|[alias=gQ]|g^*Q
\car[ru, "\crt{g}Q"']
\ar[Rightarrow, from=fQ, to=gQ, shorten <=9pt, shorten >=3pt, dashed]
\\[-5pt]
A
\ar[r, bend left=20pt, "f"{name=f}]
\ar[r, bend right=20pt, "g"'{name=g}]
&B.
\ar[Rightarrow, from=f, to=g, "\alpha", shorten <=3pt, shorten >=3pt]
\end{tikzcd}$$
To see that $\alpha^*$, thus defined, is natural, let $p:P\to{}Q$ be a morphism in $\fib{C}^B$, and observe that we have 2-cells from $p\cdot\crt{f}P$ to both $\crt{g}Q\cdot(g^*p\cdot\alpha^*_P)$ and $\crt{g}Q\cdot(\alpha^*_Q\cdot{}f^*p)$: $$\begin{tikzcd}[column sep=3pt, row sep=5pt]
f^*P
\ar[rrrd, "\crt{f}P"{name=fP}]
\ar[rrdd, "\alpha^*_P"', pos=0.7]
\ar[ddd, "f^*p"']
&[-10pt]&[-10pt]&[50pt]\\
&&&P\ar[ddd, "p"]\\
&&
|[alias=gP]|g^*P
\ar[ru, "\ct"']
\ar[from=fP, to=gP, Rightarrow, shorten <=8pt, shorten >=3pt]
&\\
f^*Q
\ar[rrrd, "\ct"{name=fQ}]
\ar[rrdd, "\alpha^*_Q"']
&&&\\
&&&Q\\
&&
|[alias=gQ]|g^*Q
\car[ru, "\crt{g}Q"']
\ar[from=fQ, to=gQ, Rightarrow, shorten <=8pt, shorten >=3pt]
\ar[from=uuu, "g^*p" {fill=white}, pos=0.4, crossing over]
&\\[20pt]
&A
\ar[rr, bend left=20pt, "f"{name=f}]
\ar[rr, bend right=20pt, "g"'{name=g}]
&&B.
\ar[Rightarrow, from=f, to=g, "\alpha", shorten <=3pt, shorten >=3pt]
\end{tikzcd}$$ Hence $(\alpha^*_Qf^*p)$ and $(\alpha^*_Qf^*p)$ are equal by Proposition \[prop:1d2f-cart-gen\].
The proofs that the association $\alpha\mapsto\alpha^*$ defines a functor $\HOM_{\B}(A,B)\to\HOM_\Cat(\fib{C}^B,\fib{C}^A)$, and that that the 2-cells $\fpsf{C}_{gf}:f^*g^*\to(gf)^*$ are natural in $f$ an $g$, are similar – in each case, it must be shown that two morphisms are equal, and this is shown by exhibiting certain 2-cells and applying Proposition \[prop:1d2f-cart-gen\].
### {#constr:groth-const-2-exp}
We now give the “reverse” construction to Construction \[constr:rev-groth-constr-ext\]. Namely, given a 2-category $\B$ with underlying category $\abs{\B}$, a cloven fibration $\fibr{C}C{\abs{\B}}$, and a pseudo-functor $\B^\op\to\Cat$ extending the pseudo-functor $\fpsf{C}:\abs{\B}^\op\to\Cat$ associated to $\fib{C}$, we will put a 2-category structure on $\C$ and extend $\fib{C}$ to a 1D2F.
Since $\fpsf{C}$ is to be a 1D2F, there can be at most one 2-cell with given domain and codomain in $\C$ over a given 2-cell in $\B$; hence, we must simply declare when there is in fact one. We declare that, given $p,q:P\to{}Q$ in $\C$ lying over $f,g:A\to{}B$ (respectively) in $\B$ and a 2-cell $\alpha:f\to{}g$, there is a 2-cell $p\to{}q$ lying over $\alpha$ if and only if the triangle $$\begin{tikzcd}[row sep=5pt]
&f^*Q\ar[dd, "\alpha^*_Q"]\\
P\ar[ru, "\cind{p}"]\ar[rd, "\cind{q}"']&\\
&g^*Q
\end{tikzcd}$$ in $\fib{C}^A$ commutes, where we write $\alpha^*$ for $\fpsf{C}\alpha$.
The composition operations in $\C$ are uniquely determined. The composite of 2-cells $p\tox{\sigma}q\tox{\tau}r$ lying over $f\tox{\alpha}g\tox{\beta}h$ in $\B$ must be the unique 2-cell lying over $\beta\cdot\alpha$, if such a 2-cell exists. That such a 2-cell does in fact exist follow from the commutativity of $$\begin{tikzcd}[row sep=12pt]
&f^*Q\ar[d, "\alpha^*_Q"']\ar[dd, bend left=45pt, "(\beta\cdot\alpha)^*Q"]\\
P\ar[ru, "\cind{p}"]\ar[r, "\cind{q}"]\ar[rd, "\cind{r}"']&
g^*Q\ar[d, "\beta^*_Q"']\\
&h^*Q,
\end{tikzcd}$$ where we have used the functoriality of $\fpsf{C}:\HOM_\B(A,B)\to\HOM_\B(\fib{C}^B,\fib{C}^A)$ – i.e., that $(\beta\cdot\alpha)^*=\beta^*\cdot\alpha^*$.
That this composition is associative is immediate. That it is unital follows immediately from the fact that there is a 2-cell $p\to{}p$ lying over $\id_f:f\to{}f$ for each $f:A\to{}B$ in $\B$, which in turn follows from the commutativity of $$\begin{tikzcd}[row sep=5pt]
&f^*Q\ar[dd, "\id^*_Q"]\\
P\ar[ru, "\cind{p}"]\ar[rd, "\cind{p}"']&\\
&f^*Q
\end{tikzcd}$$ where we have again used the functoriality of $\fpsf{C}:\HOM_\B(A,B)\to\Hom_\B(\fib{C}^B,\fib{C}^A)$.
The “horizontal” composite of 2-cells $\sigma$ and $\tau$ lying over $\alpha$ and $\beta$ as in $$\begin{tikzcd}[column sep=30pt]
P
\ar[r, bend left, "p"{name=p}]
\ar[r, bend right, "q"'{name=q}]
&Q
\ar[Rightarrow, from=p, to=q, shorten <=3pt, shorten >=3pt]
\ar[r, bend left, "r"{name=rar}]
\ar[r, bend right, "s"'{name=s}]
&Q
\ar[Rightarrow, from=rar, to=s, shorten <=3pt, shorten >=3pt]
\ar[r, bend left, "h"{name=h}]
\ar[r, bend right, "k"'{name=k}]
&B
\ar[Rightarrow, from=h, to=k, "\beta", shorten <=3pt, shorten >=3pt]
\end{tikzcd}$$ must, again, be the unique 2-cell $rp\to{}sq$ over $\beta\circ\alpha$. That such a 2-cell exists follows from the commutativity of the following diagram, in which we use the naturality of $\fpsf{C}_{ABC}$ (see [@sentai Definition 9.1] for this notation). $$\begin{tikzcd}[row sep=10pt]
&&f^*h^*R\ar[dd, "f^*\beta^*_R"]\ar[r, "(\fpsf{C}_{fh})_R"]&
(hf)^*R\ar[dddd, "(\beta\circ\alpha)^*_R"]\\
&f^*Q\ar[ru, "f^*\cind{r}"]\ar[rd, "f^*\cind{s}"']\ar[dd, "\alpha^*_Q"']&&\\
P\ar[ru, "\cind{p}"]\ar[rd, "\cind{q}"']
\ar[rrruu, "\cind{rp}", bend left=50pt]
\ar[rrrdd, "\cind{sq}"', bend right=50pt]
&&f^*k^*R\ar[dd, "\alpha^*_{k^*R}"]
&\\
&g^*Q\ar[rd, "g^*\cind{s}"']&&\\
&&g^*k^*R\ar[r, "(\fpsf{C}_{gk})_R"]&(kg)^*R,
\end{tikzcd}$$ The remaining conditions for $\C$ to be a 2-category – namely, the associativity and unitality of the horizontal composition, and that horizontal composition is functorial – all follow immediately from the “1-discreteness” of $\fib{C}$ – i.e., the fact that there is at most one 2-cell in $\C$ with given domain and codomain over a given 2-cell in $\B$.
### {#defn:canonical-extension}
Given any $\wedgeq$-cloven $\wedgeq$-fibration $\fibr{C}CB$, we have by [@sentai Theorems 8.5 and 9.12] a 2-categorical structure on $\B$ and an extension of the pseudo-functor $\fpsf{C}:\B^\op\to\Cat$ associated to the cleavage to a pseudo-functor of 2-categories. Hence, Construction \[constr:groth-const-2-exp\] extends $\fib{C}$ to a 1D2F. We call this 1D2F the *canonical extension of $\fib{C}$*.
### {#defn:psnt-over-psnt}
Let $\fibr{C}CB$ be a fibration, $\fibr{C'}{C'}{B'}$ be a 1D2F, $\phi,\psi:\B\to{}\B'$ functors, and $\Phi,\Psi:\C\to{}\C'$ functors lying over $\phi$ and $\psi$. We say that a pseudonatural transformation $\beta:\Phi\to\Psi$ *lies over* a pseudonatural transformation $\alpha:\phi\to\psi$ if $\fib{C'}\circ\beta=\alpha\circ\fib{C}$ (here we are using the “whiskering” operations from \[defn:psnt-whisker\]).
The abstract invariance theorem {#sec:abstract-invariance-theorem}
-------------------------------
We now come to the proof of the abstract version of the homotopy-invariance property. In this generality, the theorem states that a *free* $h^=$-fibration satisfies with respect to *pseudonatural* transformations the property which it is required by definition to satisfy with respect to natural transformations.
We make a small digression to discuss *freeness*. In classical (“0-categorical”) algebra, a *free* object (group, ring, etc.) is required to satisfy a certain universal property, which then determines it up to isomorphism. In the case of categorical structures, it is often more natural to impose conditions that determine the object under consideration up to *equivalence*. However, there are usually different conditions which do this.
In the case at hand, namely that of $\fibr{C}CB$ being a *free $h^=$-fibration over $\B$*, the weakest such condition one could impose is that, for any other $h^=$-fibration $\fibr{C'}{C'}B$ over $\B$, there exists a morphism $\fib{C}\to\fib{C'}$ of fibrations over $\B$, and that,any two such morphisms are isomorphic; and this is in fact the definition we use.
A slightly stronger condition one could demand is that, given any $h^=$-fibration $\fib{C'}$, any f.p. functor $\phi:\B\to{}\B'$ can be extended to a morphism $(\phi,\Phi):\fib{C}\to\fib{C'}$ of $h^=$-fibrations, and for any natural isomorphism $\phi\to{}\psi$ to another f.p. functor, there is a natural isomorphism $\Phi\to\Psi$ lying over it. In fact, a free $h^=$-fibration (in the above sense) automatically satisfies this stronger universal property; the invariance theorem then says that, when $\fib{C'}$ is a 1D2F, $\fib{C}$ satisfies this property with respect to pseudonatural equivalences and not just natural isomorphisms.
The proof that $\fib{C}$ satisfies the stronger universal property proceeds (roughly speaking) by pulling back the fibration $\fib{C'}$ along the given functors $\phi,\psi:\B\to{}\B'$, showing that the natural isomorphism $\phi\to\psi$ induces an equivalence of fibrations over $\B$ between these pullbacks, and then appealing to the universal property of $\fib{C}$ with respect to $h^=$-fibrations over $\B$.
It is easy to see why this should work for pseudonatural transformations and not just natural transformations. Namely, from the “pseudo-functor to $\Cat$” perspective, the pullback $\phi^*\fib{C'}$ of a fibration $\fib{C'}$ along a morphism $\phi:\B\to\B'$ is just given by composing $\phi^\op:\B^\op\to\B'^\op$ with the pseudo-functor $\fpsf{C'}:\B^\op\to\Cat$. On the other hand, a morphism of fibrations over $\B$ corresponds to a pseudonatural transformation of pseudo-functors from $\B^\op$ to $\Cat$. Hence, given a natural isomorphism $\alpha:\phi\to\psi$, the induced equivalence $\phi^*\fib{C'}\simeq\psi^*\fib{C'}$ between the pullbacks along $\phi$ and $\psi$ is obtained as the whiskering $$\begin{tikzcd}[row sep=0pt, column sep=15pt]
&\ar[from=dd, shorten >=-3pt, shorten <=1pt, Rightarrow, "\ \alpha^\op"', pos=0.7]&
&&\\[-5pt]
\B^\op\ar[rr, "\psi^\op", bend left]\ar[rr, "\fpsf{C'}"', bend right]&&
\B'^\op\ar[r, "\psi^\op"]&\Cat.\\
&{}&&{}&
\end{tikzcd}$$ The point is now that this whiskering can be carried out just as well if $\alpha$ is a *pseudo*-natural transformation.
### {#defn:pullback-fib}
Given a prefibration $\fibr{C'}{C'}{B'}$ and a functor $F:\B\to\B'$, we define the *pullback $\fibr{\mathit{f^*}C}{\mathit{f^*}C}{B}$ of $\fib{C}$ along $F$* to be the usual pullback $$\begin{tikzcd}
F^*\C'\ar[r, "\crt{F}{\C'}"]\ar[d, "f^*\fib{C'}"']
\ar[rd, phantom, "\lrcorner", pos=0.1]
&\C'\ar[d, "\fib{C'}"]\\
\B\ar[r, "F"]&\B'
\end{tikzcd}$$ in the category of categories, where we write $\crt{F}{\C'}$, as indicated, for the associated functor $F^*\C'\to\C'$. Explicitly, the objects of $F^*\C'$ are pairs $(A,P)$ with $A\in\Ob\B$ and $P\in\Ob\fib{C'}^{FA}$, and the morphisms are pairs $(f,p)$ with $f\in\Ar\B$ and $p\in\Ar\C'$ lying over $Ff$.
The prefibration $F^*\fib{C'}$ is a fibration if $\fib{C'}$ is, and inherits many properties from $\fib{C'}$ – in particular, if $\B$ is an f.p. category and $F$ is an f.p. functor, then if $\fib{C'}$ is an $h^=$-fibration, $F^*\fib{C'}$ is as well. The reason is that $\crt{F}{\C'}$ induces isomorphisms $(F^*\fib{C'})^A\to\fib{C'}^{FA}$ on fibers for each $A\in\Ob\B$, and – if $\fib{C'}$ is a fibration – a morphism in $F^*\C'$ is cartesian or cocartesian if and only if its image under $\crt{F}\C'$ is, and similarly the $\prod$-diagrams in $F^*\C'$ are exactly those whose images under $\crt{F}C'$ are $\prod$-diagrams. In particular, $(F,\crt{F}\C')$ is a morphism of $(h^=)$-fibrations.
By the universal property of the pullback, given any other prefibration $\fibr{C}{C}{B}$ over $\B$ and any morphism $(F,\Phi):\fib{C}\to\fib{C'}$ of prefibrations, we have a unique morphism $\cind{\Phi}:\fib{C}\to{}F^*\fib{C'}$ of prefibrations over $\B$ such that $\crt{F}\C'\cdot\cind{\Phi}=\Phi$. It follows from the above observations that $\cind{\Phi}$ is a morphism of ($h^=$-)fibrations whenever $\fib{C'}$ and $\fib{C}$ are ($h^=$-)fibrations and $(\phi,\Phi)$ is a morphism thereof.
If $\fib{C'}$ is a 1D2F, then $F^*\fib{C'}$ is defined as the pullback of the underlying fibration of $\fib{C'}$.
### {#defn:psnt-induced-stuff}
Let $\B$ be a category, $\fibr{C'}{C'}{B'}$ a cloven 1D2F, let $F,G:\B\to{}\B'$ be functors, and let $\alpha:F\to{}G$ be a pseudonatural transformation. We will construct from this a morphism $\pnif\alpha:G^*\fib{C'}\to{}F^*\fib{C'}$ of fibrations over $\B$ between the associated pullback fibrations, as well as a pseudonatural transformation $\pnipn\alpha:\crt{F}\C'\cdot\pnif{\alpha}\to\crt{G}\C'$ over $\alpha$, as shown below. We will define these simultaneously. $$\begin{tikzcd}
|[alias=FC]|F^*\C'\ar[rrrd, "\crt{F}\C'"]\ar[rddd, "F^*\fib{C'}"']
&[-30pt]&[-30pt]&[20pt]\\[-10pt]
&&&\C'\ar[dd, "\fib{C'}"]\\[-10pt]
&&G^*\C'\ar[lluu, dashed, "\pnif\alpha"]
\ar[ru, "\crt{G}\C'"' name=GC, near start]
\ar[ld, "G^*\fib{C'}", near start]
\ar[from=FC, to=GC, Rightarrow, dashed, "\pnipn\alpha", shorten <=10pt, shorten >=5pt]
&\\[30pt]
&\B
\ar[rr, bend left, "F"{name=F}]
\ar[rr, bend right, "G"'{name=G}]
\ar[from=F, to=G, Rightarrow, "\alpha", shorten <=5pt, shorten >=5pt]
&&\B'
\end{tikzcd}$$
Given $A\in\Ob\B$ and $(A,P)\in(G^*\fib{C'})^A$ (i.e., $P\in\fib{C'}^{GA}$), we set $\pnif\alpha(A,P)=(A,(\alpha_A)^*P)\in(F^*\fib{C'})^A$, and we set $\pnipn\alpha_{(A,P)}$ to be the morphism $\crt{(\alpha_A)}P:\alpha_A^*P\to{}P$.
Next, let $(f,p):(A,P)\to(B,Q)$ be a morphism in $G^*\C'$. Since we want $\pnif\alpha(f,p)$ to be a morphism over $f:A\to{}B$ in $\B$, it must be of the form $(f,p')$ for some $p':(\alpha_A)^*P\to{}(\alpha)_A^*Q$ over $Ff$. Seeing as we want there to be a 2-cell $$\pnipn\alpha_{(f,p)}:
\pnipn\alpha_{(B,Q)}\cdot(\crt{F}\C')(\pnif\alpha(f,p))
=\crt{(\alpha_B)}Q\cdot{}p'
\longrightarrow
p\cdot\crt{(\alpha_A)}P
=(\crt{G}\C')(f,p)\cdot\pnipn\alpha_{(A,P)}$$ lying over $\alpha_f$, we see that we are forced to define $p'$ to be the unique morphism over $Ff$ for which there exists a 2-cell $$\begin{tikzcd}
(\alpha_A)^*P\ar[rr, dashed, "p'"]\ar[rd, "\crt{(\alpha_A)}P"']
&&(\alpha_B)^*Q\car[rd, "\crt{(\alpha_B)}Q"]
\ar[ld, Rightarrow, dashed, shorten <=5pt, shorten >=5pt]
&\\
&P\ar[rr, "p"']&&Q\\
FA\ar[rr, "Ff"]\ar[rd, "\alpha_A"']
&&FB\ar[rd, "\alpha_B"]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\alpha_f"']
&\\
&GA\ar[rr, "Gf"]&&GB
\end{tikzcd}$$ lying over $\alpha_f$ (such a $p'$ exists by Proposition \[prop:1d2f-cart-gen\] since $\alpha_{(f,p)}$ is invertible), and we must take $\pnipn\alpha_f$ to be this 2-cell lying over $\alpha_f$.
We now prove simultaneously that $\pnif\alpha$ is a functor and that $\pnipn\alpha$ is a pseudonatural transformation.
Let $(A,P)\tox{(f,p)}(B,Q)\tox{(g,q)}(C,R)$ be morphisms in $G^*\C'$. Let us write $(f,p')$ and $(g,q')$ for $\pnif\alpha(f,p)$ and $\pnif\alpha(g,q)$, as well as $(gf,(qp)')$ for $\pnif\alpha(gf,qp)$. We must show that the 2-cells $$\begin{tikzcd}[column sep=10pt, baseline=(afp.base)]
\alpha_A^*P\ar[rr, "p'"]\ar[rd, "\crt{\alpha_A}P"']
&&\alpha_B^*Q\ar[rd, "\crt{\alpha_B}Q"', near start]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\pnipn\alpha_{(f,p)}"' name=afp]
\ar[rr, "q'"]
&&\alpha_C^*R\ar[rd, "\crt{\alpha_C}R"]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\pnipn\alpha_{(g,q)}"']
&
\\
&P\ar[rr, "p"']&&Q
\ar[rr, "q"']&&Q\\
FA\ar[rr, "Ff"]\ar[rd, "\alpha_A"']
&&FB\ar[rd, "\alpha_B"', near start]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\alpha_f"']
\ar[rr, "Fg"]
&&FC\ar[rd, "\alpha_C"]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\alpha_g"']
&\\
&GA\ar[rr, "Gf"]&&GC
\ar[rr, "Gg"]&&GC
\end{tikzcd}
\quad\text{and}\quad
\begin{tikzcd}[column sep=10pt, baseline=(agfqp.base)]
\alpha_A^*P\ar[rr, "(qp)'"]\ar[rd, "\crt{\alpha_A}P"']
&&\alpha_C^*R\car[rd, "\crt{\alpha_C}R"]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\pnipn\alpha_{(gf,qp)}"' name=agfqp]
&\\
&P\ar[rr, "qp"']&&R\\
FA\ar[rr, "F(gf)"]\ar[rd, "\alpha_A"']
&&FC\ar[rd, "\alpha_C"]
\ar[ld, Rightarrow, shorten <=5pt, shorten >=5pt, "\alpha_{gf}"']
&\\
&GA\ar[rr, "G(gf)"]&&GC
\end{tikzcd}$$ are equal – i.e., that $(\id_q\circ{}\pnipn\alpha_{(f,p)})(\pnipn\alpha_{(g,q)}\circ{}\id_{p'})=\pnipn\alpha_{(gf,qp)}$ – and also that $q'p'=(qp')$. The first claim follows at once from the second by 1-discreteness. The second claim is true since $(qp)'$ is by definition the unique morphism $\alpha_A^*P\to\alpha_C^*R$ over $F(gf)$ for which there exists a 2-cell as above on the right, and $q'p'$ also has this property.
The proof of the remaining (unitality) property of $\pnif\alpha$ and $\pnipn\alpha$ is similar.
Finally, we must see that $\pnif{\alpha}$ preserves cartesian morphisms. This follows from the definition of $\pnif{\alpha}$, Proposition \[prop:1d2f-2-of-3\], and the fact that a morphism $(f,p)$ in $F^*\C'$ is cartesian if and only if $p$ is.
### {#prop:induced-stuff-equivalence}
Let $\fib{C}$, $F$, $G$, and $\alpha$ be as in Construction \[defn:psnt-induced-stuff\]
If $\alpha$ is a pseudonatural equivalence, then $\pnif{\alpha}$ is a fiberwise-equivalence and $\pnipn\alpha$ is a pseudonatural equivalence. To see that $\pnif{\alpha}$ is a fiberwise equivalence, note that the induced functor $\pnif\alpha:(G^*\fib{C})^A\to(F^*\fib{C})^A$ is (with respect to the identifications $(F^*\fib{C})^A\cong\fib{C}^{FA}$ and $(G^*\fib{C})^A\cong\fib{C}^{FA}$) just the pullback functor $(\alpha_A)^*:\fib{C}^{GA}\to\fib{C}^{FA}$. This is an equivalence, since, given a quasi-inverse $\beta_A$ for $\alpha_A$, we obtain a quasi-inverse $(\beta_A)^*$ for $(\alpha_A)^*$.
We now prove that $\pnipn\alpha$ is a pseudonatural equivalence. Since each $\alpha_A$ is an equivalence in $\C$ and each $\pnipn\alpha_A$ is a cartesian lift of $\alpha_A$, it suffices to prove that any cartesian lift of an equivalence in a 1D2F is again an equivalence.
Let $f:A\to{}B$ be an equivalence with quasi-inverse $g:B\to{}A$, so that there exist isomorphism 2-cells $\alpha:\id_A\toi{}gf$ and $\beta:\id_B\toi{}fg$, and let $p:P\to{}Q$ be a cartesian lift of $f$. By Proposition \[prop:1d2f-cart-gen\], there exists a unique morphism $q:Q\to{}P$ over $g$ for which there exists a (necessarily invertible) 2-cell $\id_Q\toi{}pq$ over $\beta$. It remains to see that $qp\cong\id_P$. By Proposition \[prop:1d2f-2-of-3\], $q$ is cartesian, and hence, by the argument we just gave, there exists $p':P\to{}Q$ with $qp'\cong\id_P$. We then have $p\cong{}pqp'\cong{}p'$ and hence $qp\cong{}qp'\cong\id_P$.
### {#defn:free-hfib}
An $h^=$-fibration $\fibr{C}CB$ is *free over $\B$* if, for any $h^=$-fibration $\fibr{C'}{C'}{B}$ over $\B$, there is up to isomorphism a unique morphism $\fib{C}\to\fib{C'}$ of $h^=$-fibrations over $\B$; i.e., there exists such a morphism, and for any two such, there exists a natural isomorphism of morphisms of fibrations over $\B$ between them.
### {#section-11}
If $\fibr{C}CB$ is a free $h^=$-fibration over $\B$ and $\fibr{C'}{C'}{B'}$ is any $h^=$-fibration, then for any f.p. functor $\phi:\B\to\B'$, there exists a morphism of $h^=$-fibrations $(\phi,\Phi):\fib{C}\to\fib{C}'$ over $\B$. Taking the pullback $(\phi,\crt{\phi}{\C'}):\phi^*\fib{C'}\to\fib{C}$ of $\fib{C'}$ along $\phi$, we have by the discussion in Definition \[defn:pullback-fib\] that $\phi^*\fib{C'}$ is an $h^=$-fibration. Hence, by the freeness of $\fib{C}$, we then have a morphism $\Psi:\fib{C}\to\phi^*\fib{C'}$ of $h^=$-fibrations over $\B$. Since the composite of a morphism of $h^=$-fibrations is again one, $(\Phi\circ\Psi,\phi)$ is as desired.
### {#thm:abstract-invariance}
Suppose $\fibr{C}CB$ is a free $h^=$-fibration, $\fibr{C'}{C'}{B'}$ is a 1D2F which is also an $h^=$-fibration, $(\phi,\Phi)$ and $(\psi,\Psi)$ are morphisms $\fibr{C}{C}{B}\to\fibr{C'}{C'}{B'}$ of $h^=$-fibrations, and $\alpha:\phi\to\psi$ is a pseudonatural equivalence.
There exists a pseudonatural equivalence $\Phi\to\Psi$ lying over $\alpha$.
In particular, for each $P\in\Ob\C$ over some $A\in\Ob\B$, there is an equivalence $p:\Phi{}P\to\Psi{}P$ lying over the equivalence $\alpha_A:\phi{}A\to\psi{}A$ (i.e., $p$ lies over $\alpha_A$, there is a quasi-inverse $q$ of $p$ lying over a quasi-inverse of $\alpha_A$, and the associated 2-cells $\id_{\Phi{}P}\toi{}qp$ and $\id_{\Psi{}P}\toi{}pq$ in $\C$ lie over the corrsponding ones in $\B$). To begin, we choose a cleavage of $\fib{C'}$ (which we can do by the axiom of choice – otherwise, we must assume explicitly that $\fib{C'}$ admits a cleavage). Consider the pullbacks of $\fib{C'}$ along $\phi$ and $\psi$. We then have the situation depicted in Construction \[defn:psnt-induced-stuff\], with $F=\phi$ and $G=\psi$. By Proposition \[prop:induced-stuff-equivalence\] the morphism $\pnif\alpha:\psi^*\fib{C'}\to{}\phi^*\fib{C'}$ of fibrations is a fiberwise equivalence and hence, by Proposition \[prop:equivs-are-hfib-homos\] below, a morphism of $h^=$-fibrations.
By the definition of the pullback, we have morphisms of fibrations $\cind{\Phi}:\fib{C}\to\phi^*\fib{C}$ and $\cind{\Psi}:\fib{C}\to\psi^*\fib{C}$ over $\B$ such that $\crt{\phi}\C'\circ\cind\Phi=\Phi$ and $\crt{\psi}\C'\circ\cind\Psi=\Psi$ – and by the discussion in Definition \[defn:psnt-induced-stuff\], these are morphisms of $h^=$-fibrations. Hence, since the composite of morphisms of $h^=$-fibrations over $\B$ is again one, we have that $\pnif\alpha\circ\cind{\Psi}:\fib{C}\to\phi^*\fib{C}'$ is a morphism of $h^=$-fibrations. Hence, by the freeness of $\fib{C}$, we have a natural isomorphism $\eta:\cind{\Phi}\to\pnif{\alpha}\cind{\Psi}$ of morphisms of $h^=$-fibrations over $\B$: $$\begin{tikzcd}
&|[alias=FC]|\phi^*\C'\ar[rrrd, "\crt{\phi}\C'"]\ar[rddd, "\phi^*\fib{C'}"' pos=0.45]
&[-30pt]&[-30pt]&[20pt]\\[-10pt]
\C\ar[ru, "\cind{\Phi}" name=Phi]\ar[rrrd, "\cind{\Psi}"' pos=0.4, crossing over]
\ar[rrdd, "\fib{C}"']
&&&&\C'\ar[dd, "\fib{C'}"]\\[-10pt]
&&&
|[alias=psiC]|\psi^*\C'\ar[lluu, dashed, "\pnif\alpha"' pos=0.1]
\ar[from=Phi, to=psiC, Rightarrow, shorten <=12pt, shorten >=8pt, crossing over,
"\eta" pos=0.3, "\sim"' sloped]
\ar[ru, "\crt{\psi}\C'"' name=GC, near start]
\ar[ld, "\psi^*\fib{C'}", near start]
\ar[from=FC, to=GC, Rightarrow, dashed, "\pnipn\alpha", shorten <=10pt, shorten >=5pt]
&\\[30pt]
&&\B
\ar[rr, bend left, "\phi"{name=F}]
\ar[rr, bend right, "\psi"'{name=G}]
\ar[from=F, to=G, Rightarrow, "\alpha", shorten <=5pt, shorten >=5pt]
&&\B'.
\end{tikzcd}$$ Hence, the “pasting” of $\eta$ and $\pnipn{\alpha}$ – i.e. the composite pseudonatural transformation $(\pnipn\alpha\circ\id_{\cind{\Psi}})\circ(\id_{\crt{\phi}{\C'}}\circ\eta)$ – is as desired.
The final statement in the theorem follows from the definition of “pseudonatural equivalence over $\alpha$” (where, to arrange the described situation with the quasi-inverse and 2-cells in $\C$ lying over those in $\B$, we simply choose the ones in $\C$ first, and let the ones in $\B$ be their image under $\fib{C}$).
### {#prop:equivs-are-hfib-homos}
Suppose $\fibr{C}CB$ and $\fibr{C'}{C'}{B'}$ are $h^=$-fibrations, and $(\Phi,\phi):\fibr{C}CB\to\fibr{C'}{C'}{B'}$ is a morphism of fibrations which is a fiberwise equivalence and such that $\phi$ is product-preserving.
$(\Phi,\phi)$ is a morphism of $h^=$-fibrations. $\phi$ is product-preserving by assumption, and the induced functors $\fib{C}^B\to\fib{C'}^{\phi{B}}$ are clearly bi-cartesian closed since they are equivalences. Hence, it only remains to see that the relevant co-cartesian morphisms and $\prod$-diagrams are preserved.
In fact, all co-cartesian morphisms and $\prod$-diagrams are preserved. This follows easily from the fact that, in the present situation, given a morphism $f:A\to{B}$ in $\B$ and objects $P$ and $Q$ in $\fib{C}^A$ and $\fib{C}^B$, $\Phi$ induces a bijection between morphisms $P\to{Q}$ lying over $f$ and morphisms $\Phi{P}\to\Phi{Q}$ lying over $\phi{f}$, and moreover that $p:P\to{Q}$ is cartesian if and only if $\Phi{}p$ is.
Homotopy homomorphisms
----------------------
The abstract invariance theorem proven in the previous section was stated with respect to an arbitrary free $h^=$-fibration $\fibr{C}CB$. However, in practice, we are interested in particular base categories $\B$, namely the free f.p. category $\TM_\sigma$ built out of the terms of a given signature $\sigma$ (see §\[subsec:intro-predicate-logic\]).
In this situation, the invariance theorem admits a certain refinement, which we will ultimately need to recover, in §\[subsec:special-invariance\] below, the invariance property of the homotopical semantics as described in §\[subsec:properties-and-invariance\]. Namely, in Theorem \[thm:abstract-invariance\], we begin with two f.p. functors $\B\to\C$ and a pseudonatural equivalence between them. When $\B=\TM_\sigma$, we know, by the freeness property of $\TM_\sigma$, that each of these functors comes from an interpretation $\sigma\to{}\C$. However, given a homotopy-equivalence between these interpretations (which is what we are ultimately interested in), its not clear a priori that the induced f.p. functors $\TM_\sigma\to\C$ will be pseudonaturally equivalent. This is what we prove in this section.
The proof proceeds by considering a modified arrow category $\peqat\C$, such that the f.p. functors into $\peqat\C$ are pseudonatural equivalences of f.p. functors into $\C$, and the $\sigma$-interpretations in $\peqat\C$ are homotopy-equivalences of $\sigma$-interpretations into $\C$; this reduces the claim to the original freeness property of $\TM_\sigma$.
### {#section-12}
Given a 2-category $\C$, we define $\pacat\C$, the *pseudo-arrow category* of $\C$, to have objects functors $\mathbf{2}\to\C$ – i.e., 1-cells in $\C$ – and to have morphisms pseudonatural transformations, with composition being given by composition of pseudonatural transformations which, as we leave to the reader to verify, is associative and has identities
In other words a morphism $\alpha$ from $f_1:A_1\to{}B_1$ to $f_2:A_2\to{}B_2$ in $\pacat\C$ is a triple $(\alpha_A,\alpha_B,\alpha_f)$ with $\alpha_A:A_1\to{}A_2$, $\alpha_B:B_1\to{}B_2$, and $\alpha_f:\alpha_B\circ{}f_1\tox{\sim}f_2\circ{}\alpha_A$.
There are obvious domain and codomain functors $\dom,\cod:\pacat\C\to\C$.
We define $\peqat\C$ to be the full subcategory of $\pacat\C$ with objects the equivalences in $\C$.
### {#section-13}
Given a 2-category $\C$, if $\C$ has finite 2-categorical products (see [@sentai Definition 10.1]), then the categories $\pacat\C$ and $\peqat\C$ have finite products. Moreover, the inclusion functor $\peqat\C\hookrightarrow\pacat\C$, as well as the functors $\dom,\cod:\pacat\C\to\C$, are f.p. functors. Given 1-cells $g:A\to{}C$ and $h:B\to{}D$ and products $A\times{}B$ and $C\times{}D$ in $\C$, we will show that the morphisms $g\times{}h\to{}g$ and $g\times{}h\to{}h$ in $\pacat\C$ given by $(\pi_1,\pi_1,\id_{g\pi_1})$ and $(\pi_2,\pi_2,\id_{h\pi_2})$ exhibit $g\times{}h$ as a product of $g$ and $h$. It follows that $\dom$ and $\cod$ are f.p.
Given a 1-cell $f:X\to{}Y$ in $\C$ and morphisms $(s,t,\alpha):f\to{}g$ and $(u,v,\beta):f\to{}h$ in $\pacat\C$, we must show that there is a unique morphism $(w,x,\gamma):f\to{}g\times{}h$ such that $\pi_1w=s$, $\pi_2w=u$, $\pi_1x=t$, $\pi_2x=v$, $\id_{\pi_1}\circ\gamma=\alpha$, and $\id_{\pi_2}\circ\gamma=\beta$. Clearly, we must take $w=\br{s,u}$ and $x=\br{t,v}$. Then, since $C\times{}D$ is a 2-categorical product, there is a unique 2-cell $\gamma:\br{gs,hu}\to\br{tf,vf}$ such that $\id_{\pi_1}\circ\gamma=\alpha$ and $\id_{\pi_2}\circ\gamma=\beta$.
Finally, we must see that if $f$ and $g$ are equivalences in $\C$, then $f\times{}g$ is one as well. In fact, if $f\I$ and $g\I$ are quasi-inverses to $f$ and $g$, then $f\I\times{}g\I$ is easily seen to be a quasi-inverse to $f\times{}g$.
### {#prop:htpy-equiv-funs}
Given a category $\C$, a 2-category $\D$, and functors $F,G:\C\to\D$, the functors $F$ and $G$ are pseudonaturally equivalent if and only if there exists a functor $H:\C\to\peqat\D$ such that $\dom\circ{}H=F$ and $\cod\circ{}H=G$. The proof is by inspection, the point being that the data of a functor $H:\C\to\peqat\D$ is precisely the data of a pseudonatural equivalence $\dom\circ{}H\to\cod\circ{}H$, and the condition that $H$ be a functor is equivalent to the given data defining a pseudonatural transformation.
### {#defn:signature}
A *(multi-sorted) algebraic signature* $\sigma$ is given by a set $\Ob\sigma$ of *sorts* and, for each finite sequence $\vec{A}$ of sorts and each sort $B$, a set $\sigma(\vec{A},B)$ of *function symbols* (with “arity” $\vec{A}$ and “codomain sort” $B$). We denote the set of finite sequences in a set $X$ by $X^{<\omega}$, and write $\len{\vec{A}}$ for the length of a finite sequence.
Given a finite product category $\C$, an *interpretation $M$ of $\sigma$ in $\C$* consists of the following data (i)-(iii):
1. A map $M:\Ob\sigma\to\Ob\C$
2. A choice $\set{\pi_i^M:M\vec{A}\to{}MA_i}_{i=1}^{\len{\vec{A}}}$ of product diagram on $MA_1,\ldots,MA_{\len{\vec{A}}}$ for each sequence $\vec{A}\in(\Ob\sigma)^{<\omega}$ (where we require that $M\seq{A}=MA$ and $\pi_1^{M}=\id_{MA}:M\seq{A}\to{}MA$ for each $A\in\Ob\sigma$)
3. A morphism $Mf:M\vec{A}\to{}MB$ for each $f\in\sigma(\vec{A},B)$.
We write $M:\sigma\to\C$ to indicate that $M$ is an interpretation of $\sigma$ in $\C$.
Given two interpretations $M,N:\sigma\to\C$, a *homomorphism* $\alpha:M\to{}N$ consists of morphisms $\alpha_A:MA\to{}NA$ for each sort $A\in\Ob\sigma$, such that for each $f\in\sigma(\vec{A},B)$, the following diagram commutes, where we write $\alpha_{\vec{A}}$ for $\alpha_{A_1}\times\cdots\times\alpha_{A_{\len{\vec{A}}}}$. $$\begin{tikzcd}[column sep=60pt]
M\vec{A}\ar[r, "\alpha_{\vec{A}}"]\ar[d, "Mf"']&
N\vec{A}\ar[d, "Nf"]\\
MB\ar[r, "\alpha_{B}"]&NB
\end{tikzcd}$$ Interpretations $\sigma\to\C$ and homomorphisms form a category $\fpints(\sigma,\C)$ in an obvious manner.
Given an interpretation $M:\sigma\to\C$ and an f.p. functor $F:\C\to\D$, we obtain an interpretation $F\circ{}M:\sigma\to\D$ by setting $(F\circ{}M)(A)=F(MA)$ for $A\in\Ob\sigma$; $(F\circ{}M)(\vec{A})=F(M\vec{A})$ and $\pi_i^{F\circ{}M}=F\pi_i^M:(F\circ{}M)(\vec{A})\to(F\circ{}M)A_i$ for $\vec{A}\in(\Ob\sigma)^{<\omega}$ and $1\le{}i\le\len{\vec{A}}$; and $(F\circ{}M)(f)=F(Mf)$ for $f\in\sigma(\vec{A},B)$.
Similarly, given another f.p. functor $G:\C\to\D$ and a natural transformation $\alpha:F\to{}G$, we obtain a homomorphism $\alpha\circ{}M:F\circ{}M\to{}G\circ{}M$ by setting $(\alpha\circ{}M)_A=\alpha_{MA}$.
This defines, for each $M:\sigma\to\C$, a functor $\circ\sigma:\FPFun(\C,\D)\to\fpints(\sigma,\D)$. Given an interpretation $M:\sigma\to\C$, we say that the f.p. category $\C$ is *free on $\sigma$ (via $M$)* if $\circ\sigma:\FPFun(\C,\D)\to\fpints(\sigma,\D)$ is an isomorphism of categories for each f.p. category $\D$[^15].
Finally, given a 2-category $\C$ with finite 2-categorical products, the underlying category of $\C$ also has finite products, and we can consider interpretations $\sigma\to\C$. Given two such interpretations $M,N:\sigma\to\C$, a *homotopy homomorphism* $\alpha:M\to{}N$ consists of a 1-cell $\alpha_A:M\to{}N$ for each $A\in\Ob\sigma$, together with an invertible 2-cell $\alpha_f:\alpha_B\circ{}Mf\to{}Nf\circ\alpha_{\vec{A}}$ for each $f\in\sigma(\vec{A},B)$. A homotopy homomorphism $\alpha:M\to{}N$ is a *homotopy-equivalence* if $\alpha_A$ is an equivalence in $\C$ for each $A\in\Ob\sigma$.
### {#prop:htpy-equiv-sig-ints}
Two interpretations $M,N:\sigma\to\C$ are homotopy-equivalent if and only if there exists an interpretation $H:\sigma\to\peqat\C$ such that $\dom\circ{}H=M$ and $\cod\circ{}H=N$. Given a homotopy-equivalence $\alpha:M\to{}N$, we define the interpretation $H:\sigma\to\peqat\C$ by setting $HA=\alpha_A:MA\to{}NA$ for $A\in\Ob\sigma$; $H\vec{A}=\alpha_{\vec{A}}:M\vec{A}\to{}N\vec{A}$ for $\vec{A}\in(\Ob\sigma)^{<\omega}$; and $Hf=\alpha_f$ for $f\in\sigma(\vec{A},B)$.
The proof of the converse is more subtle, but as we will not actually need this direction, we leave it to the reader.
### {#thm:free-htpy-equiv}
If $\C$ is a free f.p. category on the algebraic signature $\sigma$ and $M,N:\sigma\to\D$ are two interpretations into a 2-category $\D$, then $M$ and $N$ are homotopy-equivalent if and only if the induced f.p. functors $\widetilde{M},\widetilde{N}:\C\to\D$ are pseudonaturally equivalent. We have, by definition, that $\widetilde{M}\circ{}i=M$ and $\widetilde{N}\circ{}i=N$ (where $i:\sigma\to\C$ is the interpretation via which $\C$ is free on $\sigma$).
Given a pseudonatural equivalence $\alpha:\tilde{M}\to\tilde{N}$, we thus have a homotopy-equivalence $$M=\widetilde{M}\circ{}i\tox{\alpha\circ{}i}\widetilde{N}\circ{}i=N,$$ where $\alpha\circ{}i$ is the homotopy-equivalence given by $(\alpha\circ{}i)_A=\alpha_{iA}$ and $(\alpha\circ{}i)_f=\alpha_{if}$.
Conversely, given a homotopy-equivalence $\alpha:M\to{}N$, we have by Proposition \[prop:htpy-equiv-sig-ints\] an interpretation $H:\sigma\to{}\peqat\D$ with $\dom\circ{}H=M$ and $\cod\circ{}H=N$. Hence, we have an induced f.p. functor $\widetilde{H}:\C\to\D$ with $\widetilde{H}\circ{}i=H$ and hence $$(\dom\circ\widetilde{H})\circ{}i=M\quad\text{and}\quad
(\cod\circ\widetilde{H})\circ{}i=N$$ (here, we are using that composition of f.p. functors is associative with composition of an interpretation and an f.p. functor). Hence, using the freeness of $\C$ again, we have that $\dom\circ\widetilde{H}=\widetilde{M}$ and $\dom\circ\widetilde{H}=\widetilde{N}$. By Propositions \[prop:htpy-equiv-funs\], we have a pseudonatural equivalence $\tilde{\alpha}:\dom\circ\widetilde{H}\to\cod\circ\widetilde{H}$, and hence we obtain the desired pseudonatural equivalence $$\widetilde{M}=
\dom\circ\widetilde{H}\tox{\tilde\alpha}
\cod\circ\widetilde{H}=
\widetilde{N}
.
\tag*{\qed}$$
The special invariance theorem {#subsec:special-invariance}
------------------------------
We now discuss the application of the abstract invariance theorem to the particular case of the homotopical semantics.
Let us first summarize what we have done so far. Given an algebraic signature $\sigma$ and an $h^=$-fibration $\fib{C}$, we have the canonical extension (Definition \[defn:canonical-extension\]) which is a 1D2F, and can therefore apply Theorem \[thm:abstract-invariance\] to it, and to the free $h^=$-fibration $\fibr{Pf_\sigma}{Pf_\sigma}{Tm_\sigma}$ constructed in the appendix, in which $\Ob\TM_\sigma=(\Ob\sigma)^{<\omega}$ and an object in $\Pf_\sigma$ over $\vec{A}$ is a formula $\phi$ with free variables of sorts $A_1,\ldots,A_{\len{\vec{A}}}$.
In particular, we may take $\fib{C}$ to be the fibration $\fib{HoF_\cfb(\Top_\cf)}$, which is an $h^=$-fibration by Theorem \[thm:top-sset-comparison\]. Given a $\sigma$-interpretation $M:\sigma\to\Top_\cf$ (with induced functors $M:\TM_\sigma\to\Top_\cf$ and $M:\Pf_\sigma\to\Ho(\Top_\cf^\to)_\cfb$), we have by Theorem \[thm:top-sset-comparison\], the theorem of the appendix, as well as Proposition \[prop:suitable-mfib-hfib\] and Theorem \[thm:suitable-hfib-hfib\], that for each $\vec{A}\in\Ob\TM_\sigma$ and each formula $\phi$ in $\fib{Pf_\sigma}^{\vec{A}}$, the object $M(\phi)$ in $\Ho(\Top_\cf^\to)_\cfb$ is computed according to the prescription in §\[subsec:htpical-semantics-direct\].
Next, given a second interpretation $N:\sigma\to\Top_\cf$ and a homotopy-equivalence $\alpha:M\to{}N$, we have by Theorems \[thm:abstract-invariance\] and \[thm:free-htpy-equiv\], for each $\vec{A}\in\TM_\sigma$ and each formula $\phi$ in $\Pf_\sigma^{\vec{A}}$, an equivalence $h:M(\vec{A})\to{}N(\vec{A})$ in $\B$, and an equivalence $M(\phi)\to{}N(\psi)$ over $h$.
Hence, to deduce the homotopy-invariance property promised in §\[subsec:properties-and-invariance\], it remains to verify the following two facts, the proofs of which are the goal of this section. The first is that a homotopy-equivalence of $\sigma$-interpretations in $\sigma\to\Top_\cf$ is the same thing as a homotopy-equivalence of $\sigma$-structures as described in §\[subsec:properties-and-invariance\]. This is quite easy based on what we have already done. The second is to show that an equivalence in $\Ho(\Top_\cf^\to)_\cfb$ over an equivalence in $\Top_\cf$ gives a homotopy-equivalence over a homotopy-equivalence in the sense of in the sense of §\[subsec:properties-and-invariance\].
### {#prop:htpy-equiv-is-htpy-equiv}
Two interpretations $M,N:\sigma\to\Top_\cf$ of $\sigma$, with the 2-categorical structure coming from the canonical extension of $\fib{HoF_\cfb(\Top_\cf)}$, are homotopy-equivalent if and only if they are homotopy-equivalent structures in the sense of §\[subsec:properties-and-invariance\]. The same statement holds with $\Kan$ (or $\C_\cfb$ for any right-proper model category $\C$) instead of $\Top_\cf$. By [@sentai § 19], we know that a 2-cell $
\begin{tikzcd}[row sep=0pt, column sep=10pt]
&\ar[dd, shorten >=2pt, shorten <=-2pt, Rightarrow, "\ \alpha", pos=0.4]&\\[-5pt]
A\ar[rr, "f", bend left]\ar[rr, "g"', bend right]&&B\\
&{}&
\end{tikzcd}
$ in $\Top_\cf$ is given by a homotopy class of homotopies between the continuous maps $f$ and $g$. Accordingly, an equivalence in this 2-category is a homotopy-equivalence in the usual sense.
Hence, a homotopy-equivalence $\alpha:M\to{}N$ of $\sigma$-interpretations is given by a homotopy-equivalence $\alpha_A:MA\to{}NA$ for each $A\in\Ob\sigma$ and, for each $f\in\sigma(\vec{A},B)$, a homotopy (class of homotopies) $$\begin{tikzcd}[column sep=60pt, row sep=30pt]
M\vec{A}\ar[r, "Mf"]\ar[d, "\alpha_{\vec{A}}"']\ar[d]&
MB\ar[d, "\alpha_B"]
\ar[dl, Rightarrow, shorten <=13pt, shorten >=13pt,
"\text{\tiny{homotopy}}" near end, "\sim"' pos=0.45, sloped]
\\
N{\vec{A}}\ar[r, "Nf"]&
NB,
\end{tikzcd}$$ which is precisely the definition of a homotopy-equivalence of $\sigma$-structures from §\[subsec:properties-and-invariance\].
The proof is the same for $\Kan$ or $\C_\cfb$.
### {#lem:induced-path-fibration}
Given any factorization $B\tox{s}B^I\tox{\br{d_1,d_2}}B\times{}B$ of a diagonal $\Delta_B:B\to{}B\times{}B$ as a weak equivalence followed by a fibration in a model category $\C$, and any fibration $p:E\to{}B$, there exists a factorization $E\tox{s}E^I\tox{\br{d_1,d_2}}E\times{}E$ of a diagonal $\Delta_E:E\to{}E\times{}E$ as a trivial cofibration followed by a fibration, and a fibration $p:E^I\to{}B^I$, making the following diagram commute. $$\begin{tikzcd}[]
E\ar[r, "s"]\ar[d, "p"']&
E^I\ar[r, "\br{d_1,d_2}"]\ar[d, dashed, "p^I"]&[15pt]
E\times{}E\ar[d, "p\times{}p"]\\
B\ar[r, "s"]&
B^I\ar[r, "\br{d_1,d_2}"]&
B\times{}B
\end{tikzcd}$$ Moreover, if $\C$ is $\Top$ with the mixed model structure, and $B^I$ denotes the actual exponential object, with $I=[0,1]$, then $E^I$ can be taken to be the corresponding exponential object and $p^I$ the induced map. The following construction comes from [@quillen-ha I.3.1].
We factor $E\tox{\br{\id_E,sp,\id_E}}E\times_BB^I\times_BE$ as a trivial cofibration $E\tox{s}E^I$ followed by a fibration $E^I\tox{\br{d_1,p^I,d_2}}E\times_BB^I\times_BE$. Here $E\times_BB^I\times_BE$ is defined as the pullback $$\begin{tikzcd}
E\times_BB^I\times_BE\ar[r, "\pi_2"]\ar[d, "\br{\pi_1,\pi_3}"']
\ar[rd, phantom, "\lrcorner", pos=0.1]&
B^I\ar[d, "\br{d_1,d_2}"]\\
E\times{}E\ar[r, "p\times{}p"]&B\times{}B.
\end{tikzcd}$$ Since the bottom and right morphisms in the above diagram are fibrations, so are the top and left morphisms. Hence $p^I:E^I\to{}B^I$ and $\br{d_1,d:2}:E^I\to{}E\times{}E$ are fibrations as desired, each being the composite of fibrations.
The “Moreover” statement is much simpler and simply amounts to the two well-known facts that $s:E\to{}E^I$ is a trivial cofibration (it is a deformation retract with closed image) and that $p^I$ is a Hurewicz fibration (this follows directly from the lifting property).
### {#thm:2-cells-in-hofb-tot}
Let $\C$ be a suitable model category *or* the category of topological spaces with the mixed model structure, and consider the $h^=$-fibration $\fib{Ho_\cfb(\C_\cfb)}$ as a 1D2F via its canonical extension (Definition \[defn:canonical-extension\]).
Let $
\begin{tikzcd}[row sep=0pt, column sep=10pt]
&\ar[dd, shorten >=2pt, shorten <=-2pt, Rightarrow, "\ \alpha", pos=0.4]&\\[-5pt]
A\ar[rr, "f", bend left=30pt]
\ar[rr, "g"', bend right=30pt]&&
B\\
&{}&
\end{tikzcd}
$ be a 2-cell in $\C_\cfb$. According to [@sentai §18], such a 2-cell is given by an equivalence class of homotopies $f\to{}g$. Let $h:A\to{}B^I$ be a representative of $\alpha$.
Next, let $(p,f),(q,g):(X,A,x)\to{}(Y,B,y)$ be morphisms in $\Ho(\C^\to)_\cfb$ lying over $f$ and $g$, respectively: $$\begin{tikzcd}
X\ar[r, "p", shift left]\ar[r, "q"', shift right]\ar[d, "x"']&Y\ar[d, "y"]\\
A\ar[r, "f", shift left]\ar[r, "g"', shift right]&B.
\end{tikzcd}\quad\quad$$
If there exists a 2-cell $(p,f)\to{}(q,g)$ in $\Ho(\C^\to)_\cfb$ lying over $\alpha$, then there is a homotopy from $p$ to $q$ lying over $h$ – i.e., there is a homotopy $k:X\to{}Y^I$ from $p$ to $q$ such that the diagram $$\begin{tikzcd}
X\ar[r, "k"]\ar[d, "x"']&Y^I\ar[d, "y^I"]\\
A\ar[r, "h"]&B^I
\end{tikzcd}$$ commutes, where $y^I$ is as in Lemma \[lem:induced-path-fibration\]. We first unwind the definition of the 2-cells in $\Ho(\C^\to)_\cfb$.
Let us write $P,Q$ for $(X,x)$ and $(Y,y)$. By Construction \[constr:groth-const-2-exp\], there exists a 2-cell $(p,f)\to(q,g)$ if and only if the diagram $$\begin{tikzcd}[row sep=3pt]
&f^*Q\ar[dd, "\alpha^*_Q"]\\
P\ar[ru, "\cind{(p,f)}" near end]\ar[rd, "\cind{(q,g)}"' near end]\\
&g^*Q
\end{tikzcd}$$ commutes. Now, referring to [@sentai Definition 9.4], $\alpha^*_Q$ is defined to be the unique morphism making the diagram $$\begin{tikzcd}[column sep=50pt]
f^*Q\ar[r, "\brr{\cind\ct,\alpha!}"]
\ar[d, "\alpha^*_Q"', dashed]&
\pi_1^*Q\wedge{}\Eq_B\ar[d, "\nat^Q_B"]\\
g^*Q\car[r, "\cind\ct"]&\pi_2^*Q\\[-10pt]
A\ar[r, "\br{f,g}"]&B\times{}B
\end{tikzcd}$$ commute. Putting these together, we see the existence of a 2-cell amounts to the commutativity of $$\begin{tikzcd}
&\pi_1^*Q\wedge\Eq_B\ar[d, "\nat_B^Q"]\\
P\ar[r, "\cind{(q,g)}"']\ar[ru, "\brr{\cind{(p,f)},\alpha!}"]&\pi_2^*Q\\[-5pt]
A\ar[r, "\br{f,g}"]&B\times{}B.
\end{tikzcd}$$
Now, inserting the definitions of $P$ and $Q$, and of the pullback functors $\pi_1^*$ and $\pi_2^*$ and the equality object $\Eq_B$ in the $\wedgeq$-cloven $\wedgeq$-fibration $\fib{HoF_\cfb(\C_\cfb)}$, the above amounts to the commutativity of $$\begin{tikzcd}
&Y\times_B{}B^I\ar[d, "\nu"]\\
X\ar[r, "\br{fx,q}"']\ar[ru, "\br{p,hx}"]&B\times{}Y
\end{tikzcd}$$ up to fiberwise-homotopy over $\br{f,g}$, where $\nu$ is any morphism such that $(\nu,\id_{B\times{}B})$ is a representative of $\nat^Q_B$ – in fact, we will want to pick a particular $\nu$ below – and where the fiber product $Y\times_B{}B^I$ is taken with respect to $d_1:B^I\to{}B$.
Now, we will show that the morphisms $Y\times_BB^I\tox{\pi_1}Y$ and $Y\times_BB^I\tox{\nu}B\times{}Y\tox{\pi_2}Y$ are homotopic, and in fact, that there is a homotopy $j:Y\times_BB^I\to{}Y^I$ from $\pi_1$ to $\pi_2\nu$ making $$\begin{tikzcd}
&Y^I\ar[d, "y^I"]\\
Y\times_BB^I\ar[r, "\pi_2"']\ar[ru, "j"]&B^I
\end{tikzcd}$$ commute, from which the proposition follows by precomposing $j$ with $\br{p,hx}:X\to{}Y\times_BB^I$.
To define $j$ making the above triangle commute, we consider the solid commutative square $$\begin{tikzcd}
Y\ar[r, "s"]\ar[d, "\br{\id_Y,sy}"']&Y^I\ar[d, "\br{d_1,y^I}"]\\
Y\times_BB^I\ar[r, "\id"]\ar[ru, dashed]&Y\times_BB^I.
\end{tikzcd}$$ We wish to show that there exists a dashed morphism making the diagram commute. We know from Lemma \[lem:induced-path-fibration\] that $y^I$ is a fibration, hence so is the pullback $\br{d_1,y^I}$ of $y^I$ along $\pi_2:Y\times_B{}B^I$. Thus, it remains to see that $\br{1_Y,sy}$ is a trivial cofibration. If $\C$ is a suitable model category, then it is a cofibration since it is a monomorphism, and is a weak-equivalence by the right-properness of $\C$, since it is the pullback of $s:B\to{}B^I$ along $\pi_2:Y\times_BB^I\to{}B^I$. In case $\C=\Top$, then $\br{1_Y,sy}$ is a trivial cofibration since it is a deformation retract with closed image.
Finally, it remains to see that $j$ is a homotopy form $\pi_1$ to $\pi_2\nu$. It is immediate from the definition that $d_1j=\pi_1$. To see that $d_2j=\pi_2\nu$, we use our freedom in the choice $\nu$. Namely, we define it to be the composite $Y\times{}B^I\tox{j}Y^I\tox{\br{yd_1,d_2}}B\times{}Y$. For this to be a legitimate definition, we need $(\nu,\id_{B\times{}B})$ to be a representative of $\nat_B^Q$; by definition, this means that the triangle $$\begin{tikzcd}
&Y\times_B{}B^I\ar[d, "\nu"]\\
Y\ar[r, "\br{y,\id_Y}"']\ar[ru, "\br{\id_Y,sy}"]&B\times{}Y
\end{tikzcd}$$ commutes up to fiberwise homotopy. In fact, it commutes on the nose.
Examples and further questions
==============================
We now give some examples of sentences and their interpretation under the homotopical semantics. In each case, we fix some algebraic signature $\sigma$ and a $\sigma$-structure in $\Top_\cf$. We then take some first-order sentence over this signature and see what it means for it to be *true* in the $h^=$-fibration $\fib{HoF_\cfb(\Top_\cf)}$; i.e., for its interpretation to be a non-empty space.
By the discussion in the introduction to Part \[sec:h-fib-of-spaces\], this just amounts to using the homotopical semantics, as defined directly in §\[subsec:htpical-semantics-direct\] – i.e., we first apply the singular simplicial set functor to obtain a structure in $\sSet$, and then interpret the formula using the locally cartesian closed structure on $\sSet$, except for equality, which is interpreted as the path-space fibration.
Hence, in each case, we will first see what the sentence means for an arbitrary $\sigma$-structure in $\Kan$, and from this draw a conclusion about an arbitrary $\sigma$-structure in $\Top_\cf$ to which $\Sing$ has been applied (note that $\Sing(X)$ is in $\Kan$ for every $X$ in $\Top$).
After the examples, we consider some further questions regarding the material of this paper.
Examples of interpretations of sentences {#subsec:examples}
----------------------------------------
### {#section-14}
\[examples-contractibility\] First, we consider the signature $\sigma$ consisting of a single sort $A$ and having no operation symbols, and the sentence in this language $$\exists x\ \forall y\ (x=y).$$ We claim that this is interpreted under the semantics as “$A$ is contractible”.
Fix a structure for $\sigma$ in $\Kan$, i.e., a Kan complex $X$.
Now, the formula $x=y$ in the context $\seq{x,y}$ is interpreted as a path-space fibration $X^I\tox{\br{d_1,d_2}}X\times{}X$. Next, the formula $\forall{}y\,(x=y)$ is interpreted as an image $\prod_{\pi_1}(X^I,\br{d_2,d_2})$ under a right-adjoint to the pull-back functor $\pi_1^*:\C/X\to\C/(X\times{}X)$. Finally, $\exists{}x\,\forall{}y{}\,(x=y)$ is interpreted (as always, up to isomorphism) as the domain of $\prod_{\pi_1}(X^I,\br{d_2,d_2})$.
Hence, we are interested in when the domain of $\prod_{\pi_1}(X^I,\br{d_1,d_2})$ is non-empty. This will hold if and only if there is a morphism $(\tm_\sSet,x)\to{}\prod_{\pi_1}(X^I,\br{d_1,d_2})$ in $\sSet/X$ for some $x:\tm_\sSet\to{}X$. By the adjunction, this is equivalent to having a morphism from $\pi_1^*(\tm_\sSet,x)\cong(X,\br{x!,\id_X})$ to $(X^I,\br{d_1,d_2})$ in $\C/X\times{}X$: $$\begin{tikzcd}
&X^I\ar[d, "\br{d_1,d_2}"]\\
X\ar[r, "\br{x!,\id_X}"']\ar[ru, dotted]&X\times{X}.
\end{tikzcd}$$ But this is by definition a (right-)homotopy between $X$ and the constant map $x!$, i.e. a contraction of $X$ onto $x$.
If we start with a topological space $X\in\Ob\Top_\cf$ instead of a Kan complex, then the above shows that $X$ satisfies the sentence in question if and only if the singular simplicial set of $X$ is contractible. But as is well-known, this holds if and only if $X$ itself is contractible.
### {#section-15}
\[subsubsec:examples-homotopies\] Now let $\sigma$ be the signature consisting of two sorts $A,B$ and two function symbols $f,g:A\to{B}$. We consider the sentence $$\forall{x\in{A}}\ (f(x)=g(x)).$$ We claim that this is interpreted as “$f$ is homotopic to $g$”.
Suppose we have a structure for $\sigma$ in $\Kan$; that is, two Kan complexes $X,Y$, and two morphisms $f,g:X\to{Y}$. The formula $y_1=y_2$ (in the context $\br{y_1,y_2}$) will (again) be interpreted as the path-space fibration $Y^I\tox{\br{d_1,d_2}}Y\times{}Y$. Now, we have the morphism $\br{f,g}:X\to{Y\times{Y}}$, and $f(x)=g(x)$ (in the context $\br{x}$) will be interpreted as $\br{f,g}^*(Y^I,\br{d_1,d_2})$. Finally, the above sentence will be interpreted as $\prod_!\br{f,g}^*(Y^I,p)$, the points which are (by the adjunction) in bijection with the sections of $\br{f,g}^*(Y^I,\br{d_1,d_2})$, which are in turn in bijection with the lifts $$\begin{tikzcd}
&Y^I\ar[d, "\br{d_1,d_2}"]\\
X\ar[r, "\br{f,g}"']\ar[ru, dotted]&Y\times{Y}
\end{tikzcd}$$ which are, of course, by definition (right-)homotopies $f\sim{g}$.
For a $\sigma$-structure in $\Top_\cf$, i.e., a pair of maps $f,g:X\to{}Y$, we thus see that the above sentence is satisfied if and only $\Sing(f)$ and $\Sing(g)$ are homotopic and, again, this is the case if and only if $f$ and $g$ are homotopic.
Now considering a signature with two sorts $A,B$ and two function symbols $f:A\to{B}$ and $g:B\to{A}$, we have by the same reasoning as above that $$\forall{x\in{A}}\ (g(f(x))=x)\,\wedge\,
\forall{y\in{B}}\ (f(g(y))=y)$$ is interpreted (in both $\Top_\cf$ and $\Kan$) as “$f$ and $g$ constitute a homotopy equivalence” (i.e., both composites are homotopic to the identity).
Similarly, for the signature consisting of a single sort $A$ and binary function symbol $f:A\times{A}\to{A}$, $$\forall{}x,y,z\in{}A\ [f(f(x,y),z)=f(x,f(y,z))]$$ is interpreted as “$f$ is homotopy-associative”.
### {#section-16}
We now give an example showing that the homotopical semantics are not sound for classical logic. By this we mean that there is a formula of the form $\neg\neg{P}\To{P}$ over some signature $\sigma$ and a structure for $\sigma$ (in $\Top_\cf$ and $\Kan$) under which the interpretation of this formula is empty.
First, we note that it is important that $P$ is not a closed formula. Indeed, a closed formula is interpreted as a Kan complex $X$, and its negation is interpreted as an empty Kan complex or one-point Kan complex according to whether $X$ is non-empty or empty. From this it follows that the interpretation of $\neg\neg{P}\To{P}$ (and similarly $P\vee\neg{P}$) is always non-empty. This circumstance is familiar, for example, from Kleene’s realizability semantics for intuitionistic arithmetic.
Now, for our example, we consider, in the signature $\sigma$ consisting of a single sort $A$ and no function symbols, the sentence $$(\exists x
\ \forall y
\ (\neg\neg{x=y}))
\To
(\exists x
\ \forall y
\ (x=y)).$$ Now, given a structure $X$ (in $\Kan$ or $\Top_\cf$) for $\sigma$ we have already seen that the interpretation of the right side of this implication is inhabited if and only if $X$ is contractible. Let us consider the left side. We first consider the case of $X$ in $\Kan$. We will show that this sentence is satisfied if and only if $X$ is non-empty and path-connected (i.e., for any vertices $x,y\in{X}_0$ there is an edge $e\in{X}_1$ from $x$ to $y$).
We have, again, that $x=y$ is interpreted as the path space $(X^I,p,X)$. We recall that $\neg\neg x=y$ is an abbreviation of $(x=y\To\bot)\to\bot$. Here, $\bot$ is interpreted as the initial Kan fibration $(\emptyset,\text{!`},X)$.
Now, it easy to see that in any category (such as $\Kan/X$) with a strong initial object $0$ (i.e. every morphism with codomain $0$ is an isomorphism), any exponential object $A\To0$ is a subsingleton (i.e, the morphism $!_{A\To0}$ to the terminal object is a monomorphism). Since there exists a morphism $A\to((A\To0)\To0)$, it follows that the unique morphism $A\to1$ factors through $(A\To0)\To0$.
In the case of a Kan fibration $(E,e)$ in $\Kan/X$, this tells us that $\neg\neg({E},e)$ is a monomorphism into $X$ whose image contains the image of $e$. In particular, if $E$ is surjective onto $X$, then $\neg\neg{(E,e)}$ is an isomorphism.
Now, if $X$ is path-connected, then the path space $X^I$ is clearly surjective on vertices. But as an easy inductive argument shows, any Kan fibration which is surjective on vertices is surjective. Hence, for $X$ path-connected, $\neg\neg{x=y}$ is interpreted as an isomorphism, whence it follows that, for $X$ non-empty and path-connected, the following sentence is satisfied. $$\label{eq:path-conn-sentence}
\exists x
\ \forall y
\ (\neg\neg{x=y}).$$ For the other direction, it suffices to see that if a Kan fibration $(E,e)$ in $\Kan/X$ is not surjective, then neither is $\neg\neg(E,e)$, since if $\neg\neg{}x=y$ is interpreted as a non-surjective morphism, the interpretation of (\[eq:path-conn-sentence\]) must be empty. Suppose $e$ is not surjective and let $p\in{X_0}$ be a vertex not in the image of $e$. Then the minimal sub-simplicial set $\br{p}$ of $X$ containing $p$ is disjoint from the image of $e$. Hence $(\br{p},i)\wedge(E,e)\cong\init$, where $i:\br{p}\to{}E$ is the inclusion, so we have a morphism $(\br{p},i)\to\neg({E},e)$. In particular, $(\br{p},i)\wedge\neg({E},e)$ is non-empty, so there cannot be a morphism $(\br{p},i)\to\neg\neg({E},e)$.
Note that by the same kind of argument as in the previous examples, we also have that the interpretation of the sentence (\[eq:path-conn-sentence\]) in $\Top_\cf$ is “$A$ is path-connected”.
Now suppose $X$ is a Kan complex or a topological space in $\Top_\cf$ which is path-connected but not contractible (for instance, the circle). Then the antecedent of the above sentence is interpreted as a non-empty Kan complex, whereas the conclusion is interpreted as the empty Kan complex. Hence the implication is empty.
Finally, we note that this implies that $$\neg\neg{x=y}\To{x=y}$$ cannot be satisfied for such an $X$ since this would imply (by the soundness of the interpretation with respect to intuitionistic logic) that the above sentence would also be satisfied.
### {#section-17}
We saw above that we can easily express that two morphisms constitute a homotopy equivalence, in the same way as we would classically express that they constitute a bijection. We can also classically express that a single function $f:A\to{B}$ is a bijection by $$\forall{b\in{B}}\ \exists{a\in{A}}\ (
fa=b\wedge
\forall{a'\in{A}}\ (fa'=b\To{a'=a})
).$$ Let us see that the interpretation of this sentence (in $\Kan$, and hence in $\Top_\cf$) is non-empty if and only if the interpretation of $f$ is a homotopy equivalence.
First of all, the sentence is equivalent to the conjunction of $$\forall{b\in{B}}\ \exists{a\in{A}}\ (
fa=b)
\quad\text{and}\quad
\forall{b\in{B}}\ \exists{a\in{A}}
\ \forall{a'\in{A}}\ (fa'=b\To{a'=a}
).$$ The same reasoning as in §\[subsubsec:examples-homotopies\] shows that the first part is satisfied by a map $f:X\to{}Y$ if and only if there exists a map $g:Y\to{}X$ such that $f\circ{}g$ is homotopic to $\id_Y$.
The second part, with the quantifiers removed, is interpreted as a certain fibration over $Y\times{}X\times{}X$. By making use of the relevant adjunctions, we can see that the space which is the interpretation of the quantified sentence is inhabited if and only if there exists a map $g:Y\to{}X$ and a dotted lift in the following diagram. $$\begin{tikzcd}
&Y\times{}X^I\ar[d, "\br{\pi_1,d_2\pi_2,d_1\pi_2}"]\\
(Y\times{}X)\times_{Y\times{}X\times{}X}(X\times(X\times_Y{Y^I}))
\ar[r]\ar[ru, "k", dashed]&Y\times{}X\times{}X
\end{tikzcd}$$ Here, in $X\times_YY^I$, $X$ is mapping to $Y$ via $f$ and $Y^I$ is mapping to $Y$ via $d_1$; and in the object on the bottom-left of the diagram, $Y\times{}X$ is mapping to $Y\times{}X\times{}X$ via $\br{\pi_1,g\pi_1,\pi_2}$, and $X\times(X\times_YY^I)$ is mapping to $Y\times{}X\times{}X$ via $\br{d_2\pi_2\pi_2,\pi_1,\pi_1\pi_2}$.
We claim that such a lift exists if and only if $g\circ{}f$ is homotopic to $\id_X$.
In one direction, we have a map $q:X\to{}(Y\times{}X)\times_{Y\times{}X\times{}X}(X\times(X\times_Y{Y^I}))$ which is given by $\br{\br{f,\id_X},\br{gf,\br{\id_X,sf}}}$ (where $s$ is, as usual, the canonical map $Y\to{}Y^I$). Hence, given a lift $k$ as above, the composite $\pi_2kq$ gives a homotopy $X\to{}X^I$ from $\id_X$ to $gf$.
In the other direction, suppose we have a homotopy $h:X\to{}X^I$ from $\id_X$ to $gf$. We then define $k$ as $\br{\pi_1\pi_1,h'}$, where $h'$ is the composite $$(Y\times{}X)\times_{Y\times{}X\times{}X}(X\times(X\times_Y{Y^I}))
\tox{\br{h\pi_2\pi_1,g^I\pi_2\pi_2\pi_2}}
X^I\times_XX^I\to{}X^I$$ in which the second map is composition of paths.
Further problems and questions {#subsec:questions}
------------------------------
We mention some possible further directions.
*Completeness*. This is probably the most natural question to ask about the homotopical semantics: are they a complete semantics for intuitionistic logic? I.e., is it the case that, if a sentence $\phi$ over a signature $\sigma$ is interpreted as a non-empty space in every $\sigma$-structure in $\Kan$, then $\phi$ is intuitionistically provable?
*Limited expressivity*. We mentioned at the end of §\[subsubsec:intro-hott\] that first-order homotopical logic is much less expressive than homotopy type theory. However, we have not *proven* that any particular property is inexpressible, and it would be interesting to do so; for example, to prove that over the trivial signature with a single sort $A$ and no function symbols, there is no sentence $\phi$ satisfied by exactly those spaces $X$ which are simply connected. Or to give another example, there should be no sentence over the signature consisting of one sort and a single binary operation which is satisfied exactly by those operations satisfying the “$A_4$” (or “Stasheff pentagon”) condition.
*Two-dimensional universal algebra*. The construction in the appendix not only produces a free $h^=$-fibration, but one which is described explicitly in terms of first-order formulas. However, if the question is simply that of the *existence* of a free $h^=$-fibration – without requiring it to have anything to do with syntax – we might hope to be able to prove it on purely formal grounds, as we can in universal algebras, using the adjoint functor theorem. One difficulty is that we are dealing with a “two-dimensional” universal property. There has, however, been substantial work on “two-dimensional universal algebra” (see, e.g., [@bkp-2-monad-thy]), and perhaps this could be brought to bear on this problem.
*Higher-dimensional generalizations*. Given the lack of expressiveness mentioned above, it is natural to seek extensions of first-order logic which increase this expressiveness. For example, one could add some, but not all, of what is present in type theory – say, an additional sort $s=_At$ for any two terms $s$ and $t$ of sort $A$, so that one could express that “two homotopies are homotopic”: $e=_{s=_At}e'$.
One would hope to have a nice categorical formulation of the corresponding semantics, as we have for first-order homotopical logic. Indeed, it is also natural to seek “higher-dimensional” generalizations of the fibrational semantics. For example, the fact that we can only express “one level” of homotopies in the language seems to correspond to the fact that the fibrations we are considering are (“only”) two-dimensional. On the “semantic” side, there are natural higher-dimensional categories close at hand – for example, instead of having the fibers of $\fib{HoF(\C)}$ be the *homotopy categories* of the slices $\C/A$, one could try to take the corresponding *$\infty$-categories* (or some truncation thereof). We might then seek a higher-dimensional analogue of the syntactic fibration, morphisms out of which would give the semantics for such “higher-dimensional” extensions of first-order logic.
Appendix: Construction of the free $h^=$-fibration
==================================================
In this appendix, we construct, for each signature $\sigma$, a free finite product category $\TM_\sigma$ on $\sigma$, and a free $h^=$-fibration $\fibr{Pf_\sigma}{Pf_\sigma}{Tm_\sigma}$ over $\TM_\sigma$, out of the terms and formulas (and proofs) of $\sigma$.
The category $\TM_\sigma$ is the well-known “Lawvere algebraic theory” associated to the signature $\sigma$, and its construction is described, for example, in [@makkai-lauchli2 p. 475]. However, as our construction of $\fib{\Pf_\sigma}$ relies delicately on the nature of $\TM_\sigma$, we give a careful construction of the latter as well.
In [@makkai-harnik-lauchli], a construction (due to Lambek) analogous to the present one but for propositional logic is carried out, and our construction proceeds along the same lines (also, see [@makkai-harnik-lauchli] for a thorough discussion of and motivation behind the construction).
However, there are added difficulties in the case of predicate logic. In addition to the general increased complexity due to the presence of variables and quantifiers, there is also the following problem. In the case of propositional logic, the associated syntactic category happens to be free in a stricter sense, since the propositional connectives correspond precisely to the associated categorical operations (namely, finite products and coproducts and exponentials).
In the case of predicate logic, however, this is not quite so. The point is that, from the perspective of the fibration, “substitution” is a primitive operation, whereas in the syntax it is not. Thus, in the “strictly” free $h^=$-fibration, $(x=y)[y:=z]$ and $x=z$ would be two different objects.
Finally, we note that the above references consider the more general – and very familiar – situation in which one has not only a *language*, but a *theory* over that language, and constructs an associated fibration so that the morphisms out of it are models of the theory. We do not do this, as it makes things more complicated, and it is not necessary for our purposes, though it would be desirable in general.
We now summarize the main properties of the syntactic fibration, which are need to deduce the homotopy invariance of the homotopical semantics explained in §\[subsec:properties-and-invariance\] from the special invariance theorem of §\[subsec:special-invariance\]. The proof of this theorem will occupy the remainder of the appendix. For each algebraic signature $\sigma$, there exists an $h^=$-fibration $\fibr{Pf_\sigma}{Pf_\sigma}{Tm_\sigma}$ and an interpretation $M:\sigma\to\TM_\sigma$ (see Definition \[defn:signature\]) with the following properties:
1. $\TM_\sigma$ is a free f.p. category on $\sigma$ via $M$ and $\fib{Pf_\sigma}$ is a free $h^=$-fibration over $\TM_\sigma$.
2. Each object of $\TM_\sigma$ is of the form $M(\vec{A})$ for a unique $\vec{A}\in(\Ob\sigma)^{<\omega}$.
3. \[item:apx-ugly-thm-formulas\] The objects in the fiber $\fib{Pf_\sigma}^{M(\vec{A})}$ over $M(\vec{A})$ are formulas whose free variables have sorts $A_1,\ldots,A_{\len{\vec{A}}}$, up to renaming of variables. More precisely, an object in $\fib{Pf_\sigma}^{M(\vec{A})}$ is an equivalence class of pairs $(\phi,\vec{x})$, in which $\vec{x}$ is a sequence of length $\len{\vec{A}}$ of distinct variables, containing all the free variables of $\phi$ and where $x_i$ has sort $A_i$ for $1\le{}i\le{}\len{\vec{A}}$; and where two such pairs $(\phi,\vec{x})$ and $(\psi,\vec{y})$ are equivalent if $\psi$ is obtained by (possibly renaming the bound variables) and renaming the variables in $\vec{x}$ to those in $\vec{y}$.
4. The morphisms $M(\vec{A})\to{}M(B)$ in $\TM_\sigma$ are given by terms over $\sigma$ of sort $B$ whose free variables have sorts $A_1,\ldots,A_{\len{\vec{A}}}$, up to renaming of variables (in the same sense as in \[item:apx-ugly-thm-formulas\]); in particular, the morphism $\pi_i^{M}:M(\vec{A})\to{}M(A_i)$ is given by the term $x_i$ consisting of a single variable of sort $A_i$; and for any morphisms $p_i:M(\vec{A})\to{}M(B_i)$ given by terms $t_i$ for $1\le{}i\le\len{\vec{B}}$ and a function symbol $f\in\sigma(\vec{B},C)$, the composite $M(\vec{A})\tox{\br{p_1,\ldots,p_n}}M(\vec{B})\tox{M(f)}M(C)$ is given by the term $ft_1\ldots{}t_{\len{\vec{B}}}$.
5. The logical operations on formulas are given by the various $h^=$-fibration operations, in the sense of properties \[item:apx-ugly-thm-first-property\]-\[item:apx-ugly-thm-last-property\] below.
6. \[item:apx-ugly-thm-first-property\] For any two objects $P$ and $Q$ in $\fib{Pf}_\sigma^{M(\vec{A})}$ given by formulas $\phi$ and $\psi$, the conjunction $\phi\wedge\psi$, disjunction $\phi\vee\psi$, and implication $\phi\To\psi$ of these formulas represent objects in $\fib{Pf}_\sigma^{M(\vec{A})}$ which are a product $P\wedge{}Q$, coproduct $P\vee{}Q$, and exponential object $P\To{}Q$ of $P$ and $Q$, respectively.
7. The formulas “true” $\top$ and “false” $\bot$ are terminal and initial objects in $\fib{Pf}_\sigma^{M(\vec{A})}$.
8. For any object $P$ in $\fib{Pf_\sigma}^{M(\vec{A}\seq{B})}$ (here, $\vec{A}\seq{B}$ denotes concatenation of finite sequences) given by a formula $\phi$, the objects $Q$ and $R$ in $\fib{Pf_\sigma}^{M(\vec{A})}$ given by $\exists{}x\phi$ and $\forall{}x\phi$ (where $x$ is the specified variable of sort $B$ in $\phi$) are, respectively, the codomain $\sum_{\pi^{\vec{A}\seq{B}}}P$ of a cocartesian lift of $\pi^{\vec{A}\seq{B}}$ with domain $P$, and the object $\prod_{\pi^{\vec{A}\seq{B}}}P$ associated to a $\prod$-diagram over $\pi^{\vec{A}\seq{B}}$ based on $P$, where $\pi^{\vec{A}\seq{B}}$ is the canonical projection $\br{\pi_1^{M},\ldots,\pi_{\len{\vec{A}}}^M}:M(\vec{A}\seq{B})\to{}M(\vec{A})$.
9. \[item:apx-ugly-thm-last-property\] Given a pair of terms $s,t$ over $\sigma$ of sort $B$ with free variables in $\vec{A}$, the object $P$ in $\fib{Pf_\sigma}^{M(\vec{A})}$ given by the formula $s=t$ is the domain $\br{s,t}^*\Eq_B$ of a cartesian lift $P\to\Eq_B$ of the morphism $M(\vec{A})\to{}M(B)\times{}M(B)=M(\seq{B,B})$ given the terms $s$ and $t$, where $\Eq_B$ is an equality object (i.e., there exists a cocartesian morphism $\top\to\Eq_B$ over $\Delta_B:B\to{}B\times{}B$).
In addition, we have the following property:
1. For any two objects $P$ and $Q$ in $\fib{Pf_\sigma}^{\vec{A}}$ given by formulas $\phi$ and $\psi$, there is a morphism $P\to{}Q$ in $\fib{Pf_\sigma}^{\vec{A}}$ if and only if $\phi\To\psi$ is an intuitionistic validity.
The syntax of first-order logic
-------------------------------
To begin with, we define the syntax of the first-order language over a given algebraic signature $\sigma$ – i.e., the set of terms and the set of formulas over $\sigma$ – upon which the rest of the construction will be based. The particular identity of the sets of terms and formulas is not important, but rather only the universal properties defining these sets (Proposition \[prop:recursion-principle\]) – namely, that they are *freely generated* by the various syntactic operations. Nonetheless, for definiteness, we will define these sets very explicitly.
For the rest of the appendix, fix an algebraic signature $\sigma$, as well as an arbitrary infinite set $\Varn$ of “variable names” (for definiteness, we could take $\Varn=\N$). Since we are fixing $\sigma$, we will sometimes omit the prefix “$\sigma$-” or the subscript $_\sigma$ from expressions such as “$\sigma$-term” and $\Tm_\sigma$ below.
We maintain from Definition \[defn:signature\] the notation $\len{\vec{t}}$ for the length of a finite sequence. Also, we write $\vec{s}\vec{t}$ for the concatenation of finite sequences $\vec{s}$ and $\vec{t}$, and we will sometimes conflate a finite sequence with the set of its elements.
### {#defn:raw-syntax}
Given a sort $A\in\Ob\sigma$ and any thing $s$, we write $s:A$ to indicate that $s$ is a non-empty sequence whose first element is $A$. Given a sequence $\vec{A}\in(\Ob\sigma)^{<\omega}$ of sorts and any sequence $\vec{t}$, we write $\vec{t}:\vec{A}$ to indicate that $\len{\vec{t}}=\len{\vec{A}}$ and that $t_i:A_i$ for $1\le{}i\le{}\len{\vec{A}}$.
Next, we define the set $\Tm^n_\sigma$ of *level-$n$ $\sigma$-terms* for $n\in\N$ inductively as follows. The set $Tm^0_\sigma$ is empty. A level-$(n+1)$ $\sigma$-term is either (i) a length-2 sequence $\seq{A,v}$ where $A\in\Ob\sigma$ is a sort, and $v\in\Varn$ is a variable name, or (ii) a sequence $\seq{B,f,t_1,\ldots,t_k}$ where $f\in\sigma(\vec{A},B)$ for some $\vec{A}$ with $\len{\vec{A}}=k$, each $t_i$ is a level-$n$ $\sigma$-term, and $t_i:A_i$ for $1\le{}i\le{}k$. The set $\Tm=\Tm_\sigma$ of *$\sigma$-terms* is $\bigcup_{n\in\N}\Tm^n_\sigma$.
We define the set $\Var=\Var_\sigma$ of *$\sigma$-variables* to be $\Ob\sigma\times\Varn\subseteq\Tm_\sigma$, and for each $A\in\Ob\sigma$, we set $\Var_A=\{A\}\times\Varn\subseteq\Var$.
Note that every term is a non-empty sequence whose first element is a sort. For a term $t$, we write $\tp(t)$ for the first element of $t$; hence $t:\tp(t)$. For a sequence $\vec{t}$ of terms, we write $\tp(\vec{t})$ for the sequence $\seq{\tp(t_1),\ldots,\tp(t_{\len{\vec{t}}})}$.
Next, fix a set $\set{\top,\bot,\wedge,\vee,\To,\forall,\exists,=}$ with 8 distinct elements (for definiteness, we could take $\top=0$, $\bot=1$, etc.).
We define the set $\Formset^n_\sigma$ of *level-$n$ $\sigma$-formulas* for $n\in\N$ inductively as follows. The set $\Formset^0_\sigma$ is empty. A level-$(n+1)$ $\sigma$-formula is either (i) a length-3 sequence $\seq{=,s,t}$, where $s$ and $t$ are terms with $\tp(s)=\tp(t)$, (ii) one of the length-1 sequences $\seq{\top}$ or $\seq{\bot}$, (iii) a triple $(X,\phi,\psi)$, where $X\in\set{\wedge,\vee,\To}$ and $\phi,\psi$ are level-$n$ $\sigma$-formulas, or (iv) a triple $(X,v,\phi)$ with $X\in\set{\forall,\exists}$, $v\in\Var$, and $\phi$ a level-$n$ $\sigma$-formula. The set $\Formset=\Formset_\sigma$ of *$\sigma$-formulas* is $\bigcup_{n\in\N}\Formset_\sigma^n$. We call level-$1$ formulas *atomic formulas*.
For $B\in\Ob\sigma$, we denote by $\Tm_B\subseteq\Tm$ the subset consisting of $t\in\Tm$ with $t:B$. For each $f\in\sigma(\vec{A},B)$, we denote by $f$ the function $\Tm_{A_1}\times\cdots\times{}\Tm_{A_n}\to\Tm_B$ (where $n=\len{\vec{A}}$) taking $t_1,\ldots,t_n$ to $ft_1\ldots{}t_n:=\seq{B,f,t_1,\ldots,t_n}$.
Similarly, we have binary operations $\wedge,\vee,\To:\Formset\times\Formset\to\Formset$, “0-ary” operations $\top,\bot\in\Formset$, unary operations $\forall{}v,\exists{}v:\Formset\to\Formset$ for $v\in\Var$, and a binary operation $=_A:\Tm_A\times\Tm_A\to\Formset$ for each sort $A$. For the above binary operations, we use infix notation ($\phi\wedge\psi$, $s=_At$, etc.).
### {#prop:induction-principle}
[**Proposition (*Principle of induction*).**]{} $\Tm$ is the least subset $S\subseteq{}\Tm$ containing $\Var$ and closed under the operations $f:(\Tm_{A_1}\cap{}S)\times\cdots\times(\Tm_{A_{\len{\vec{A}}}}\cap{}S)\to\Tm$ for each $f\in\sigma(\vec{A},B)$.
Similarly $\Formset$ is the least subset of $\Formset$ containing the image of $=_A:\Tm\times\Tm\to\Formset$, and closed under each of the operations $\top,\bot,\wedge,\vee,\To,\forall{}v,\exists{}v$. Immediate from the definitions of $\Tm$ and $\Formset$.
### {#prop:unique-readability}
[**Proposition (*Unique readability*).**]{} Each element of $\Formset$ is in the image of exactly one of the maps $\wedge,\vee,\To,\top,\bot,\forall{}v,\exists{}v,=_B$ and has a unique preimage under this map. In other words, the map $$\Formset^2\sqcup
\Formset^2\sqcup
\Formset^2\sqcup
\Formset^0\sqcup
\Formset^0\sqcup
\Formset\times\Var\sqcup\,
\Formset\times\Var\sqcup\,
\textstyle\bigsqcup_{B\in\Ob\sigma}(\Tm_B\times\Tm_B)
\to
\Formset$$ obtained by putting all of these maps together is a bijection.
Similarly, the map $$\Var\sqcup\,
\Bigg(
\bigsqcup_{\substack{\vec{A}\in(\Ob\sigma)^{<\omega}\\{}B\in\Ob\sigma}}
\bigsqcup_{f\in\sigma(\vec{A},B)}
\Tm_{A_1}\times\cdots\times\Tm_{A_{\len{\vec{A}}}}
\Bigg)\to\Tm$$ induced from the inclusion $\Var\hookrightarrow\Tm$ and the maps $f:\Tm_{A_1}\times\cdots\times\Tm_{A_{\len{\vec{A}}}}\to\Tm$ is a bijection. Immediate from the definitions of $\Tm$ and $\Formset$.
### {#prop:recursion-principle}
[**Proposition (*Principle of recursion*).**]{} Given any set $X$ with maps $\tilde=_A:\Tm_A\times\Tm_A\to{}X$ for $A\in\Ob\sigma$, elements $\tilde\top,\tilde\bot\in{}X$, binary operations $\tilde\wedge,\tilde\vee,\tilde\To:X\times{}X\to{}X$, and unary operations $\tilde\forall{}v,\tilde\exists{}v:X\to{}X$ for each $v\in\Var$, there is a unique map $\Formset\to{}X$ taking the operation $\wedge$ to $\tilde\wedge$, $\vee$ to $\tilde\vee$, and so on.
Similarly, given any family $\set{X_A}_{A\in\Ob\sigma}$, together with maps $\beta:\Var_A\to{}X_A$ for each $A\in\Ob\sigma$, and operations $\tilde{f}:X_{A_1}\times\cdots{}X_{A_{\len{\vec{A}}}}\to{}X_B$ for each $f\in\sigma(\vec{A},B)$, there is a unique map $\Tm\to\bigcup_{A\in\Ob\sigma}X_A$ taking $\Tm_A$ to $X_A$ for each $A$, taking each operation $f$ to $\tilde{f}$ for $f\in\sigma(\vec{A},B)$, and taking each variable $v\in\Tm$ to $\beta(v)$. By induction on $n$, we can show that there is a unique $\Formset^n_\sigma\to{}X$ (or $\Tm^n_\sigma\to\bigcup_{A\in\Ob\sigma}X_A$) satisfying the required condition; in the induction step, we use Proposition \[prop:unique-readability\]. From this, we conclude that there is a unique function $\Formset\to{}X$ (or $\Tm\to\bigcup_{A\in\Ob\sigma}X$) as desired.
### {#defn:substitution}
Given a sequence $\vec{x}\in\Var^{<\omega}$ of distinct variables and a sequence $\vec{t}\in\Tm^{<\omega}$ of terms with $\tp(\vec{x})=\tp(\vec{t})$, we define *substitution of $\vec{t}$ for $\vec{x}$*, which is a map $\Tm\to\Tm; s\mapsto{}s[\vec{x}:=\vec{t}]$, taking $\Tm_A$ to $\Tm_A$ for each $A\in\Ob\sigma$, by recursion on $s$. We set $$v[\vec{x}:=\vec{t}]=
\begin{cases}
t_i&\text{if $v=x_i$ for some (necessarily unique) $1\le{}i\le\len{\vec{x}}$}
\\
v&\text{otherwise},
\end{cases}$$ for a variable $v$, and we define $(ft_1\ldots{}t_n)[\vec{x}:=\vec{t}]$ to be $f(t_1[\vec{x}:=\vec{t}])\ldots(t_n[\vec{x}:=\vec{t}])$. Given a sequence $\vec{s}\in\Tm^{<\omega}$ of terms, we define $\vec{s}[\vec{x}:=\vec{t}]$ to be $\seq{s_1[\vec{x}:=\vec{t}],\ldots,s_{\len{\vec{s}}}[\vec{x}:=\vec{t}]}$.
We similarly define substitution for formulas by recursion. We set $\phi[\vec{x}:=\vec{t}]$ to be (i) $\phi$ if $\phi$ is $\top$ or $\bot$, (ii) $s[\vec{x}:=\vec{t}]=_As'[\vec{x}:=\vec{t}]$ if $\phi$ is $s=_As'$, (iii) $\psi[\vec{x}:=\vec{t}]\xbop\psi'[\vec{x}:=\vec{t}]$ if $\phi$ is $\psi{}\xbop\psi'$, where $\xbop$ is one of $\wedge,\vee,\To$, (iv) $\xop{}v(\psi[\vec{x}^{(\vec{x},v)}:=\vec{s}^{(\vec{x},v)}])$ if $\phi$ is $\xop{}v\psi$, where $\xop$ is one of $\forall,\exists$. Here, for any sequence $\vec{r}$, we write $\vec{r}^{(\vec{x},y)}$ for the sequence which is obtained from $\vec{r}$ by removing the $i$-th entry in case $y=x_i$ for some (necessarily unique) $1\le{}i\le\len{\vec{x}}$, and which is just $\vec{r}$ itself otherwise.
Next, we define the set of free variables $\FV(\tau)\subseteq\Var$ and bound variables $\BV(\tau)\subseteq\Var$ in a term or a formula $\tau$ by recursion. If $\tau$ is a term, we set $\BV(\tau)=\emptyset$, and we define $\FV(\tau)$ to be $\set{v}$ if $\tau=v\in\Var$, and to be $\bigcup_{i=1}^n\FV(t_i)$ if $\tau=ft_1\ldots{}t_n$. If $\tau$ is a formula, we define (i) $\FV(\tau)=\FV(s)\cup\FV(t)$ if $\tau$ is $s=_At$, (ii) $\FV(\tau)=\emptyset$ if $\tau$ is $\top$ or $\bot$, (iii) $\FV(\tau)=\FV(\phi)\cup\FV(\psi)$ if $\tau$ is $\phi{}\xbop\psi$ where $\xbop$ is one of $\wedge,\vee,\To$, and (iv) $\FV(\tau)=\FV(\phi)\setminus\set{v}$ if $\tau$ is $\xop{}v\phi$ where $\xop$ is one of $\forall,\exists$. The defining clauses for $\BV$ are the same as those of $\FV$, except we replace (iv) by $\BV(\xop{}v\phi)=\BV(\phi)\cup\set{v}$.
We define the set $\V(\tau)$ of variables in a term or formula $\tau$ to be $\FV(\tau)\cup\BV(\tau)$.
Free finite product categories
------------------------------
We now construct the free finite product category $\TM_\sigma$ on a given algebraic signature $\sigma$. As an intermediate step, we will construct an equivalent category $\OTM_\sigma$. The difference between the two is that the objects of $\OTM_\sigma$ are arbitrary sequences of (distinct) *variables*, while those of $\TM_\sigma$ are sequences of *sorts*. Thus, $\TM_\sigma$ is more canonical, as it does not depend on the choice of the set $\Varn$ of variable names. It also satisfies the “strict” universal property of being free on $\sigma$ from Definition \[defn:signature\], which determines it up to *isomorphism*, while $\OTM_\sigma$ does not.
### {#prop:tm-cat-sub-props}
Given any sequence $\vec{s}$ of terms:
1. \[item:tm-cat-fv\] If $\FV(\vec{s})\subseteq\vec{x}$, then $\FV(\vec{s}[\vec{x}:=\vec{t}])\subseteq\FV(\vec{t})$.
2. \[item:tm-cat-id\] $\vec{t}[\vec{x}:=\vec{x}]=\vec{t}$
3. \[item:tm-cat-assoc\] If $\FV(\vec{s})\subseteq\vec{x}$, then $\vec{s}[\vec{x}:=\vec{t}][\vec{y}:=\vec{u}]=\vec{s}[\vec{x}:=(\vec{t}[\vec{y}:=\vec{u}])]$.
By induction.
### {#section-18}
We define a category $\OTM=\OTM_\sigma$ as follows. The objects of $\OTM$ are (possibly empty) finite sequences of distinct variables. A morphism $\vec{x}\to\vec{y}$ is a sequence $\vec{t}\in\Tm^{<\omega}$ such that $\tp(\vec{t})=\tp(\vec{y})$ and $\FV(\vec{t})\subseteq\vec{x}$.
The composite of $\vec{s}:\vec{x}\to\vec{y}$ and $\vec{t}:\vec{y}\to\vec{z}$ is $\vec{t}[\vec{y}:=\vec{s}]$, which by Proposition \[prop:tm-cat-sub-props\] \[item:tm-cat-fv\] has free variables in $\vec{x}$ as required. Associativity and the existence of identities follow from Proposition \[prop:tm-cat-sub-props\] \[item:tm-cat-id\]-\[item:tm-cat-assoc\].
Let us call a morphism $\vec{t}:\vec{x}\to\vec{y}$ in $\OTM$ a *renaming* if $\vec{t}=\vec{x}$ (and hence $\tp(\vec{x})=\tp(\vec{y})$). It follows that $\vec{t}$ is an isomorphism with inverse $\vec{y}:\vec{y}\to\vec{x}$. Note that there exists a renaming $\vec{x}\to\vec{y}$ if and only if $\tp(\vec{x})=\tp(\vec{y})$, and in this case there exists exactly one. We say that two morphisms $p$ and $q$ in $\OTM$ (not necessarily having the same domain and codomain) are *equivalent* if $p=r\cdot{}q$ or $p=q\cdot{}r$ for some renaming $r$.
We now define $\TM=\TM_\sigma$ to be the category obtained by identifying equivalent morphisms in $\OTM$; the objects of $\TM$ are finite sequences of sorts, and a morphism $\vec{A}\to\vec{B}$ is an equivalence class of morphisms $\vec{x}\to\vec{y}$ in $\OTM$ with $\tp(\vec{x})=\vec{A}$ and $\tp(\vec{y})=\vec{B}$. We leave to the reader the easy verification that the composition on $\OTM$ descends to $\TM$, and that the canonical functor $\OTM\to\TM$ is initial among functors taking each renaming to an identity morphism.
Note that the canonical functor $\OTM\to\TM$ is in fact an equivalence; it is clearly full and surjective on objects, and it is faithful, since if two morphisms $\vec{s},\vec{t}:\vec{x}\to\vec{y}$ in $\OTM$ have the same image in $\TM$, then $\vec{t}=\vec{r}\vec{s}$ or $\vec{t}=\vec{s}\vec{r}$ for a renaming $\vec{r}$, which can only be the identity morphism. Choosing a section $\TM\to\OTM$ (note that we can find a section even without the axiom of choice, assuming there exists an injection $\N\to\Varn$), we then have that, given a functor $F:\OTM\to\D$ taking renamings to identity morphisms, the induced functor $\TM\to\D$ is *equal* (not just isomorphic) to $F$ composed with the section $\TM\to\OTM$.
We denote by $[\vec{t},\vec{x}]$ the morphism $\tp(\vec{x})\to\tp(\vec{y})$ in $\TM$ represented by $\vec{t}:\vec{x}\to\vec{y}$. Composition in $\TM$ is then given by $[\vec{t},\vec{y}]\cdot[\vec{s},\vec{x}]=[\vec{t}[\vec{y}:=\vec{s}],\vec{x}]$.
For each $\vec{A}\in\Ob\TM$ and $1\le{}i\le\len{\vec{A}}$, we denote by $\pi_i^{\vec{A}}$ the morphism $[x_i,\vec{x}]:\vec{A}\to\seq{A_i}$ in $\TM$, where $\vec{x}:\vec{A}$. For a term $t$, we also write $[t,\vec{x}]$ for $[\seq{t},\vec{x}]$.
For each function symbol $f\in\sigma(\vec{A},B)$, we denote simply by $f$ the morphism $[fx_1\ldots{}x_{\len{\vec{A}}},\vec{x}]:\vec{A}\to\seq{B}$ in $\TM$, where again $\vec{x}:\vec{A}$.
### {#section-19}
Each $\vec{A}\in\Ob\TM$, is the product of the objects $\seq{A_i}$ for $1\le{}i\le{}\len{\vec{A}}$, with product projections the $\pi_i^{\vec{A}}:\vec{A}\to\seq{A_i}$. Given $\vec{B}\in\Ob\TM_\sigma$ and morphisms $\vec{B}\to\seq{A_i}$ – i.e., terms $t_i:A_i$ with $\FV(t_i)\subseteq\vec{x}$ for some $\vec{x}:\vec{B}$ – we want to show that there is a unique morphism $\vec{B}\to\vec{A}$ – i.e., a sequence $s_1,\ldots,s_{\len{\vec{A}}}$ of terms with $\FV(s_i)\subseteq\vec{x}$ and $s_i:A_i$ – whose composites with the $\pi_i^{\vec{A}}$ are the given morphisms $t_i$ – i.e., such that $x_i[\vec{x}:=\vec{s}]=t_i$. Clearly, we can and must take $s_i=t_i$ for all $i$.
### {#section-20}
The category $\TM$ has binary products given by $\vec{A}\times\vec{B}=\vec{A}\vec{B}$ (concatenation of sequences) with projections $\pi_1=[\vec{x},\vec{x}\vec{y}]$ and $\pi_2=[\vec{y},\vec{x}\vec{y}]$, where $\vec{x}:\vec{A}$ and $\vec{y}:\vec{B}$. Also the empty sequence is a (actually, the unique) terminal object.
Hereafter, we will consider $\TM$ to be endowed with these specified binary products and terminal object. This follows from the fact that $\pi_1$ and $\pi_2$ are the maps into the products $\vec{A}$ and $\vec{B}$ induced by the product projections $\pi_i^{\vec{A}\vec{B}}$.
### {#section-21}
We define an interpretation $M=M_\sigma:\sigma\to\TM_\sigma$ by setting $MA=\seq{A}$ for $A\in\Ob\Sigma$, defining $M\vec{A}$ to be the product $\set{\pi_i^{\vec{A}}:\vec{A}\to{}A_i}_{i=1}^{\len{\vec{A}}}$, and setting $Mf=f:M\vec{A}\to{}MB$ for $f\in\sigma(\vec{A},B)$.
We now want to show that $\TM_\sigma$ is free on $\sigma$ via $M_\sigma$.
### {#prop:unary-mor-fregen}
The $(\Ob\TM\times\Ob\sigma)$-indexed family of sets $(\Hom_\TM(\vec{A},\seq{B}))_{\vec{A},B}$ is freely generated by the operations
1. $\pi_i^{\vec{A}}\in\Hom_\TM(\vec{A},\br{A_i})$ for $\vec{A}\in\Ob\TM$ and $1\le{}i\le\len{\vec{A}}$
2. The operation $\Hom_{\TM}(\vec{C},\seq{A_1})\times\cdots\times\Hom_{\TM}(\vec{C},\seq{A_{\len{\vec{A}}}})\to
\Hom_\TM(\vec{C},\seq{B})$ taking $t_1,\ldots,t_{\len{\vec{A}}}$ to the composite $\vec{C}\tox{\br{t_1,\ldots,t_{\len{\vec{A}}}}}\vec{A}\tox{f}\seq{B}$ for each $f\in\sigma(\vec{A},B)$ and each $\vec{C}\in\Ob\TM$
in the following sense:
1. \[item:unary-mor-freegen-gen\] For any $(\Ob\TM\times\Ob\sigma)$-indexed family of subsets $(X_{\vec{A}B}\subseteq\Hom_\TM(\vec{A},\seq{B}))_{\vec{A}B}$ which is closed under these operations, we have $X_{\vec{A}B}=\Hom_\TM(\vec{A},\seq{B})$ for all $\vec{A},B$.
2. \[item:unary-mor-freegen-uniq\] We have a “unique readability” property as in Proposition \[prop:unique-readability\]; that is, the function $$\bigsqcup_{\substack{\vec{A}\in\Ob\TM}}\!\!\set{1,\ldots,\len{\vec{A}}}\ \sqcup\!\!
\bigsqcup_{\substack{\vec{A}\in\Ob\TM_\sigma\\B\in\Ob\sigma\\\vec{C}\in\Ob\TM}}
\bigsqcup_{f\in\sigma(\vec{A},B)}
\prod_{i=1}^{\len{\vec{A}}}
\Hom_\TM(\vec{C},\seq{A_i})
\to\!\!
\bigsqcup_{\substack{\vec{A}\in\Ob\TM_\sigma\\B\in\Ob\sigma}}\!\!
\Hom_\TM(\vec{A},\seq{B})$$ induced by these operations is a bijection.
3. \[item:unary-mor-freegen-rec\] Given an $(\Ob\TM\times\Ob\sigma)$-indexed family of sets $(X_{\vec{A}B})_{\vec{A}B}$ together with operations as in (1)-(2), there is a unique family of maps $(\Hom_\TM(\vec{A},\seq{B})\to{}X_{\vec{A}B})_{\vec{A}B}$ preserving all the operations.
To prove (i), given any such family $(S_{\vec{A}B})_{\vec{A}B}$ we can show by induction on $t$ that $[t,\vec{x}]\in{}S_{\tp(\vec{x})\tp{t}}$ for every $\vec{x}$ with $\V(t)\subseteq\vec{x}$.
As for (ii), surjectivity follow from (i), and injectivity follows from unique readability for terms (Proposition \[prop:unique-readability\]) and the definition of renaming.
We prove (iii) in the same way as in Proposition \[prop:recursion-principle\]. Suppose we are given a family $(X_{\vec{A}B})_{\vec{A}B}$ together with the indicated operations. We now let $\Hom_\TM^0(\vec{A},\seq{B})=\emptyset$ for each $\vec{A}$ and $B$, and let $\Hom_\TM^{n+1}(\vec{A},\seq{B})\subseteq\Hom_\TM(\vec{A},\seq{B})$ consist of all the elements obtained by applying one of the operations to the elements of the sets $\Hom_\TM^n(\vec{A},\seq{B})$. By (i), $\bigcup_{n\in\N}\Hom_\TM^n(\vec{A},\seq{B})=\Hom_\TM(\vec{A},\seq{B})$. We can now show by induction on $n$ that for each $n$, there exists a unique family of functions $(\Hom_\TM^n(\vec{A},\seq{B})\to{}X_{\vec{A}\seq{B}})_{\vec{A}\seq{B}}$ respecting the operations, where the induction step uses unique readability. The claim follows.
### {#prop:freefp-fun}
Given an f.p. category $\D$ and an interpretation $\bar{F}:\sigma\to\D$, there is a unique f.p. functor $F:\TM\to\D$ such that $F\circ{}M=\bar{F}$.
For every object $\vec{A}$ of $\TM$, we have $\vec{A}=M\vec{A}$, and hence we are forced to take $F\vec{A}=\bar{F}\vec{A}$.
We next define the action of $F$ on morphisms of the form $t:\vec{A}\to\seq{B}$ by recursion on $t$ (using Proposition \[prop:unary-mor-fregen\] \[item:unary-mor-freegen-rec\]).
If $t$ is $\pi^M_i$, then the requirement $F\circ{}M=\bar{F}$ forces us to take $Ft$ to be the product projection $\pi^{\bar{F}}_i:\bar{F}\vec{A}\to\bar{F}A_i$.
Suppose $t=f\br{t_1,\ldots{},t_n}$ for some $f\in\sigma(\vec{C},B)$. Then the requirements that $F\circ{}M=\bar{F}$ and that $F$ be a functor force us to take $Ft$ to be the composite of $\bar{F}f$ with $\br{Ft_1,\ldots,Ft_n}:F\vec{A}\to{}F\vec{C}$.
Finally, any morphism $\vec{A}\to\vec{B}$ with $\len{\vec{B}}\ne{1}$ is of the form $\br{t_1,\ldots,t_n}:\vec{A}\to\vec{B}$. Hence, we are forced to take $Ft$ to be $\br{Ft_1,\ldots,Ft_n}:F\vec{A}\to{}F\vec{B}$.
It remains to see that $F$, so defined, is an f.p. functor. The identity morphism $\vec{A}\to\vec{A}$ in $\TM$ is $\br{\pi_1^{\vec{A}},\ldots,\pi_{\len{\vec{A}}}^{\vec{A}}}$. This is taken by $F$ to $\br{F\pi_1^{\vec{A}},\ldots,F\pi_{\len{\vec{A}}}^{\vec{A}}}:F\vec{A}\to{}F\vec{A}$, which is the identity morphism, since $F\pi_i^{\vec{A}}$ is by definition the product projection $\pi^{\bar{F}}_i:F\vec{A}\to{}F\seq{A_i}$.
Next, consider a composite $\vec{A}\tox{s}\vec{B}\tox{t}\seq{C}$, where $s=\br{s_1,\ldots,s_n}$. We show by induction on $t$ that this is to taken by $F$ to the corresponding composite in $\D$.
First, we recall that $Fs$ is by definition $\br{Fs_1,\ldots,Fs_n}:F\vec{A}\to{}F\vec{B}$.
Now, if $t=\pi_i^{\vec{B}}$, then $ts$ is just $s_i$, and $Ft=\pi^{\bar{F}}_i$, hence $
Ft\cdot{}Fs
=
\pi_i^{\bar{F}}\cdot{}\br{Fs_1,\ldots,Fs_n}
=
Fs_i
$ as desired.
If $t=f\br{t_1\ldots{}t_m}$, where $f\in\sigma(\vec{D},C)$, then $ts$ is the composite of $f$ with $\br{t_1s,\ldots,t_ms}:\vec{A}\to\vec{D}$. By definition, $F$ takes this composite to the composite of $Ff:F\vec{D}\to{}F\seq{C}$ with $\br{F(t_1s),\ldots,F(t_ms)}$. By induction, we have that $F(t_is)=Ft_i\cdot{}Fs$ for each $i$. Hence, $F(ts)$ is $$Ff\cdot
\br{
Ft_1\cdot{}Fs,\ldots,Ft_n\cdot{}Fs
}=
Ff\cdot
\br{
Ft_1,\ldots,Ft_n
}\cdot{}Fs$$ which is by definition $Ft\cdot{}Fs$. The proof that $F$ preserves composites of the form $\br{t_1,\ldots,t_n}s$ with $n>1$ is similar.
Finally, to see that $F$ is f.p., it suffices to see that it preserves the specified terminal object and binary products. It preserves the terminal object by definition. By definition, $F$ preserves each of the products $\vec{A}$ of the $\seq{A_i}$ with projections $\pi_i^{M}$. It follows that $F$ also preserves the specified products $\vec{A}\xot{\pi_1}\vec{A}\vec{B}\tox{\pi_2}\vec{B}$ since $\pi_1$ and $\pi_2$ are induced by the projections $\pi_i^{M}$.
### {#prop:freefp-nt}
Given an f.p. category $\D$, two f.p. functors $F,G:\TM\to\D$, and a homomorphism $\bar\alpha:F\circ{}M\to{}G\circ{}M$, there is a unique natural transformation $\alpha:F\to{G}$ with $\bar\alpha=\alpha\circ{}i$. For objects in $\TM$ of the form $\seq{A}$, we must take $\alpha_{\seq{A}}=\bar\alpha_{A}$. For any other object $\vec{A}$, the diagram $$\begin{tikzcd}
F\vec{A}\ar[d, "F\pi_i^{\vec{A}}"']\ar[r, "\alpha_{\vec{A}}"]
&G\vec{A}\ar[d, "G\pi_i^{\vec{A}}"]\\
F\seq{A_i}\ar[r, "\alpha_{\seq{A_i}}"]&G\seq{A_i}
\end{tikzcd}$$ must commute for each $1\le{i}\le\len{\vec{A}}$, and (having already defined the $\alpha_{\seq{A_i}}$) there is exactly one morphism $\alpha_{\vec{A}}$ satisfying this condition, since $G\vec{A}$ is a product of the $GA_i$ with projections $G\pi^{\vec{A}}_i$.
Hence, $\alpha$ is determined uniquely, and it remains to see that the above prescription really defines a natural transformation. We first prove, by induction on $t$, that the naturality square commutes for morphisms $t:\vec{A}\to\seq{B}$ in $\TM$.
The case $t=\pi_i^{\vec{A}}$ follows from the definition. Suppose $t=f\br{t_1\ldots{}t_n}$ for some $f\in\sigma(\vec{C},B)$. The naturality squares for each $t_i$ commute by the induction hypothesis. This means that the outside of each diagram $$\begin{tikzcd}
F\vec{A}\ar[r, "{F\br{t_1,\ldots,t_n}}"]\ar[d, "\alpha_{\vec{A}}"']&[50pt]
F\vec{C}\ar[r, "F\pi_i^{\vec{C}}"]\ar[d, "\alpha_{\vec{C}}"']&[20pt]
F\seq{C_{i}}\ar[d, "\alpha_{\seq{C_{i}}}"]\\
G\vec{A}\ar[r, "{G\br{t_1,\ldots,t_n}}"]
&G\vec{C}\ar[r, "G\pi_i^{\vec{C}}"]&G\seq{C_{i}}
\end{tikzcd}$$ commutes, since the morphisms on the top and bottom are $Ft_i$ and $Gt_i$. Since the right square also commutes by the base case of the induction, it follows that the left square commutes by the universal property of the product $G\vec{C}$.
Hence, we have the commutativity of the naturality square for $t$, since this is the outside of the diagram $$\begin{tikzcd}[column sep=60pt]
F\vec{A}\ar[r, "{F\br{t_1,\ldots,t_n}}"]
\ar[d, "\alpha_{\vec{A}}"']&
F\vec{C}\ar[r, "Ff"]\ar[d, "\alpha_{\vec{C}}"']&
F\seq{B}\ar[d, "\alpha_{\seq{B}}"]\\
G\vec{A}\ar[r, "{G\br{t_1,\ldots,t_n}}"]
&G\vec{C}\ar[r, "Gf"]&G\seq{B},
\end{tikzcd}$$ and the right square commutes since $\bar\alpha$ is a homomorphism.
It remains to show naturality for morphisms $\vec{A}\to\vec{B}$ with $\len{\vec{B}}\ne1$, but in fact we essentially already gave the argument for this above.
### {#section-22}
$\TM$ is free on $\sigma$ via $M$. This is precisely the content of Propositions \[prop:freefp-fun\] and \[prop:freefp-nt\].
The syntactic fibration
-----------------------
Our next task is to define a functor $\OTM^\op\to\Set$ sending $\vec{x}$ to the set $\OOForm(\vec{x})$ of formulas with free variables in $\vec{x}$, the action on morphisms being given by substitution.
Unfortunately, if we do this in the most obvious way, the result is not functorial. For example, consider the morphisms (in fact, renamings) $\seq{x}\tox{\seq{x}}\seq{y}\tox{\seq{y}}\seq{z}$ in $\OTM$, and the formula $\phi$ given by $\forall{}y(y=_Az)$ (where $x,y,z:A$). Then $\phi[z:=x]$ is $\forall{}y(y=_Ax)$ whereas $\phi[z:=y][y:=x]$ is $\forall{}y(y=_Ay)$. To solve this, we must employ the usual device of identifying formulas when they differ only by renaming of bound variables.
This construction can be described in a more abstract manner as follows. Though the mapping $\OTM^\op\to\Set$ described above does not yield a functor, it *does* yield what we might call a “partial functor” $\OOForm:\OTM^\op\to\Set$ – i.e., we assign to each morphism $\vec{t}:\vec{x}\to\vec{y}$ the partial function $\OOForm(\vec{y})\to\OOForm(\vec{x})$ given by the restriction of $[\vec{y}:=\vec{t}]$ to those formulas $\phi$ with $\BV(\phi)\cap\vec{x}=\emptyset$; by Proposition \[prop:form-sub-functoriality\] \[item:form-sub-funct-funct\], this then satisfies the functoriality condition $\phi[\vec{z}:=\vec{t}][\vec{y}:=\vec{s}]=\phi[\vec{z}:=(\vec{t}[\vec{y}:=\vec{s}])]$ whenever both sides of the equation are defined[^16].
Given such a “partial functor” $F:\C\to\Set$, let us say that a *saturation* of $F$ is a (total) functor $G:\C\to\Set$ together with a natural transformation $\alpha:F\to{}G$ (i.e. (total) functions $\alpha_A:FA\to{}GA$ for $A\in\Ob\C$, satisfying the naturality equations whenever both sides of the equation are defined). Now, the functor $\OForm:\OTM^\op\to\Set$ obtained by identifying formulas differing by a renaming of bound variables will be a saturation of the partial functor $\OOForm$, and we would like to say that it is in some sense the universal such saturation. This is true in the following sense (we will not prove this, though it will be more or less implicit in what follows). The partial functor $\OOForm$, and the functor $\OForm$, have the additional structure of the operations $\wedge,\vee,\To,\forall,\exists$ on and between the sets in its image, as well as the distinguished elements $\top,\bot,s=_At$, and these are all preserved, in an appropriate sense, by the action of the functor on morphisms, as well as by the natural transformation $\OOForm\to\OForm$. If we consider only functors equipped with such additional structure, and natural transformations preserving it, then every saturation $\OOForm\to{}F$ will factor uniquely through the saturation $\OOForm\to\OForm$.
The functor $\OForm:\OTM^\op\to\Set$ will give rise to a functor $\Form:\TM\to\Set$. Our next task will then be to lift this to a functor $\Form:\TM^\op\to\Cat$. The morphisms of the category $\Form(\vec{A})$ will be given by (equivalence classes of) “deductions” of one formula from another. In particular, such a morphism will exist if and only if the codomain is an intuitionistic consequence of the domain.
### {#section-23}
We define the relation $\phi\sim_\alpha\psi$ on formulas (“$\phi$ and $\psi$ are alphabetic variants”) by recursion on $\phi$:
1. If $\phi$ is $s=_At$ or $\top$ or $\bot$, then $\phi\sim_\alpha\psi$ if and only if $\phi=\psi$.
2. If $\phi$ is $\phi_1\xbop\phi_2$ with $\xbop$ one of $\wedge,\vee,\To$, then $\phi\sim_\alpha\psi$ if and only if $\psi$ is $\psi_1\xbop\psi_2$ where $\phi_1\sim_\alpha\psi_1$ and $\phi_2\sim_\alpha\psi_2$.
3. If $\phi$ is $\xop{}x\tilde\phi$ with $\xop$ one of $\forall,\exists$, then $\phi\sim_\alpha\psi$ if and only if $\psi$ is $\xop{}y\tilde\psi$ where $\tilde\phi^x_u\sim_\alpha\tilde\psi^y_u$ for every $u\in\Var_{\tp(x)}\setminus(\V(\phi)\cup\V(\psi))$, where we write $\omega^a_b$ for $\omega[\seq{a}:=\seq{b}]$. (It will follow from Proposition \[prop:av-props\] \[item:av-props-fv-every-some\] below that we can replace “every” with “some”.)
### {#prop:av-props}
We record some important properties of the relation $\simm_\alpha$:
1. \[item:av-props-fv\] If $\phi\sim_\alpha\psi$, then $\FV(\phi)=\FV(\psi)$.
2. \[item:av-props-fv-every-some\] If $\phi^x_u\sim_\alpha\phi^y_u$ for *some* $u\notin\{x,y\}\cup\V(\phi)\cup\V(\psi)$, then this holds for *every* such $u$.
3. $\sim_\alpha$ is an equivalence relation.
4. \[item:av-props-sub\] If $\phi\sim_\alpha\psi$ and $\FV(\vec{t})\cap\BV(\phi)=\FV(\vec{t})\cap\BV(\psi)=\emptyset$, then $\phi[\vec{x}:=\vec{t}]\sim_\alpha\psi[\vec{x}:=\vec{t}]$.
5. \[item:av-props-freshness\] For any formula $\phi$ and any finite $S\subseteq{}\Var$, there is a formula $\psi$ with $\phi\sim_\alpha\psi$ and $\BV(\psi)\cap{}S=\emptyset$.
The proofs of these statements are all straightforward inductions. However, some of them are somewhat tricky, as they require lemmas which themselves need to be proven by induction. Also, in some cases, rather than using the induction principle from Proposition \[prop:induction-principle\], we must do an induction on the *length*/*complexity* of a formula (i.e., the claim is proven for level-$n$ formulas by induction on $n$ – see Definition \[defn:raw-syntax\]).
The needed lemmas are:
1. $\FV(\phi^x_u)\setminus\set{u}=\FV(\phi)\setminus\set{x}$ whenever $u\notin\V(\phi)$.
2. $(\phi^w_{v})^x_u=(\phi^x_u)^w_v$ whenever $w\ne{}x$, $v\ne{}x$, and $u\ne{}w$.
3. \[item:av-props-proof-general-switcheroo\] More generally, $(\phi[\vec{w}:=\vec{t}])^x_u=(\phi^x_u)[\vec{w}:=\vec{t}]$ whenever $x\notin\vec{w}$, $u\notin\vec{w}$, and $x\notin\FV(\vec{t})$.
4. $u\in\FV(\phi^x_u)$ whenever $u\notin\BV(\phi)$ and $x\in\FV(\phi)$.
5. $\phi^x_u=\phi$ whenever $x\notin\FV(\phi)$.
6. More generally, $\phi[\vec{x}:=\vec{t}]=\phi[\vec{x}^{(\vec{x},v)}:=\vec{t}^{(\vec{x},v)}]$ whenever $v\notin\FV(\phi)$ (see Definition \[defn:substitution\] for the notation).
7. $(\phi^x_u)^u_v=\phi^x_v$ whenever $u\notin\V(\phi)$.
### {#defn:av-class-subst}
Let $\phi\in\Formset/\simm_\alpha$. We denote by $\phi[\vec{x}:=\vec{t}]$ the $\simm_\alpha$-equivalence class of $\tilde{\phi}[\vec{x}:=\vec{t}]$ where $\tilde\phi$ is a representative of $\phi$ with $\BV(\tilde\phi)\cap\FV(\vec{t})=\emptyset$. By Proposition \[prop:av-props\] \[item:av-props-freshness\], such a representative exists, and by Proposition \[prop:av-props\] \[item:av-props-sub\], the result is independent of the choice of $\tilde\phi$.
### {#prop:form-sub-functoriality}
1. \[item:form-sub-funct-fv\] If $\FV(\phi)\subset\vec{x}$, then $\FV(\phi[\vec{x}:=\vec{t}])\subseteq\FV(\vec{t})$.
2. \[item:form-sub-funct-funct\] If $\phi$ is a formula and $\BV(\phi)\cap\FV(\vec{s})=\BV(\phi)\cap\FV(\vec{t})=\emptyset$, then $$\label{eq:form-sub-funct-eq}
\phi[\vec{x}:=\vec{s}][\vec{y}:=\vec{t}]=\phi[\vec{x}:=(\vec{s}[\vec{y}:=\vec{t}])].$$ In particular, if $\phi$ is an $\simm_\alpha$-equivalence class of formulas, then (\[eq:form-sub-funct-eq\]) holds.
3. \[item:form-sub-funct-id\] $\phi[\vec{x}:=\vec{x}]=\phi$
By induction.
### {#section-24}
We define a functor $\OForm=\OForm_\sigma:\OTM_\sigma^\op\to\Set$ as follows. For each $\vec{x}\in\Ob\OTM$, $\OForm(\vec{x})$ is $\set{\phi\in\Formset/\simm_\alpha:\FV(\phi)\subseteq\vec{x}}$. Given a morphism $\vec{t}:\vec{x}\to\vec{y}$ and $\phi\in\OForm(\vec{y})$, we set $\OForm(\vec{t})(\phi)=\phi[\vec{y}:=\vec{t}]$ which by Proposition \[prop:form-sub-functoriality\] \[item:form-sub-funct-fv\] is in $\OForm(\vec{x})$ as required. That this is a functor follows from Proposition \[prop:form-sub-functoriality\] \[item:form-sub-funct-funct\]-\[item:form-sub-funct-id\].
We now “force” $\OForm$ to descend to a functor $\Form=\Form_\sigma:\TM^\op\to\Set$. Namely, we define $\Form(\vec{A})$ to be $$\Bigg(\bigsqcup_{\substack{\vec{x}\in\Ob\OTM\\\vec{x}:\vec{A}}}\OForm(\vec{x})\Bigg)
\Big/\simm,$$ where $(\vec{x},\phi)\sim(\vec{y},\psi)$ if and only if there exists a renaming $r:\vec{x}\to\vec{y}$ with $\OForm(r)(\psi)=\phi$ – i.e., if and only if $\psi=\phi[\vec{x}:=\vec{y}]$. The quotient maps $\OForm(\vec{x})\to\Form(\tp\vec{x})$ assemble in a unique (and obvious) way into a natural isomorphism from $\OForm$ to a functor $\OTM^\op\to\Set$, and this latter functor takes renaming to identities, and hence factors uniquely through a functor $\Form:\TM^\op\to\Set$.
We denote the image of $\phi\in\OForm(\vec{x})$ in $\Form(\tp(\vec{x}))$ by $[\phi,\vec{x}]$. Note that for $[\vec{t},\vec{x}]:\vec{A}\to\vec{B}$ and $[\phi,\vec{y}]\in\Form(\vec{B})$, we have that $\Form([\vec{t},\vec{x}])([\phi,\vec{y}])$ is given by $[\phi[\vec{y}:=\vec{t}],\vec{x}]$. For $t\in\Hom_\TM(\vec{A},\vec{B})$ and $\phi\in\Form(\vec{B})$, we will also write $t^*\phi$ for $\Form(t)(\phi)$.
### {#section-25}
We note that the operations $\wedge,\vee,\To:\Formset\times\Formset\to\Formset$ and $\forall{}v,\exists{}v:\Formset\to\Formset$ are well-defined on $\simm_\alpha$-equivalence classes (the last two cases follow from Proposition \[prop:av-props\] \[item:av-props-sub\]), and we use the same notation to denote the induced operations $\wedge,\vee,\To:\Formset/\simm_\alpha\times\Formset/\simm_\alpha\to\Formset/\simm_\alpha$, and so on. Similarly, we write $\top$, $\bot$ for $[\top],[\bot]\in\Form/\simm_\alpha$, and write $\FV$ for the induced map $\Formset/\simm_\alpha\to\pow(\Var)$ (which is well-defined by Proposition \[prop:av-props\] \[item:av-props-fv\]).
Next, for each $\vec{A}\in\Ob\TM$, we have restricted operations $\wedge,\vee,\To:\OForm(\vec{x})\times\OForm(\vec{x})\to\OForm(\vec{x})$, as well as operations $\forall{}y,\exists{}y:\OForm(\vec{x}y)\to\OForm(\vec{x})$, and elements $\top,\bot\in\OForm(\vec{x})$, and $s=_At\in\OForm(\vec{x})$, where $\FV(s),\FV(t)\subseteq\vec{x}$.
### {#prop:oform-stab}
The operations $\top,\bot,\wedge,\vee,\To,\forall{}y,\exists{}y,s=_Bt$ on the sets $\OForm(\vec{x})$ are compatible with the substitution maps $[\vec{x}:=\vec{t}]=\OForm(\vec{t}):\OForm(\vec{x})\to\OForm(\vec{z})$ for $\vec{t}:\vec{z}\to\vec{x}$. Specifically, we mean
1. $\top[\vec{x}:=\vec{t}]=\top$ and $\bot[\vec{x}:=\vec{t}]=\bot$.
2. $(\phi\xbop{}\psi)[\vec{x}:=\vec{t}]=\phi[\vec{x}:=\vec{t}]\xbop{}\psi[\vec{x}:=\vec{t}]$ for $\xbop$ any of $\wedge,\vee,\To$ and $\phi,\psi\in\OForm(\vec{x})$.
3. $(\xop{}y\phi)[\vec{x}:=\vec{t}]=\xop{}w(\phi[\vec{x}\seq{y}:=\vec{t}\seq{w}])$ for $\xop$ one of $\forall,\exists$, and $\phi\in\OForm(\vec{x}\seq{y})$ and $w\in\Var_{\tp(y)}\!\!\setminus\vec{z}$.
4. $(s=_Bs')[\vec{x}:=\vec{t}]=(s[\vec{x}:=\vec{t}]=_Bs'[\vec{x}:=\vec{t}])$ for any $s,s'\in\Tm_B$ with $\FV(s),\FV(s')\subseteq\vec{x}$.
The cases (i),(ii),(iv) are immediate from the definition of substitution.
Let us prove the case (iii). Let $U=\FV(\vec{t})\cup\vec{x}\cup\vec{z}\cup\set{y,w}$ (the set of variables that occurred so far). Let $\tilde{\phi}$ be a representative of $\phi$ with $\BV(\tilde\phi)\cap{}U=\emptyset$, so that the left hand side in (iii) is represented by $$\label{eq:quant-subst-pf-1}
(\xop{}y\tilde\phi)[\vec{x}:=\vec{t}].$$ Let $\Psi\sim_\alpha(\xop{}y\tilde{\phi})$ be such that $\BV(\Psi)\cap{}U=\emptyset$, so in particular $\Psi=\xop{}y'\psi$ for some $\psi$ and some $y'\in\Var_{\tp(y)}\!\setminus\,U$, and we have (by Proposition \[prop:av-props\] \[item:av-props-sub\]) that (\[eq:quant-subst-pf-1\]) is alphabetically equivalent to $$\label{eq:quant-subst-pf-2}
(\xop{}y'\psi)[\vec{x}:=\vec{t}]=
\xop{}y'(\psi[\vec{x}:=\vec{t}]).$$ We also have that the right-hand side in (iii) is represented by $$\label{eq:quant-subst-pf-3}
\xop{}w(\tilde\phi[\vec{x}\seq{y}:=\vec{t}\seq{w}]).$$ We need to show that (\[eq:quant-subst-pf-2\]) is alphabetically equivalent to (\[eq:quant-subst-pf-3\]), i.e., that $$\label{eq:quant-subst-pf-4}
\tilde\psi[\vec{x}:=\vec{t}]^{y'}_v\sim_\alpha
\tilde\phi[\vec{x}\seq{y}:=\vec{t}\seq{w}]^w_v$$ for some $v\in\Var_{\tp(y)}$ distinct from all the variables in all the terms and formulas thus far considered.
Note that, for any such $v$, we have $\psi^{y'}_v\sim_\alpha\tilde\phi^y_v$ by the definition of $\psi$. Hence, the left-hand side of (\[eq:quant-subst-pf-4\]), which is equal to $\tilde\psi^{y'}_v[\vec{x}:=\vec{t}]$ by Proposition \[prop:av-props\] \[item:av-props-proof-general-switcheroo\], is alphabetically equivalent to $\tilde\phi^{y}_v[\vec{x}:=\vec{t}]$ by Proposition \[prop:av-props\] \[item:av-props-sub\].
On the other hand, the right-hand side of (\[eq:quant-subst-pf-4\]) is equal to $\tilde\phi[\vec{x}\seq{y}:=\vec{t}\seq{v}]$ by Proposition \[prop:form-sub-functoriality\] \[item:form-sub-funct-funct\] since $v\notin\vec{z}\supset\FV(\vec{t})$. Hence, it remains to see that $$\tilde\phi^{y}_v[\vec{x}:=\vec{t}]=
\tilde\phi[\vec{x}\seq{y}:=\vec{t}\seq{v}].$$ We have thus reduced the claim to the following lemma, which can be proven by induction:
Given any formula $\omega$, a sequence $\vec{x}\seq{y}$ of distinct variables, a sequence $\vec{t}$ of terms with $\tp(\vec{t})=\tp(\vec{x})$, and a variable $v\in\Var_{\tp(y)}\setminus(\vec{x}\cup\{y\}\cup\FV(\vec{t}))$, and assuming $\BV(\omega)\cap(\vec{x}\cup\set{y})=\emptyset$, we have $$\omega^{y}_v[\vec{x}:=\vec{t}]=
\omega[\vec{x}\seq{y}:=\vec{t}\seq{v}].
\tag*{\qed}$$
### {#defn:form-ops}
Because the operations $\wedge,\vee,\To:\OForm(\vec{x})\times\OForm(\vec{x})\to\OForm(\vec{x})$ are compatible with substitutions (Proposition \[prop:oform-stab\]), and in particular with renamings $\vec{y}\to\vec{x}$, they descend to give operations $\Form(\vec{A})\times\Form(\vec{A})\to\Form(\vec{A})$ (where $\vec{x}:\vec{A}$), which we denote by the same symbols; these are given by $[\phi,\vec{x}]\xbop[\psi,\vec{x}]=[\phi\xbop\psi,\vec{x}]$.
Similarly, we have induced operations $\forall,\exists:\Form(\vec{A}\times\seq{B})=\Form(\vec{A}\seq{B})\to\Form(\vec{A})$ given by $\square[\phi,\vec{x}\seq{y}]=[\square{}y\phi,\vec{x}]$ where $\square$ is one of $\forall,\exists$.
We also have an induced map $=_B:\Hom_{\TM}(\vec{A},\seq{B})\times\Hom_{\TM}(\vec{A},\seq{B})\to\Form(\vec{A})$ taking $[s,\vec{x}]$ and $[t,\vec{x}]$ to $[s=_Bt,\vec{x}]$. We denote by $\Eq_{B}$ the object $\pi_1^{\seq{B,B}}=_B\pi_2^{\seq{B,B}}\in\Form(\seq{B})$, and we note that we then have $(s=_Bt)=\br{s,t}^*\Eq_B$ for any $s,t:\vec{A}\to\seq{B}$.
Finally, we denote by $\top_{\vec{A}},\bot_{\vec{A}}$ the elements $[\top,\vec{x}],[\bot,\vec{x}]\in\Form(\vec{A})$ (where $\vec{x}:\vec{A}$).
### {#prop:form-stab}
The operations $\top_{\vec{A}},\bot_{\vec{A}},\wedge,\vee,\To,\forall{},\exists{},s=_Bt$ on the sets $\Form(\vec{A})$ are compatible with the substitution maps $t^*=\Form(t):\Form(\vec{A})\to\Form(\vec{B})$ for $t:\vec{B}\to\vec{A}$. Specifically, we mean
1. $t^*\top_{\vec{A}}=\top_{\vec{B}}$ and $t^*\bot_{\vec{A}}=\bot_{\vec{B}}$.
2. $t^*(P\xbop{}Q)=t^*P\xbop{}t^*Q$ for $\xbop$ any of $\wedge,\vee,\To$ and $P,Q\in\Form(\vec{A})$.
3. $t^*(\xop{}P)=\xop(t\times\id_{\seq{C}})^*P$ for $\xop$ either of $\forall,\exists$ and $P\in\Form(\vec{A}\times\seq{C})=\Form(\vec{A}\seq{C})$.
4. $t^*(s=_Cs')=(s\cdot{}t=_Cs'\cdot{}t)$ for any $s,s':\vec{A}\to\seq{C}$.
Immediate from Proposition \[prop:oform-stab\].
### {#prop:form-freegen}
The family of sets $(\Form(\vec{A}))_{\vec{A}\in\Ob\TM}$ are freely generated by the operations $\top_{\vec{A}},\bot_{\vec{A}},\wedge,\vee,\To,\forall,\exists,=_{B}$ described in Definition \[defn:form-ops\], in the following sense:
1. \[item:form-freegen-gen\] For any family $(S_{\vec{A}}\subseteq\Form(\vec{A}))_{\vec{A}\in\Ob\TM}$ of subsets of this family which is closed under all of these operations, we have $S_{\vec{A}}=\Form(\vec{A})$ for all $\vec{A}\in\Ob\TM$.
2. \[item:form-freegen-uniq\] We have “unique readability” as in Proposition \[prop:unique-readability\] and Proposition \[prop:unary-mor-fregen\] \[item:unary-mor-freegen-uniq\]; the map $$\begin{split}
\Bigg(
&\bigsqcup_{\vec{A}\in\Ob\TM}\hspace{-5pt}
\Big(
\Form(\vec{A})^0\sqcup
\Form(\vec{A})^0\sqcup
\Form(\vec{A})^2\sqcup
\Form(\vec{A})^2\sqcup
\Form(\vec{A})^2
\Big)
\sqcup\\
&\bigsqcup_{\substack{\vec{A}\in\Ob\TM\\\seq{B}\in\Ob\TM}}\hspace{-2pt}
\Big(
\Form(\vec{A}\seq{B})\sqcup
\Form(\vec{A}\seq{B})
\Big)
\sqcup\hspace{-12pt}
\bigsqcup_{\substack{\vec{A}\in\Ob\TM\\\seq{B}\in\Ob\TM}}\hspace{-8pt}
\Hom_{\TM}(\vec{A},\seq{B})^2
\Bigg)
\to\hspace{-8pt}
\bigsqcup_{\vec{A}\in\Ob\TM}\hspace{-8pt}
\Form(\vec{A})
\end{split}$$ induced from these operations is a bijection.
3. \[item:form-freegen-map\] Given any family $(X_{\vec{A}})_{\vec{A}\in\Ob\TM}$ of sets, together with operations $\wedge,\vee,\To:X_{\vec{A}}\times{}X_{\vec{A}}\to{}X_{\vec{A}}$ and elements $\top_{\vec{A}},\bot_{\vec{A}}\in{}X_{\vec{A}}$ for each $\vec{A}\in\Ob\TM$, operations $\forall,\exists{}:X_{\vec{A}\seq{B}}\to{}X_{\vec{A}}$ for each $\vec{A},\seq{B}\in\Ob\TM$, and operations $\Hom_{\TM}(\vec{A},\seq{B})\times\Hom_{\TM}(\vec{A},\seq{B})\to{}X_{\vec{A}}$ for each $\vec{A},\seq{B}\in\Ob\TM$, there is a unique family of maps $(f_{\vec{A}}:\Form(\vec{A})\to{}X_{\vec{A}})_{\vec{A}}$ preserving all of these operations.
The claims (i) and (iii) are proven as in Proposition \[prop:unary-mor-fregen\].
The claim (ii) is also proven similarly. Surjectivity follows from (i), and injectivity follows from unique readability for formulas (Proposition \[prop:unique-readability\]) and the definitions of renaming and of alphabetical equivalence.
### {#section-26}
\[defn:deductions\] We now define a set $\Ded=\Ded_\sigma$ (the set of “deductions”). Each deduction has an object $\vec{A}$ of $\TM_\sigma$ associated with it, as well as two objects of $\Form_{\sigma}(\vec{A})$ (its “premise” and “conclusion”). We write $f:P\to{}Q$ to indicate that $f$ is a deduction with premise $P$ and conclusion $Q$.
In fact, we will define the set of deductions as the *initial structure* (or free structure with no generators) for a certain algebraic signature, whose sorts are triples $(\vec{A},P,Q)$ with $\vec{A}$, $P$, and $Q$ as above. Recall from Definition \[defn:raw-syntax\] that the set $\Tm_{\sigma'}$ of terms over a signature $\sigma'$ was defined relative to a fixed infinite set $\Varn$ of variable names. In fact, the set thus constructed – or better, the $\Ob\sigma'$-indexed family of sets – is precisely the *free $\sigma'$-algebra on the ($\Ob\sigma'$-indexed) set $\Var_{\sigma'}=\Ob\sigma'\times\Var$*. If we instead take $\Varn=\emptyset$ – or, what amounts to the same thing – if we omit from the definition of $\Tm_{\sigma'}$ the reference to variables – we obtain the definition of the initial $\sigma'$-structure. This structure is characterized up to isomorphism by an obvious universal property – namely, the one obtained from Proposition \[prop:recursion-principle\] by eliminating the reference to variables.
It now remains to define the signature $\sigma'$ for which the set of deductions is the initial $\sigma'$-structure. As we said above, we take $\Ob\sigma'$ to be the set of triples $(\vec{A},P,Q)$ with $\vec{A}\in\Ob\TM_{\sigma}$ and $P,Q\in\Ob\Form_\sigma(\vec{A})$. The function symbols of $\sigma'$ are given schematically below. Each figure indicates a set of function symbols, one for each of the possible values of the relevant parameters $\vec{A}$, $B$, $P$, $Q$, $R$, $S$, $T$, $t$. Here, $\vec{A}$ is an (arbitrary) object of $\TM_\sigma$; $P,Q,R$ are objects in $\Form_\sigma(\vec{A})$; $S$ is an object in $\Form_\sigma(\vec{A}\setminus\set{A_{\len{A}}})$ (where in the figures involving $\forall,\exists$, it is assumed that $\len{\vec{A}}>0$); $T$ is an object in $\Form_\sigma(\seq{B,B})$; and $t$ is a morphism in $\TM_\sigma$ with codomain $\vec{A}$.
Each figure displays, above the line, the input sorts of the function symbol and below, the codomain sort, and also introduces a notation for the function symbol being defined.
We write $\pi^{\vec{A}}$ as a shorthand for $\br{\pi^{\vec{A}}_1,\ldots,\pi^{\vec{A}}_{\len{\vec{A}}-1}}=
[\vec{x}\setminus{x_{\len{\vec{A}}}},\vec{x}]:\vec{A}\to\vec{A}\setminus\set{A_{\len{\vec{A}}}}$ in the figures for $\lambda$ and $\mu$.
[**Category and fibration structure:**]{} $$\frac{{\vphantom{P}}}{1_P:P\to{}P}{\hspace{20pt}}\frac{f:P\to{}Q\quad{}g:Q\to{}R}{g\circ{}f:P\to{}R}{\hspace{20pt}}\frac{f:P\to{}Q}{t^*f:t^*P\to{}t^*Q}$$
[**Finite products and coproducts:**]{} $$\frac{}{!_P:P\to{}\top_{\vec{A}}}{\hspace{40pt}}\frac{{\vphantom{P}}}{\ex_P:\bot_{\vec{A}}\to{}P}$$$$\frac{{\vphantom{P}}}{\pi_{PQ}:P\wedge{}Q\to{}P}{\hspace{20pt}}\frac{{\vphantom{P}}}{\pi'_{PQ}:P\wedge{}Q\to{}P}{\hspace{20pt}}\frac{f:P\to{Q}\quad{}g:P\to{R}}{\br{f,g}:P\to{Q\wedge{R}}}$$$$\frac{{\vphantom{P}}}{\kappa_{PQ}:P\to{}P\vee{}Q}{\hspace{20pt}}\frac{{\vphantom{P}}}{\kappa'_{PQ}:Q\to{}P\vee{}Q}{\hspace{20pt}}\frac{f:P\to{R}\quad{}g:Q\to{R}}{\pl[]{f,g}:P\vee{}Q\to{R}}$$
[**Exponentials:**]{} $$\frac{{\vphantom{P}}}{\varepsilon_{PQ}:(P\To{}Q)\wedge{}P\to{}Q}{\hspace{20pt}}\frac{f:P\wedge{}Q\to{R}}{f^\simm:P\to{}Q\To{R}}$$
[**Adjoints to pullback along projections:**]{} $$\frac{}
{\varepsilon^\forall_P:(\pi^{\vec{A}})^*\forall{}P\to{}P}{\hspace{40pt}}\frac{f:(\pi^{\vec{A}})^*S\to{}P}{\lambda{}f:S\to{}\forall{}P}$$$$\frac{}
{\eta^\exists_P:P\to(\pi^{\vec{A}})^*\exists{}P}{\hspace{40pt}}\frac{f:P\to(\pi^{\vec{A}})^*S}{\mu{}f:\exists{}P\to{}S}$$
[**Equality objects:**]{} $$\frac{{\vphantom{P}}}{r^{B}:\top_{\seq{B}}\to{}\Delta_{\seq{B}}^*\Eq_{\seq{B}}}{\hspace{20pt}}\frac{f:\top_{\seq{B}}\to\Delta_{\seq{B}}^*T}
{\xi{}f:\Eq_{\seq{B}}\to{}T}$$
### {#defn:deduction-rels}
We next want to define an equivalence relation $\simm\subseteq\Ded\times\Ded$ on the set of deductions. These are chosen precisely in such a way that, for each $\vec{A}\in\Ob{\TM_\sigma}$, the set of deductions up to equivalence between objects in $\Form_\sigma(\vec{A})$ form a category and that, moreover, this makes $\Form_{\sigma}$ a functor $\TM_\sigma^\op\to\Cat$ whose associated fibration is an $h^=$-fibration.
The equivalence relation $\simm$ will be given by a certain set of *equations* over the language $\sigma'$ defined in Definition \[defn:deductions\], so that the resulting set of equivalence classes will be precisely the *free algebra* of the algebraic theory over $\sigma'$ given by these equations.
More concretely, we define below a certain set of *basic relations* on $\Ded$ and then define $\simm\subseteq\Ded\times\Ded$ to be the least equivalence relation which contains each of the basic relations, and which is closed under each of the operations of $\sigma'$ (in the sense that if $t_i\sim{}t_i'$ for each $i$ and $f$ is any operation symbol of $\sigma'$, then $ft_1\ldots{}t_n\sim{}ft_1'\ldots{}t_n'$).
We now define the basic relations. Each figure below represents a set of basic relations, one (or more) for each possible value of the relevant parameters $\vec{A}$, $B$, $P$, $Q$, $R$, $R'$, $S$, $T$, $s$, $t$, $u$, $f$, $g$, $h$. Here $\vec{A}$, $B$, $P$, $Q$, $R$, $S$, $T$, $t$ are as in Definition \[defn:deductions\]; $R'$ is an additional object in $\Form(\vec{A})$; $s$ is a morphism with $\cod(s)=\dom(t)$; $u$ is a morphism with codomain $\vec{A}\setminus{A_{\len{\vec{A}}}}$; and finally, $f$, $g$, $h$ are deductions whose premise and conclusion are specified above the horizontal line. Below the line are indicated one or more equations, each representing a basic relation. Note also that in the relations under “Exponentials”, we use the notation $x\wedge{}y$ for $\br{x\circ\pi_{PQ},y\circ\pi'_{PQ}}$.
[**Category:**]{} $$\frac{f:P\to{}Q{\hspace{20pt}}{}g:Q\to{}R{\hspace{20pt}}{}h:R\to{}R'}
{f\circ{1_P}=f{\hspace{20pt}}{}1_Q\circ{}f=f{\hspace{20pt}}{(h\circ{}g)\circ{}f=h\circ(g\circ{}f)}}$$
[**Fibration:**]{} $$\frac{f:P\to{}Q{\hspace{20pt}}{}g:Q\to{}R}
{t^*1_P=1_{t^*P}{\hspace{20pt}}t^*(g\circ{}f)=t^*g\circ{}t^*f{\hspace{20pt}}\id_{\vec{A}}^*f=f{\hspace{40pt}}s^*t^*f=(ts)^*f}$$
[**Finite products and coproducts:**]{} $$\frac{f:P\to\top_{\vec{A}}}
{f=!_P}{\hspace{40pt}}\frac{f:\bot_{\vec{A}}\to{}A}
{f=\ex_P}$$$$\frac{f:P\to{}Q{\hspace{20pt}}{}g:P\to{}R}
{\pi_{QR}\circ\br{f,g}=f{\hspace{20pt}}\pi'_{QR}\circ\br{f,g}=g}{\hspace{20pt}}\frac{h:P\to{}Q\wedge{}R}
{\seq{{\pi_{QR}\circ{}h,\pi'_{QR}\circ{}h}}=h}$$$$\frac{f:P\to{}R{\hspace{20pt}}{}g:Q\to{}R}
{[f,g]\circ\kappa_{PQ}=f{\hspace{20pt}}[f,g]\circ\kappa'_{PQ}=g}{\hspace{20pt}}\frac{h:P\vee{}Q\to{}R}
{[h\circ\kappa_{PQ},h\circ\kappa'_{PQ}]=h}$$
[**Exponentials:**]{} $$\frac{f:P\wedge{}Q\to{}R}
{\varepsilon_{QR}\circ{}(f^\sim\wedge{}1_Q)=f}{\hspace{40pt}}\frac{h:P\to{}(Q\Rightarrow{}R)}
{(\varepsilon_{QR}\circ(h\wedge1_Q))^\sim=h}$$ [**Adjoints to pullback along projections:**]{} $$\frac{f:(\pi^{\vec{A}})^*S\to{}P}
{\varepsilon^\forall_P\circ(\pi^{\vec{A}})^*(\lambda{}f)=f}{\hspace{40pt}}\frac{h:S\to{}\forall{}P}
{\lambda(\varepsilon^\forall_P\circ(\pi^{\vec{A}})^*h)=h}$$$$\frac{f:P\to(\pi^{\vec{A}})^*S}
{(\pi^{\vec{A}})^*(\mu{}f)\circ\eta^\exists_P=f}{\hspace{40pt}}\frac{h:\exists{}P\to{}S}
{\mu((\pi^{\vec{A}})^*h\circ\eta^\exists_P)=h}$$
[**Equality objects:**]{} $$\frac{f:\top_{\br{B}}\to\Delta_{\br{B}}^*T}
{\Delta_{\br{B}}^*(\xi{}f)\circ{}r^{B}=f}{\hspace{40pt}}\frac{h:\Eq_B\to{}T}
{\xi(\Delta_{\br{B}}^*h\circ{}r^{B})=h}$$
[**Stability of the operations under pullbacks**]{} $$t^*\pi_{QR}=\pi_{(t^*Q)(t^*R)}{\hspace{40pt}}t^*\pi'_{QR}=\pi'_{(t^*Q)(t^*R)}$$$$t^*\kappa_{PQ}=\kappa_{(t^*P)(t^*Q)}{\hspace{40pt}}t^*\kappa'_{PQ}=\kappa'_{(t^*P)(t^*Q)}$$$$t^*\varepsilon_{QR}=\varepsilon_{(t^*Q)(t^*R)}$$$$(u\times\id_{\seq{{}A_{\len{\vec{A}}}}})^*\varepsilon^\forall_P=
\varepsilon^\forall_{(u\times\id_{\seq{{}A_{\len{\vec{A}}}}})^*P}{\hspace{40pt}}(u\times\id_{\seq{{}A_{\len{\vec{A}}}}})^*\eta^\exists_P=
\eta^\exists_{(u\times\id_{\seq{{}A_{\len{\vec{A}}}}})^*P}$$
### {#prop:deduction-rels-domco}
The equivalence relation from Definition \[defn:deduction-rels\] respects premise and conclusion – i.e., if $f:P\to{}Q$ and $f\sim{}f'$, then $f':P\to{}Q$.
First, the claim is true for each of the basic relations. This is clear in most cases, but some thought is needed for the relation $t^*\pi_{QR}=\pi_{(t^*Q)(t^*R)}$ and the other six relations which are displayed without a horizontal line. In order for $t^*\pi_{QR}$ and $\pi_{(t^*Q)(t^*R)}$ to have the same premise, we need $t^*(Q\wedge{}R)=t^*Q\wedge{}t^*R$. This follows from Proposition \[prop:form-stab\]. The other six cases also follow from Proposition \[prop:form-stab\], where in the last two we must also use that $\pi^{\vec{A}}(u\times\id_{{\seq{{}A_{\len{\vec{A}}}\rangle}}})=u\pi^{\vec{B}{\seq{A_{\len{\vec{A}}}}}}$, where $\vec{B}=\operatorname{dom}u$.
Next, taking the closure of the basic relations under the operations of $\sigma'$ clearly preserves this property. Finally, taking the transitive, symmetric, reflexive closure also preserves this property.
### {#defn:syntactic-fib}
Given $\vec{A}\in\Ob\TM_\sigma$ and $P,Q\in\Form_\sigma(\vec{A})$, we define $\Hom_{\Form_{\sigma}(\vec{A})}(P,Q)$ to be the set of equivalence classes of deductions $P\to{}Q$ (where by Proposition \[prop:deduction-rels-domco\], the domain and codomain of equivalence classes of deductions are well-defined). Given $[f]\in\Hom_{\Form_{\sigma}(\vec{A})}(P,Q)$ and $[g]\in\Hom_{\Form_{\sigma}(\vec{A})}(Q,R)$, we define their composite to be $[g\circ{}f]$. Since $\sim$ is defined to be closed under $(-\circ-)$, this is well-defined.
The “Category” relations from Definition \[defn:deduction-rels\] immediately imply that this makes $\Form_\sigma(\vec{A})$ into the set of objects a category (which we again denote by $\Form_\sigma(\vec{A})$).
Given $t:\vec{A}\to\vec{B}$ and $[f]\in\Hom_{\Form_{\sigma}(\vec{B})}(P,Q)$, we define $t^*[f]\in\Hom_{\Form_{\sigma}(\vec{A})}(t^*P,t^*Q)$ to be $[t^*f]$. Again, this is well-defined since $\simm$ is defined to be closed under $t^*$.
By the “Fibration” relations, this makes $t^*$ into a functor $\Form_{\sigma}(\vec{B})\to\Form_{\sigma}(\vec{A})$.
We also have from the “Fibration” relations that $\id_{\vec{A}}^*$ is the identity functor and that $t^*s^*=(s\cdot{t})^*$. Hence, we have upgraded $\Form_{\sigma}:\TM^\op\to\Set$ to a functor $\TM_\sigma^\op\to\Cat$, which we again denote by $\Form_\sigma$ (or $\Form$).
We denote by $\fibr{Pf}{Pf}{Tm}=\fibr{Pf_\sigma}{Pf_\sigma}{Tm_\sigma}$ the fibration associated to the functor $\Form:\TM^\op\to\Cat$. Explicitly, $\Pf$ has the set of objects $\bigsqcup_{\vec{A}\in\Ob\TM}\Form(\vec{A})$, and the morphisms are given by $$\textstyle
\Hom_{\Pf}((\vec{A},P),(\vec{B},Q))=\bigsqcup_{t:\vec{A}\to\vec{B}}\Hom_{\Form(\vec{A})}(P,t^*Q).$$ Composition is given by $(t,q)\cdot(s,p)=(t\cdot{}s,s^*q\cdot{}p)$. The functor $\fib{Pf}$ takes $(\vec{A},P)$ to $\vec{A}$ and $(t,p)$ to $t$.
As the fibration $\fib{Pf}$ is the one associated to a pseudo-functor, it has a canonical cleavage – namely, the specified cartesian lift of $t:\vec{A}\to\vec{B}$ with codomain $Q$ is $(t,\id_{t^*Q})$ – and since our pseudo-functor is in fact a functor, this cleavage is even *split* (see [@sentai §14.4]).
### {#prop:syntactic-fib-is-hfib}
The fibration $\fib{Pf}$ from Definition \[defn:syntactic-fib\] is an $h^=$-fibration. We must check all the conditions in the definition of an $h^=$-fibration.
*The fibers are bicartesian closed.*
It follows from the “Finite products and coproducts” relations of Definition \[defn:deduction-rels\] that $\top_{\vec{A}},\bot_{\vec{A}}$ are terminal and initial objects of $\Form(\vec{A})$, and that $P\wedge{Q}$ and $P\vee{Q}$ are the product and coproduct of $P$ and $Q$ with projections $\pi_{PQ},\pi'_{PQ}$ and coprojections $\kappa_{PQ},\kappa'_{PQ}$, respectively. The “Exponentials” relations imply that $P\To{}Q$ is an exponential object of $P$ and $Q$ with evaluation morphism $\varepsilon_{PQ}$.
*The necessary cocartesian morphisms and $\prod$-diagrams exist.*
From the “Adjoint to pullback along projections” relations, we have that $\gamma^P$ is a universal arrow from $(\pi^{\vec{A}})^*$ to $P$, and that $\zeta^P$ is a universal arrow from $P$ to $(\pi^{\vec{A}})^*$, and hence that $(\pi^{\vec{A}})^*$ has right and left adjoints, with object functions $\exists$ and $\forall$, respectively. It follows from Remark \[subsubsec:adjoints\] that the morphisms $(\pi^{\vec{A}})$ have the required cocartesian lifts and $\prod$-diagrams. Since the specified product projections in $\TM_\sigma$ are composites of the morphisms $\pi^{\vec{A}}$, and every product projection is isomorphic to a specified one, it follows from Propositions \[prop:bccomp\] and \[prop:bcisoinv\] below that the required cocartesian lifts and $\prod$-diagrams exist over every product projection.
*Everything is stable.*
The first five “Stability of the operations under pullbacks” relations imply that each of the functors $t^*$ preserves (the specified, and hence all) products, coproducts, and exponentials.
We claim that the last two of these relations imply that (the specified and hence all) cocartesian morphisms and $\prod$-diagrams over the morphisms $\pi^{\vec{A}}$ are stable along every morphism. Again, once we show this, the same thing then follows for arbitrary product projections by Propositions \[prop:bccomp\] and \[prop:bcisoinv\].
More precisely, we claim that, for every $t:\vec{B}\to\vec{A}\setminus\set{A_{\len{\vec{A}}}}$, the cocartesian morphisms and $\prod$-diagrams over $\pi^{\vec{A}}$ satisfy the stability condition with respect to the pullback square $$\begin{tikzcd}
\vec{B}\times\seq{A_{\len{\vec{A}}}}\ar[r, "t\times\id_{\seq{A}}"]\ar[d, "\pi_1"']
&\vec{A}\ar[d, "\pi^{\vec{A}}"]\\
\vec{B}\ar[r, "t"]&\vec{A}\setminus\set{A_{\len{\vec{A}}}}
\end{tikzcd}$$ and hence are stable along $t$.
To prove the claim, let us put ourselves in the more general situation of a fibration $\fibr{C}CB$ with a fixed split cleavage. Consider a pullback square $$\label{eq:pf-hfib-stab-pbsquare}
\begin{tikzcd}
A\ar[r, "f"]\ar[d, "h"']&B\ar[d, "k"]\\
C\ar[r, "g"]&D
\end{tikzcd}$$ in $\B$, and suppose $h^*:\fib{C}^C\to\fib{C}^A$ and $k^*:\fib{C}^D\to\fib{C}^B$ have left (or right) adjoints, and that we have fixed such left adjoints $\sum_h\dashv{}h$ and $\sum_k\dashv{}k$ with units $\eta^h$ and $\eta^k$ (or right adjoints $h\dashv{}\prod_h$ and $k\dashv{}\prod_k$ with counits $\varepsilon^h$ and $\varepsilon^k$).
Since the cleavage is split, the diagram of functors below to the left strictly commutes. Suppose further that $\sum_h$ and $\sum_k$ (or $\prod_h$ and $\prod_k$) have been chosen so that the square below to the right commutes on objects (this is the case in our situation by Proposition \[prop:form-stab\]). $$\begin{tikzcd}
\fib{C}^A\ar[from=r, "f^*"']\ar[from=d, "h^*"]&\fib{C}^B\ar[from=d, "k^*"']\\
\fib{C}^C\ar[from=r, "g^*"']&\fib{C}^D
\end{tikzcd}\hspace{2cm}
\begin{tikzcd}
\fib{C}^A\ar[from=r, "f^*"']\ar[d, "\sum_h"', "\prod_h"]&\fib{C}^B\ar[d, "\sum_k"', "\prod_k"]\\
\fib{C}^C\ar[from=r, "g^*"']&\fib{C}^D
\end{tikzcd}$$ Now, to see that the cocartesian morphisms (or $\prod$-diagrams) over $k$ satisfy the stability condition with respect to the square (\[eq:pf-hfib-stab-pbsquare\]), it suffices to see that this is true for *some* cocartesian morphism with domain $P$ (or $\prod$-diagram based on $P$) for each $P\in\fib{C}^B$, so we might as well take the cocartesian morphism (or $\prod$-diagram) associated to the fixed adjoint $\sum_k$ (or $\prod_k$), namely the one shown below. $$\begin{tikzcd}
k^*\sum_kP\ar[dr, "\ct"]\\
P\ar[u, "\eta^k_P"]\rac[r, "\ct\eta^k_P"]&\sum_kP\\[-10pt]
B\ar[r, "k"]&D
\end{tikzcd}\hspace{2cm}
\begin{tikzcd}
k^*\prod_kP\ar[d, "\varepsilon^k_P"']\ar[r, "\ct"]&\prod_kP\\
P\\[-10pt]
B\ar[r, "k"]&D
\end{tikzcd}\hspace{2cm}$$ Again, to see that the stability condition holds for this particular cocartesian morphism (or $\prod$-diagram) it suffices to see that it holds with respect to *some* instance of the data involved in the stability condition (i.e., cartesian morphisms into $P$ and $\sum_kP$, and so on), so we might as well take the particular one shown below. $$\begin{tikzcd}[row sep=8pt, column sep=10pt]
h^*g^*\sum_kP\areq[d]
&[-5pt]k^*\sum_kP\ar[rrdd, "\ct"]&[-5pt]&[-5pt]\\
f^*k^*\sum_kP\ar[ru, "\ct", pos=0.9]\\[-10pt]
&P\ar[rr, "\ct\eta^k_P"]\ar[uu, "\eta^k_P"']&&\sum_kP\\[-5pt]
f^*P\ar[rr, "\ct\cdot{}f^*\eta^k_P"']\ar[ru, "\ct", near start]\ar[uu, "f^*\eta^k_P"]
&&g^*\sum_kP\ar[ru, "\ct"']
\ar[from=lluuu, "\ct", crossing over, pos=0.35]\\[15pt]
&B\ar[rr, "k"]&&D\\
A\ar[rr, "h"']\ar[ru, "f"]&&C\ar[ru, "g"']
\end{tikzcd}\hspace{0.5cm}
\begin{tikzcd}[row sep=10pt, column sep=15pt]
f^*k^*\prod_kP\areq[dd]\ar[rd, "\ct" near start]&&g^*\prod_kP
\ar[rd, "\ct" near start]\\[-15pt]
&k^*\prod_kP\ar[dd, "\varepsilon^k_P"', near end]\ar[rr, "\ct", pos=0.13]&
&\prod_kP\\[-15pt]
h^*\prod_hf^*P\ar[dd, "f^*\varepsilon^k_{P}"']\ar[rr, "\ct"' near end, crossing over]&&
\prod_hf^*P\areq[uu, crossing over]&&
\\
&P\\[-10pt]
f^*P\ar[ru, "\ct"]\\[-10pt]
&B\ar[rr, "k"]&&D\\
A\ar[rr, "h"']\ar[ru, "f"]&&C\ar[ru, "g"']&
\end{tikzcd}\hspace{2cm}$$ Hence we see that the equation $f^*\eta^k_P=\eta^h_{f^*P}$ from “Stability of the operations under pullbacks” says precisely that the morphism $\ct\cdot{}f^*\eta^k_P$ which is required by the stability condition to be cocartesian is in fact the specified cocartesian morphism $\ct\eta^h_{f^*P}$ (and similarly, $f^*\varepsilon^k_P=\varepsilon^h_{f^*P}$ is the condition that the diagram required to be a $\prod$-diagram is in fact the specified $\prod$-diagram).
*Equality objects*
Finally, the “Equality objects” relations ensure that the morphism $(\crt{(\Delta_{\seq{B}})}{\Eq_B})\cdot{}r^B:\top_B\to\Eq_B$ over $\Delta_B$ is cocartesian. Now, since $\fib{Pf}$ has $\prod$-diagrams over product projections, all cocartesian morphisms in $\fib{Pf}$ are stable along product projections by Proposition \[prop:beck-chev-swap\]. Hence, we also have cocartesian morphisms over $\id_{\vec{A}}\times\Delta_{\seq{B}}:\vec{A}\seq{B}\to\vec{A}\seq{B,B}$ with domain $\top_{\vec{A}\seq{B}}$ for every $\vec{A}$. Since every (specified and hence every) diagonal morphism in $\TM$ is isomorphic to a composite of such morphisms, it follows from Propositions \[prop:bccomp\] and \[prop:bcisoinv\] below that there is a cocartesian lift with domain $\top_A$ for every diagonal morphism $A\to{}A\times{}A$.
### {#prop:dedclasses-univ-prop}
The $(\bigsqcup_{\vec{A}\in\Ob\TM}\Ob\Form(\vec{A})\times\Ob\Form(\vec{A}))$-indexed family of sets given by $(\Hom_{\Form(\vec{A})}(P,Q))_{\vec{A},P,Q}$ is the (underlying family of sets of a) free $\sigma'$-structure (for $\sigma'$ the signature introduced in Definition \[defn:deductions\]) subject to the relations from Definition \[defn:deduction-rels\], in the following sense:
Given any $\sigma'$-structure $X$ with underlying sets $(X_{PQ})_{\vec{A},P,Q}$ satisfying all of the relations given in Definition \[defn:deduction-rels\] (these were defined as relations on the particular $\sigma'$-structure $\Ded$, but of course they make sense for any $\sigma'$-structure), there is a unique morphism $(\Hom_{\Form(\vec{A})}(P,Q)\to{}X_{PQ})_{\vec{A},P,Q}$ of $\sigma'$-structures. This is more or less obvious. Since $\Ded$ is the initial $\sigma'$-algebra (i.e., set of $\sigma'$-terms with no variables), there is by Proposition \[prop:recursion-principle\] precisely one morphism of $\sigma'$-structures from $\Ded$ to $X$, and hence there is at most one morphism from the quotient $(\Hom_{\Form(\vec{A})}(P,Q))_{\vec{A},P,Q}$ of $\Ded$ to $X$; and the assumption that $X$ satisfies the relations of Definition \[defn:deduction-rels\] implies that the unique morphism from $\Ded$ to $X$ does indeed factor through the quotient.
Lemmas for the proof that $\fib{Pf}$ is an $h^=$-fibration {#subsec:pf-is-hfib-lemmas}
----------------------------------------------------------
We now state and prove the Propositions \[prop:bccomp\] and \[prop:bcisoinv\] which were needed above in the proof that $\fib{Pf_\sigma}$ is an $h^=$-fibration. For the rest of §\[subsec:pf-is-hfib-lemmas\], let $\fibr{C}CB$ be a fibration.
### {#defn:prod-diag-comp}
Let $A\tox{f}B\tox{g}C$ be morphisms in $\B$. Suppose we are given $\prod$-diagrams $$\begin{tikzcd}
Q\ar[d, "p"']\ar[r, "q"]&R\\[-5pt]
P\\[-15pt]
A\ar[r, "f"]&B
\end{tikzcd}\quad\quad\quad
\begin{tikzcd}
S\ar[d, "s"']\ar[r, "t"]&T\\[-5pt]
R\\[-15pt]
B\ar[r, "g"]&C.
\end{tikzcd}$$
We define a *composite* of these two $\prod$-diagrams to be any diagram $$\begin{tikzcd}
U\ar[d, "u"']\ar[r, "v"]&T\\[-5pt]
P\\[-15pt]
A\ar[r, "gf"]&C
\end{tikzcd}$$ with $u$ over $A$ and $v$ over $gf$, which arises in the following way. We choose a cartesian lift $\ct:f^*S\to{}S$ of $f$, so that we obtain an induced morphism $f^*s:f^*S\to{}Q$ $$\begin{tikzcd}[row sep=12pt]
f^*S\ar[r, "\ct"]\ar[d, "f^*s"']&S\ar[r, "t"]\ar[d, "s"]&T\\
Q\car[r, "q"]\ar[d, "p"']&R\\
P\\[-8pt]
A\ar[r, "f"]&B\ar[r, "g"]&C,
\end{tikzcd}$$ and then we set $u=p\cdot{}f^*s$ and $v=t\ct$.
### {#prop:bccomp}
Any composite of cocartesian morphisms or $\prod$-diagrams in $\fib{C}$ is again a cocartesian morphism or $\prod$-diagram.
Moreover, given a cocartesian morphism (or $\prod$-diagram) over $g:B\to{}C$ which is stable along some morphism $h:D\to{}C$ for which there exists a pullback square $$\begin{tikzcd}
\cdot\ar[d, "k"']\ar[r]\ar[rd, "\lrcorner", phantom, pos=0.1]&D\ar[d, "h"]\\
B\ar[r, "g"]&C,
\end{tikzcd}$$ and given any cocartesian morphism (or $\prod$-diagram) over $f:A\to{}B$ which is stable along $k$, the composite of these cocartesian morphisms (or $\prod$-diagrams) is stable along $h$ as well. We already know that cartesian (and hence, dually, cocartesian) morphisms are closed under composition (see [@sentai Proposition 2.5]).
That $\prod$-diagrams are closed under composition follows from a straightforward but somewhat lengthy diagram chase. Suppose we have a composite of $\prod$-diagrams, as shown below, as well as some cocartesian lift $\ct:(gf)^*U\to{}U$ of $gf$ and a morphism $u:(gf)^*U\to{}P$. $$\begin{tikzcd}[row sep=12pt]
(gf)^*U\ar[rrrr, "\ct"]\ar[dddr, bend right, "u"']&&&&U\\
\end{tikzcd}$$ We obtain the requisite morphism $U\to{}T$ as follows. We first factor $\ct:(gf)^*U\to{}U$ as the composite $(gf)^*U\tox{\cind\ct}g^*U\tox{\ct}U$ of two cartesian morphisms. The first $\prod$-diagram then induces a morphism $g^*U\to{}R$, and thence, the second $\prod$-diagram induces a morphism $U\to{}T$. It is then straightforward to see that this morphism has the desired property, and is the unique such morphism.
To see that the composite of stable cocartesian morphisms is stable, suppose we have pullback squares $$\begin{tikzcd}
\cdot\ar[r, ""]\ar[d, "l"]\ar[rd, "\lrcorner", phantom, pos=0.1]&
\cdot\ar[d, "k"']\ar[r]\ar[rd, "\lrcorner", phantom, pos=0.1]&D\ar[d, "h"]\\
A\ar[r, "f"]&B\ar[r, "g"]&C,
\end{tikzcd}$$ and are considering the composite of cocartesian lifts $P\tox{p}Q\tox{q}R$ of $f$ an $g$. We need to show that, given cartesian lifts $\ct:l^*P\to{}P$ and $\ct:h^*R\to{}R$ of $l$ and $h$, the induced morphism $l^*P\to{}h^*R$ is cocartesian. We choose an additional cartesian lift $k^*Q\to{}Q$ of $k$. We then have, by the stability of $p$ and $q$, that the induced morphisms $l^*P\to{}k^*Q$ and $k^*Q\to{}h^*R$ are cartesian. But the induced morphism $l^*P\to{}h^*R$ must be the composite of these, and hence cocartesian.
The proof that the composite of stable $\prod$-diagrams is stable is similar.
### {#prop:bcisoinv}
Let $f:A\to{}B$ be a morphism in $\B$, and suppose $f':A'\to{}B'$ is isomorphic to $f$, in the sense that there exists a commutative square $$\begin{tikzcd}
A\ar[r, "\sim", "i"']\ar[d, "f"']&A'\ar[d, "f'"]\\
B\ar[r, "\sim", "j"']&B'.
\end{tikzcd}$$ Let $P\in\Ob\fib{C}^A$ and let $p:P'\to{}P$ be a cartesian lift of $i\I$ ($p$ is then also an isomorphism by [@sentai Proposition 10.2]). If there exists a cocartesian morphism with domain $P$ (or $\prod$-diagram based on $P$) over $f$, then there exists a cocartesian morphism with domain $P'$ (or $\prod$-diagram based on $P'$) over $f'$.
Moreover, if the original cocartesian morphism (or $\prod$-diagram) over $f$ is stable along some morphism $g:C\to{}B$, then the resulting one over $f'$ is stable along the composite $C\tox{g}B\tox{j}B'$. Given a cocartesian lift $q:P\to{}Q$ of $f$, we can obtain a cocartesian lift of $f'$ by composing $qp$ with an isomorphic (hence cocartesian) lift of $j$. To see that the result is stable, it suffices by Proposition \[prop:bccomp\] to see that cocartesian lifts of isomorphisms are stable along every possible morphism. In fact, using the stability of isomorphisms under pullbacks, the 2-of-3 property of cartesian morphisms [@sentai Proposition 2.5], as well as [@sentai Proposition 10.2] and its dual, one can see that the morphism required in the stability condition to be cocartesian will again be an isomorphism.
The proof for $\prod$-diagrams is similar; we need only see that we can find a $\prod$-diagram over any isomorphism and based on any object, and that these are always stable. In fact, given an isomorphism $i:A\to{}A'$ and an object $P$ over $A$, we can take any isomorphism $Q\to{}P$ (for example, $\id_P$), any isomorphism $q:Q\to{}R$ over $f$, and the result $$\begin{tikzcd}
Q\ar[d, "p"', "\vsim"]\ar[r, "q", "\sim"']&R\\[-5pt]
P\\[-15pt]
A\ar[r, "f"]&B
\end{tikzcd}$$ will be a $\prod$-diagram. The proof that this is stable is similar to the one for cocartesian morphisms; in the same way, one can show that in the stability condition, the induced diagram which is required to be a $\prod$-diagram again consists of two isomorphisms.
Freeness of the fibration
-------------------------
Our final task is to show that the $h^=$-fibration $\fib{Pf_\sigma}$ is free over $\TM_\sigma$.
This will involve, first, showing that for any other $h^=$-fibration $\fib{C}$ over $\TM$, there is a morphism $\fib{Pf}\to\fib{C}$ of $h^=$-fibrations over $\TM$, and secondly, that any two such morphisms are connected by a unique natural isomorphism.
In both cases, this will involve a construction (or proof) involving the objects of $\Pf$ and then one involving the morphisms, and these will both proceed by recursion/induction. In the case of morphisms, we note that it is only the morphisms in the *fibers* of $\fib{Pf}$ (i.e., the “deductions”) that satisfy a recursion principle. To deal with a general morphism, we factor it as a morphism in the fibers and a cartesian morphism in the canonical cleavage. The latter morphisms can be handled by a recursion on the objects of $\Pf$.
Hence, we begin with a lemma, showing that one can construct a morphism out of a (split cloven) fibration by separately specifying its action on the cleavage and on the fiber morphisms.
It is worth noting that, as will become more or less clear from the proof, the fibration $\fib{Pf_\sigma}$ also satisfies a different universal property, which determines it up to isomorphism. Namely, given another $h^=$-fibration $\fib{C}$ over $\TM_\sigma$ in which “all of the choices” corresponding to the syntactic operations on formulas – i.e., finite products, coproducts, and exponentials in the fibers, and so on – have been, there is a *unique* morphism of $h^=$-fibrations $\fib{Pf_\sigma}\to\fib{C}$ over $\sigma$ respecting all of these choices.
### {#lem:splitext}
Let $\fibr{C}CB$ and $\fibr{D}{D}{B}$ be fibrations over $\B$. Suppose $\fib{C}$ admits a split cleavage, and fix one such, so that we have a subcategory $\C_1$ of $\C$ containing all the objects in $\C$ and all the morphisms in the cleavage. Let $\C_2$ be the union of all the fibers $\fib{C}^A$ for $A\in\Ob\cat{B}$. Let $\Phi_1:\C_1\to\D$ and $\Phi_2:\C_2\to\D$ be functors over $\B$, in the sense that $\fib{D}\Phi_i=\fib{C}$ for $i=1,2$, and suppose that $\Phi_1$ and $\Phi_2$ agree on the intersection of $\C_1$ and $\C_2$ (i.e., on the objects of $\C$). Suppose further that, for each $f:A\to{}B$ in $\B$ and each $p:P\to{}Q$ in $\fib{C}^B$, the following square commutes. $$\begin{tikzcd}[column sep=huge]
\Phi_1(f^*P)\ar[r, "\Phi_1(\crt{f}P)"]\ar[d, "\Phi_2(f^*p)"']&\Phi_1P\ar[d, "\Phi_2p"]\\
\Phi_1(f^*Q)\ar[r, "\Phi_1(\crt{f}Q)"]&\Phi_1Q
\end{tikzcd}$$
There is a unique functor $\Phi:\C\to\D$ over $\B$ restricting to $\Phi_1$ and $\Phi_2$. Moreover, if $\Phi_1$ takes each morphism to a cartesian morphism, then $\Phi$ is a morphism of fibrations over $\B$. Let $p:P\to{}Q$ be a morphism in $\C$ over $f:A\to{}B$ in $\B$. Then $p$ factors as $(\crt{f}Q)\cind{p}$ for a unique $\cind{p}:P\to{}f^*Q$ over $\id_A$. Hence, if $p$ is not in $\C_1$ or $\C_2$, we are forced to take $\Phi{}p$ to be $(\Phi_1(\crt{f}Q))(\Phi_2\cind{p})$ (and if $p$ is in $\C_1$ or $\C_2$, we of course take $\Phi{}p$ to be $\Phi_1p$ or $\Phi_2p$). Note that the equation $\Phi{}p=(\Phi_1(\crt{f}Q))(\Phi_2\cind{p})$ still holds if $p$ in $\C_1$, since then $\cind{p}=\id_{f^*Q}$, and it holds if $p$ is in $\C_2$, since then $\crt{f}Q=\id_Q$.
It remains to see that $\Phi$, thus defined, is a functor. We know that $\Phi$ preserves identity morphisms since $\Phi_1$ and $\Phi_2$ do, so we need only see that it preserves composition.
Let $p:P\to{}Q$ and $q:Q\to{}R$ be morphisms in $\C$ over $f:A\to{}B$ and $g:B\to{}C$ in $\B$. We then have the following commutative diagram in $\C$ and its image under $\Phi$ in $\D$. $$\begin{tikzcd}
P\ar[dr, "p"]\ar[d, "\cind{p}"']&&[15pt]\\
f^*Q\ar[r, "\ct"']\ar[d, "f^*\cind{q}"']&Q\ar[dr, "q"]\ar[d, "\cind{q}"']\\
f^*g^*R\ar[r, "\ct"]&g^*R\ar[r, "\ct" near start]&R\\[-10pt]
A\ar[r, "f"]&B\ar[r, "g"]&C
\end{tikzcd}\quad\quad
\begin{tikzcd}
\Phi{}P\ar[dr, "\Phi{}p"]\ar[d, "\Phi_2\cind{p}"']
\ar[ddrr, bend left=30pt, "\Phi(qp)"]&&[20pt]\\
\Phi(f^*Q)\ar[r, "\Phi_1\ct"']\ar[d, "\Phi_2(f^*\cind{q})"']&
\Phi{}Q\ar[dr, "\Phi{q}", near start]\ar[d, "\Phi_2\cind{q}"]\\
\Phi(f^*g^*R)\ar[r, "\Phi_1\ct"]&\Phi(g^*R)\ar[r, "\Phi_1\ct" near start]&\Phi{}R\\[-10pt]
A\ar[r, "f"]&B\ar[r, "g"]&C
\end{tikzcd}$$ By definition, the two small triangles in the second diagram commute, as does the large triangle. By assumption, the rectangle also commutes. Hence $(\Phi{}q)(\Phi{}p)=\Phi(qp)$.
### {#prop:syntactic-fib-functor}
Given an $h^=$-fibration $\fibr{C}C{Tm_\sigma}$ over $\TM_\sigma$, there exists a morphism of $h^=$-fibrations $M:\fib{Pf_\sigma}\to\fib{C}$ over $\TM_\sigma$.
We note that this will involve an essential use of the axiom of choice. We will use Proposition \[lem:splitext\] to construct $M$.
*First part: defining $M$ on objects*
To define $M$ on the objects of $\Pf$, we need, by Proposition \[prop:form-freegen\] \[item:form-freegen-map\] an $\Ob\TM$-indexed family of sets together with operations as listed there.
Let us (arbitrarily) choose, in $\fib{C}$, fiberwise binary products and coproducts, initial and terminal objects, and exponential objects (i.e., we choose for each pair of objects a product diagram, coproduct diagram, and exponential diagram based on them). Let us also fix cocartesian morphisms and prod-diagrams over the morphisms $\pi^{\vec{A}}$, giving us functors $\sum_{\pi^{\vec{A}}}$ and $\prod_{\pi^{\vec{A}}}$ (which would be left and right adjoint to the functor $(\pi^{\vec{A}})^*$, had we already chosen a cleavage). Finally, let us choose an equality object $\Eq_B$ in $\fib{C}^{\seq{B}\times\seq{B}}$ (i.e., a cocartesian morphisms over $\Delta_B:\seq{B}\to{}\seq{B}\times{}\seq{B}$ with domain a terminal object) for each $B\in\TM$, and cartesian morphisms $\crt{\br{s,t}}{\Eq_B}:\br{s,t}^*\Eq_B\to\Eq_B$ for each pair of morphisms $s,t:\vec{A}\to{}\seq{B}$ in $\Ob\TM$.
This clearly endows the family of sets $(\Ob\fib{C}^{\vec{A}})_{\vec{A}\in\Ob\TM}$ with the operations listed in Proposition \[prop:form-freegen\], and hence gives us maps $\Ob\fib{Pf}^{\vec{A}}\to\fib{C}^{\vec{A}}$ for each $\vec{A}\in\Ob\TM$.
*Second part: defining $M$ on the cleavage*
We have thus defined $M:\Ob\Pf\to\C$. We next define $M$ on the cartesian lifts constituting the canonical cleavage of $\fib{\Pf}$.
Each such morphism has the form $\ct=\crt{t}Q:t^*Q\to{}Q$ for some $t:\vec{A}\to\vec{B}$ in $\TM_\sigma$. We define $M(\crt{t}Q)$ by induction on $Q$; i.e., we define by recursion a function taking each $Q\in\Ob\Pf$ to a function taking each $t:\vec{A}\to\vec{B}$ (where $Q\in\fib{Pf}^{\vec{B}}$) to a cartesian morphism $M(t^*Q)\to{}MQ$.
If $Q$ is $\top_{\vec{B}}$ or $\bot_{\vec{B}}$, then, by Proposition \[prop:form-stab\], $P$ is $\top_{\vec{A}}$ or $\bot_{\vec{A}}$, and $MP$ and $MQ$ are the terminal or initial objects of $\vec{A}$ and $\vec{B}$. We then let $M(\crt{t}Q)$ be the unique cartesian morphism $MP\to{}MQ$.
If $Q$ is $s^*\Eq_C$ for some $s:\vec{B}\to\br{C,C}$, so that $t^*Q=(ts)^*\Eq_C$, then we have a solid diagram $$\begin{tikzcd}
M((ts)^*\Eq_C)\car[rr, "\crt{(ts)}{(M\Eq_C)}"]\ar[rd, dashed, "M(\crt{t}Q)"']&&M\Eq_C\\
&M(s^*\Eq_C)\car[ur, "\crt{s}{(M\Eq_C)}"']&\\[-15pt]
\vec{A}\ar[r, "t"]&\vec{B}\ar[r, "s"]&\br{C,C},
\end{tikzcd}$$ where $\crt{(ts)}(M\Eq_C)$ and $\crt{s}(M\Eq_C)$ are the cartesian morphisms chosen in the definition of $M((ts)^*\Eq_C)$ and $M(s^*\Eq_C)$, and we let $M(\crt{t}Q)$ be the unique morphism making the whole diagram commute. By [@sentai Proposition 2.5], it is again cartesian.
Now suppose $Q$ is $Q'\wedge{Q''}$, so that $P=t^*Q'\wedge{}t^*Q''$. Now, $MP$ and $MQ$ have been defined as the chosen products $M(t^*Q')\wedge(t^*Q'')$ and $MQ'\wedge{}MQ''$. By induction, we have already defined the cartesian morphisms $M(\crt{t}Q'):M(t^*Q')\to{}M(Q')$ and $M(\crt{t}Q''):M(t^*Q'')\to{}M(Q'')$. We now let $M(\crt{t}Q)$ be the unique (necessarily cartesian) morphism over $t$, given by Proposition \[prop:longprod\] below, making the diagram $$\begin{tikzcd}
&[-20pt]MP\ar[ldd]\ar[rd]\ar[rrr, dashed, "M(\crt{t}{Q})"]&[-20pt]&
&[-20pt]MQ\ar[rd]&[-20pt]\\
&&M(t^*Q'')\ar[rrr, "M(\crt{t}{Q''})", near start]&&&MQ''\\
M(t^*Q')\ar[rrr, "M(\crt{t}{Q'})"]&&&MQ'\ar[from=uur, crossing over]&&
\end{tikzcd}$$ commute, where the unlabeled arrows form the chosen product diagrams for $MP$ and $MQ$. If $P$ is $Q'\vee{}Q''$ or $Q'\Rightarrow{}Q''$, then $Mp$ is defined similarly, but using Proposition \[prop:longcoprod\] or Proposition \[prop:longexp\]. Similarly, if $P$ is $\forall{}Q$ or $\exists{}Q$, then $Mp$ is defined using Proposition \[prop:longprodf\] or \[prop:longsumf\].
*Third part: $M$ is a functor on the cleavage*
Next, we must verify that, on the cartesian morphisms on which we have just defined $M$, it preserves composition and identities. That it preserves identities is proven by induction: in each case $M(\crt{t}Q)$ was defined to be the unique morphism satisfying some property, and the identity morphism always satisfies the property in question.
The proof that $M$ preserves composites is similar. We consider a composite $(ts)^*R\tox{\crt{s}{(t^*R)}}t^*R\tox{\crt{t}R}R$ of morphisms in the cleavage over $\vec{A}\tox{s}\vec{B}\tox{t}\vec{C}$ in $\TM$, and we prove by induction on $R$ that $M(\crt{(ts)}R)=M(\crt{t}R)M(\crt{s}{(t^*R)})$. The point is, again, that $M(\crt{(ts)}R)$ is defined to be the unique morphism satisfying some property, and the morphism $M(\crt{t}R)M(\crt{s}{(t^*R)})$ is always seen to satisfy this property. For example, if $R=R'\wedge{R''}$, then we have a commutative diagram $$\begin{tikzcd}
&[-47pt]M((ts)^*R)\ar[ldd]\ar[rd]\ar[rrr, "M(\crt{s}{(t^*R)})"]&[-25pt]&
&[-40pt]M(t^*R)\ar[rd]\ar[rrr, "M(\crt{t}R)"]&[-20pt]&
&[-30pt]MR\ar[rd]&[-20pt]\\
&&M((ts)^*R'')\ar[rrr, "M(\crt{s}{(t^*R'')})", near start]&&&
M(t^*R'')\ar[rrr, "M(\crt{t}{R''})", near start]&&&MR''\\
M((ts)^*R')\ar[rrr, "M(\crt{s}{(t^*R')})"]&&&
M(t^*R')\ar[rrr, "M(\crt{t}{R'})"]\ar[from=uur, crossing over]&&&
MR'.\ar[from=uur, crossing over]&&
\end{tikzcd}$$ Since, by induction, $$(M(\crt{t}{R'}))(M(\crt{s}{(t^*R')}))=M(\crt{(ts)}{R'})$$ and $$(M(\crt{t}{R''}))(M(\crt{s}{(t^*R'')}))=M(\crt{(ts)}{R''}),$$ we have by definition that $M(\crt{(ts)}R)$ is the unique morphism $M((ts)^*R)\to{}MR$ making the diagram (with the object $M(t^*R)$ and all incident arrows removed) commute. Obviously, $M(\crt{t}R)M(\crt{s}{(t^*R)})$ is the unique such morphism as desired.
*Fourth part: defining $M$ on the fibers*
We next define $M$ on morphisms in the fibers $\Form_\sigma(\vec{A})$ – i.e., on the deductions. According to Proposition \[prop:dedclasses-univ-prop\], to define such a function, we need to specify a $(\bigsqcup_{\vec{A}\in\Ob\TM}\Ob\Form(\vec{A})\times\Ob\Form(\vec{A}))$-indexed family of sets, together with the operations given in Definition \[defn:deductions\], satisfying the relations given in Definition \[defn:deduction-rels\].
For our family of sets, let us take $(\Hom_{\fib{C}^{\vec{A}}}(MP,MQ))_{\vec{A},P,Q}$ (of course, this choice is more or less forced on us since we are trying to extend $M$ to a functor). The definitions of the operations listed in Definition \[defn:deductions\] more or less suggest themselves.
For the “Category and fibration structure” operations, we use the category structure on $\C$ to define “$1_P$” and “$\circ$”, and given $f:MP\to{}MQ$ in $\fib{C}^A$ and $t:\vec{B}\to\vec{A}$ we define $t^*f:M(t^*P)\to{}M(t^*Q)$ using the chosen cartesian morphisms $M(\crt{t}P):M(t^*P)\to{}MP$ and $M(\crt{t}Q):M(t^*Q)\to{}MQ$.
The “Finite products and coproducts” and “Exponentials” operations are defined using the chosen finite products and coproducts and exponentials in the fiber of $\fib{C}$.
The “Adjoints to pullback along projections” operations are defined using the chosen cocartesian morphisms and $\prod$-diagrams, and the “Equality objects” operations are defined using the chosen Equality objects.
Now we must verify that these operations satisfy the relations in Definition \[defn:deduction-rels\]. In fact, these almost all follow immediately from how we defined the operations. The two cases that require some thought are the final two equations under “Stability of the operations under pullbacks”; but these follow from a similar analysis to the one given in the proof of Proposition \[prop:syntactic-fib-is-hfib\].
Having verified that the relations from Definition \[defn:deduction-rels\] are satisfied, we have by Proposition \[prop:dedclasses-univ-prop\] a unique family of maps $\Hom_{\fib{Pf}^{\vec{A}}}(P,Q)\to\Hom_{\fib{C}^{\vec{A}}}(MP,MQ)$ preserving all the operations from Definition \[defn:deductions\]. In particular, composition and identities are preserved, so this defines functors $\fib{Pf}^A\to\fib{C}^A$ on the fibers. The way we defined the operations $t^*$ on the family of sets $(\Hom_{\fib{C}^{\vec{A}}}(MP,MQ))_{\vec{A},P,Q}$ ensure the final condition of Lemma \[lem:splitext\] is satisfied, and so we have a morphism of fibrations $\fib{Pf}\to\fib{C}$ over $\TM$.
*Fifth part: $M$ is a morphism of $h^=$-fibrations*
Finally, we must verify that our morphism $\fib{Pf}\to\fib{C}$ of fibrations is a morphism of $h^=$-fibrations. This will follow from the preservation of (some of) the operations from Definition \[defn:deductions\].
Indeed, the preservation of the operations $!_P$, $\ex_P$, $\pi_{PQ}$, $\pi'_{PQ}$, $\kappa_{PQ}$, $\kappa'_{PQ}$, $\varepsilon_{PQ}$ from “Finite products and coproducts” and “Exponentials” operations shows that the specified – and hence all – finite products and coproducts and exponentials in the fibers are preserved. Similarly, the preservation of the operations $\varepsilon^{\forall}_P$ and $\eta^{\exists}_P$ imply that the specified – and hence all – cocartesian morphisms and $\prod$-diagrams over the product projections $\pi^{\vec{A}}$ are preserved.
To see that $M$ preserves cocartesian morphisms and $\prod$-diagrams over arbitrary product projections, we use – as in the proof of Proposition \[prop:syntactic-fib-is-hfib\] – that every product projection in $\TM$ is isomorphic to a composite of the morphisms $\pi^{\vec{A}}$. Now, it is clear that if two composable cocartesian morphisms are taken by $M$ to cocartesian morphisms, then so is their composite. Similarly, if two composable $\prod$-diagrams are taken by $M$ to $\prod$-diagrams, then so is their composite (Definition \[defn:prod-diag-comp\]). Finally, any cocartesian morphism or $\prod$-diagram which is isomorphic to one preserved by $M$ is preserved as well.
Finally, the preservation of the operation $r^B$ under “Equality objects” shows that the specified cocartesian morphisms over the diagonal morphisms $\Delta_{\seq{B}}$ are preserved and hence, by a similar argument to that in the previous paragraph, that every cocartesian morphism over a diagonal morphism is preserved. Indeed, it is easy to see that if any cocartesian morphism $p$ is preserved by $M$, then so is any morphism which is cocartesian “by virtue” of $p$ being stable along some morphism. In particular, each cocartesian lift of $\id_{\vec{A}}\times\Delta_{\seq{B}}$ with domain $\top_{\vec{A}\seq{B}}$ is preserved, and so we are done by the argument in the previous paragraph since every diagonal morphism in $\TM$ is isomorphic to a composite of the morphisms $\id_{\vec{A}}\times\Delta_{\seq{B}}$.
### {#prop:syntactic-hfib-nat-trans}
Let again $\fibr{C}C{Tm}$ be an $h^=$-fibration over $\TM$, and suppose now that we have two morphisms $M,M':\fib{Pf}\to\fib{C}$ of $h^=$-fibrations.
There is a unique natural isomorphism $M\to{}M'$ of morphisms of fibrations (which is then necessarily a natural isomorphism).
We will show, by induction on $P\in\Ob\Pf$, that there is a unique morphism $\alpha_P:MP\to{}M'P$ over $\vec{A}$ (where $P\in\Form_\sigma(\vec{A})$) which can be part of a natural transformation of morphisms of fibrations, and that $\alpha_P$ is an isomorphism. More precisely, for each $P\in\Ob\Pf$, we will define an isomorphism $\alpha_P:MP\to{}M'P$ such that, if $\beta:M\to{}M'$ is any natural transformation of morphisms of fibrations, then $\beta_P=\alpha_P$.
We will then show that the $\alpha_P$ do in fact constitute a natural transformation.
*First part: definition of $\alpha_P$*
We now define $\alpha_P$ by recursion on $P$.
According to Proposition \[prop:form-freegen\], in order to define a function on the objects of $\Ob\Pf$, we need to specify an $\Ob\TM$ indexed family of objects $(X_{\vec{A}})_{\vec{A}\in\Ob\TM}$, together with the operations listed there.
For our family of sets, let us take $\big(\bigsqcup_{P\in\Ob\fib{C}^A}\operatorname{Iso}_{\fib{C}^A}(MP,M'P)\big)_{\vec{A\in\Ob\TM}}$, where $\operatorname{Iso}$ denotes the subset of $\Hom$ consisting of isomorphisms.
We define operations $\wedge$, $\vee$, and so on, on this family of sets, by sending, for example, given isomorphisms $MP\cong{}MP'$ and $MQ\cong{}MQ'$ to the induced isomorphism $M(P\wedge{}P')\cong{}M(Q\wedge{}Q')$ between the products, and similarly with the other operations.
We therefore get, by Proposition \[prop:form-freegen\], an induced family $\alpha:\Ob\fib{Pf}^{\vec{A}}\to\bigsqcup_{P\in\Ob\fib{C}^A}\operatorname{Iso}(MP,M'P)$ of maps preserving the operations. It follows by induction that $\alpha_P:MP\to{}M'P$ for each $P$.
*Second part: uniqueness of $\alpha$*
We next show that if $\beta:M\to{}M'$ is any natural transformation over $\TM$, then $\alpha_P=\beta_P$ for every $P\in\Ob\Pf$, by induction on $P$.
If $P$ is $\top_{\vec{A}}$ or $\bot_{\vec{A}}$, then $MP$ and $M'P$ are both terminal or initial, and there is a unique morphism $MP\to{}M'P$.
Suppose $P$ is $s^*\Eq_{\br{C}}$ for some $s:\vec{A}\to\br{C,C}$. Then we have the following commuting naturality squares for $\beta$ $$\begin{tikzcd}[column sep=50pt]
M(s^*\Eq_{\br{C}})\car[r, "M(\ct)"]\ar[d, "\beta_{s^*\Eq_{\br{C}}}"']&
M(\Eq_{\br{C}})\ar[d, "\beta_{\Eq_{\br{C}}}"]&
M(\top_{\br{C}})\rac[l, "M(\ct\cdot{}r^C)"]\ar[d, "\beta_{\top_{\br{C}}}"]\\
M'(s^*\Eq_{\br{C}})\car[r, "M'(\ct)"]&M'(\Eq_{\br{C}})&
M'(\top_{\br{C}})\rac[l, "M'(\ct\cdot{}r^C)"]\\[-15pt]
\vec{A}\ar[r, "s"]&\br{C,C}&\br{C}\ar[l, "\Delta_{\br{C}}"].
\end{tikzcd}$$ We see, by cartesianness and cocartesianness, that there is a unique choice of $\beta_{\Eq_{\br{C}}}$ and $\beta_{s^*\Eq_{\br{C}}}$ making these diagrams commute; and these unique morphisms are precisely the definition of $\alpha_{\Eq_{\br{C}}}$ and $\alpha_{s^*\Eq_{\br{C}}}$.
If $P=Q\wedge{}Q'$, then by induction we have $\beta_Q=\alpha_Q$ and $\beta_{Q'}=\alpha_{Q'}$. We then have the commuting naturality squares for the morphisms $\pi_{QQ'}$ and $\pi'_{QQ'}$, and these imply that the morphism $\beta_P$ must be the isomorphism of the products $MP\cong{}M'P$ induced by the isomorphisms $\beta_Q=\alpha_Q:MQ\toi{}M'Q$ and $\beta_{Q'}=\alpha_{Q'}:MQ'\toi{}M'Q'$ – and this is precisely the definition of $\alpha_P$. The same proof applies when $P=Q\vee{}Q'$.
The argument when $P=Q\To{}Q'$ is similar. The naturality squares of $\beta$ for $\pi_{Q(Q\To{}Q')}$, $\pi'_{Q(Q\To{}Q')}$, and $\varepsilon_{QQ'}$ imply that $\beta_{P}$ is the isomorphism between exponential objects induced by the isomorphisms $\beta_Q=\alpha_Q:Q\toi{}Q'$ and $\beta_{Q'}=\alpha_{Q'}:Q\toi{}Q'$ – which is the definition of $\alpha_P$.
The arguments when $P=\forall{}Q$ or $P=\exists{}Q$ are also similar; we use the naturality squares for $\varepsilon^\forall_{Q}$ and $\crt{(\pi^{\vec{A}})}{(\forall{}Q)}$ (where $Q\in\fib{Pf}^{\vec{A}}$), or for $\ct\cdot\eta^{\exists}_Q$, and conclude that $\beta_P$ is the isomorphism $M(\forall{}Q)\cong{}M'(\forall{}Q)$ or $M(\exists{}Q)\cong{}M'(\exists{}Q)$ induced by the isomorphism $\beta_Q=\alpha_Q:MQ\toi{}M'Q$.
*Third part: naturality of $\alpha$*
Next, we must check that the family $\alpha=(\alpha_P)_{P\in\Ob\Pf}$ constitutes a natural transformation.
We first note that $\alpha$ is natural with respect to the composite $qp$ of morphisms in $\Pf$ whenever it is natural with respect to $p$ and $q$. Hence, since each morphism in $\Pf$ is a composite a morphism in a fiber and a morphism in the canonical cleavage of $\fib{Pf}$, we need only check naturality for morphisms in these two classes.
*Cleavage morphisms*
We first show that $\alpha$ is natural with respect to the morphisms $\crt{t}P:t^*P\to{}P$ in the cleavage by induction on $P$, where $t:\vec{A}\to\vec{B}$.
Suppose $P$ is one of $\top_{\vec{B}}$ or $\bot_{\vec{B}}$. Then the naturality square for $\crt{t}P$ consists of two morphisms $M(t^*P)\to{}M'(P)$ over $t$, which are equal, since there is only one such morphism.
Next, suppose $P$ is $s^*\Eq_{\br{C}}$ for some $s:\vec{B}\to\br{C,C}$. We then have the following diagram, where the rectangle on the right and the outer rectangle commute by the definitions of $\alpha_{(ts)^*\Eq_C}$ and $\alpha_{s^*\Eq_C}$. Hence, the rectangle on the left commutes by a diagram chase using [@sentai Proposition 2.8]. $$\begin{tikzcd}[column sep=60pt]
M(t^*s^*\Eq_{C})\ar[r, "M(\crt{t}{(s^*\Eq_{C})})"]
\ar[d, "\alpha_{t^*s^*\Eq_{C}}"']&
M(s^*\Eq_{C})\ar[r, "M(\crt{s}{(\Eq_{C})})"]
\ar[d, "\alpha_{s^*\Eq_{C}}"']&
M(\Eq_{C})\ar[d, "\alpha_{\Eq_{C}}"]\\
M'(t^*s^*\Eq_{C})\ar[r, "M'(\crt{t}{(s^*\Eq_{C})})"]&
M'(s^*\Eq_{C})\car[r, "M'(\crt{s}{(\Eq_{C})})"]&
M'(\Eq_{C})
\end{tikzcd}$$
Now suppose that $P$ is $Q\wedge{}R$. Consider the following diagram. $$\begin{tikzcd}[column sep=-20pt]
&&&[-20pt]M'(t^*Q\wedge{}t^*R)\ar[rrrrrr, "\ct"]\ar[dll]&&&[40pt]&[24pt]&&
M'(Q\wedge{}R)\ar[dll]\ar[ddrr]\ar[from=lddd, "\alpha_{Q\wedge{}R}"]&[15pt]&[20pt]\\
&M'(t^*Q)\ar[rrrrrr, "\ct" pos=0.65]&&&&&&M'Q\ar[from=lddd, "\alpha_Q"]&&&&\\
&&&&&M'(t^*R)\ar[rrrrrr, "\ct", crossing over]\ar[from=lluu, crossing over]&&&&&&M'R\\
&&M(t^*Q\wedge{}t^*R)\ar[rrrrrr, "\ct", crossing over]\ar[dll]
\ar[ruuu, crossing over, "\alpha_{t^*Q\wedge{}t^*R}", near start]&&&&&&
M(Q\wedge{}R)\ar[dll]\ar[ddrr]&&\\
M(t^*Q)\ar[rrrrrr, "\ct"]\ar[ruuu, "\alpha_{t^*Q}"]&&&&&&MQ&&&&\\
&&&&M(t^*R)\ar[rrrrrr, "\ct"]\ar[ruuu, crossing over, "\alpha_{t^*R}"]
\ar[from=uull, crossing over]&&&&&&MR\ar[ruuu, "\alpha_R"]\\
\end{tikzcd}$$ By Proposition \[prop:longprod\], there is a unique morphism $M(t^*Q\wedge{}t^*R)\to{}M'(Q\wedge{}R)$ making the diagram (with the objects $M'(t^*Q\wedge{}t^*R)$ and $M(Q\wedge{}R)$ and all arrows incident to them removed) commute. By a diagram chase using the commutativity (which we know by induction) of the naturality squares of $\alpha$ for $\pi^1_{QR}$, $\pi^2_{QR}$, $t^*\pi^1_{QR}$, and $t^*\pi^2_{QR}$, as well as the definitions of $M(\crt{t}{(Q\wedge{}R)})$ and $M'(\crt{t}{(Q\wedge{}R)})$, we have that the two composite morphisms $M(t^*Q\wedge{}t^*R)\to{}M'(Q\wedge{}R)$ in the diagram satisfy this condition, hence they must be equal, as desired.
The case in which $P$ is $Q\vee{}R$ is similar.
Next, suppose $P$ is $\exists{}Q$ for some $Q\in\Form_\sigma(\vec{B}\seq{C})$ and some sort $C$. We then have the following diagram, in which we wish to show that the back face commutes. $$\begin{tikzcd}
&[-20pt]M(t^*\exists{}Q)\ar[dd, "\alpha_{t^*\exists{}Q}", near start]&[10pt]&[-5pt]M(\exists{}Q)
\ar[from=ll, "M\ct"]\\
M((t\times\id_{\br{C}})^*Q)\rac[ru, "M(\ct\cdot{}\eta^\exists_{(t\times\id_{\br{C}})^*Q})"]
&&MQ\ar[from=ll, crossing over, "M\ct", near end]
\ar[ru, "M(\ct\cdot\eta^\exists_Q)", near start]&\\
&M'(t^*\exists{}Q)&&M'(\exists{}Q)\ar[from=ll, "M'\ct", near start]
\ar[from=uu, "\alpha_{\exists{}Q}"]\\
M'((t\times\id_{\br{C}})^*Q)\ar[from=uu, "\alpha_{(t\times\id_{\br{C}})^*Q}"']
\ar[ru, "M'(\ct\cdot{}\eta^\exists_{(t\times\id_{\br{C}})^*Q})"', near end]&&
M'Q\ar[from=ll, "M'\ct"']
\ar[from=uu, crossing over, "\alpha_Q"', near end]
\ar[ru, "M'(\ct\cdot\eta^\exists_Q)"']&
\end{tikzcd}$$ By induction, we know that the front face commutes; by the definitions of $\alpha_{\exists{}Q}$ and $\alpha_{t^*\exists{}Q}=\alpha_{\exists(t\times\id_{\br{C}})^*Q}$, we have that the sides commute; and by the definitions of $M(\crt{t}{(\exists{}Q)})$ and $M'(\crt{t}{(\exists{}Q)})$, we know that the top and bottom faces commute. Hence, by a diagram chase using the dual of [@sentai Proposition 2.8], the back face commutes as desired.
The case in which $P$ is $Q\to{}R$ is handled in the same way as the case $Q\wedge{}R$ above. Specifically, in the diagram for that case, replace “$R$” with “$Q\To{}R$”, and adjoin the square $$\begin{tikzcd}
M'(t^*R)\ar[r, "\ct"]\ar[from=d, "\alpha_{t^*R}"]&M'R\ar[from=d, "\alpha_R"']\\
M(t^*R)\ar[r, "\ct"]&MR
\end{tikzcd}$$ as well as the evaluation morphisms $M(\varepsilon_{QR})$, $M'(\varepsilon_{QR})$, $M(\varepsilon_{(t^*Q)(t^*R)})$, and $M'(\varepsilon_{(t^*Q)(t^*R)})$. We then have by Proposition \[prop:longexp\] that there is a unique morphism $M(t^*(Q\To{}R))\to{}M'(Q\To{}R)$ for which there exists a morphism $M(t^*Q\wedge{}t^*(Q\To{}R))\to{}M'(Q\wedge{}Q\To{}R)$ making the whole diagram commute. A diagram chase using the induction hypothesis and the definitions of $M(\crt{t}{(Q\To{}R)})$ and $M'(\crt{t}{(Q\To{}R)})$ then shows that both of the composite morphisms $M(t^*(Q\To{}R))\to{}M'(Q\To{}R)$ make the diagram commute, and hence are equal as desired.
The final case in which $P$ is $\forall{}Q$ is handled similarly using \[prop:longprodf\].
*Fiber morphisms*
We next show, by induction, that $\alpha$ is natural with respect to the morphisms in the fibers.
We begin with the base cases of the induction.
Naturality with respect to $1_P$, $!_P$, $\ex_P$ is clear.
Naturality with respect to $\pi_{PQ}$, $\pi'_{PQ}$, $\kappa_{PQ}$, $\kappa'_{PQ}$ follows immediately from the definitions of $\alpha_{P\wedge{}Q}$, $\alpha_{P\vee{}Q}$.
Next, consider naturality with respect to $\varepsilon_{QR}$. By definition, $\alpha_{P\To{}Q}$ is the unique morphism for which there exists a (necessarily unique) morphism $p$ making the following diagram commute. $$\begin{tikzcd}[column sep=0pt]
&[-20pt]&[-15pt]MQ\ar[rrr, "\alpha_Q"]&&[-23pt]&[-15pt]M'Q\\
&M((P\To{}Q)\wedge{}P)\ar[ldd, "M\pi_{(P\To{}Q)P}"']\ar[rd, "M\pi'_{(P\To{}Q)P}", near end]
\ar[ru, "M\varepsilon_{PQ}"]\ar[rrr, "p"]
&&&M'((P\To{Q})\wedge{}P)\ar[rd, "M'\pi_{(P\To{}Q)P}", near end]
\ar[ru, "M'\varepsilon_{PQ}", near start]&\\
&&M(P\To{}Q)\ar[rrr, "\alpha_{P\To{}Q}", near end]&&&M'(P\To{}Q)\\
MP\ar[rrr, "\alpha_P"]&&&M'P\ar[from=ruu, crossing over, "M'\pi'_{(P\To{}Q)P}", near end]&&
\end{tikzcd}$$ Since $\alpha_{(P\To{}Q)\wedge{}P}$ is by definition the unique morphism $p$ making the above diagram (with the objects $MQ$ and $MQ'$ and all incident arrows removed) commute, it follows that $p=\alpha_{(P\To{}Q)\wedge{}P}$, and hence that we have the desired commutativity of the naturality square for $\varepsilon_{PQ}$.
Let us next consider naturality with respect to $\eta^\exists_P$ and $\varepsilon^\forall_P$, and let us assume $P\in\Ob\fib{Pf}^{\vec{A}}$. By definition, $\alpha_{\exists{}P}$ is the unique morphism making the square on the bottom of the diagram below to the left commute. It then follows from a diagram chase using [@sentai Proposition 2.8] and the already-proven naturality of $\alpha$ with respect to $\crt{(\pi^{\vec{A}})}{(\exists{}P)}$ that the desired naturality square for $\eta^\exists_P$ commutes. $$\begin{tikzcd}[row sep=8pt, column sep=10pt]
&M'((\pi^{\vec{A}})^*\exists{}P)\car[rrdd, "\ct"]&[-45pt]&[10pt]\\
M((\pi^{\vec{A}})^*\exists{}P)\ar[ru, "\alpha_{(\pi^{\vec{A}})^*\exists{}P}"]\\[-15pt]
&M'P\ar[rr, "\ct\cdot{}M'\eta^\exists_P"']\ar[uu, "M'\eta^\exists_P"]&&M'(\exists{}P)\\[5pt]
MP\ar[rr, "\ct\cdot{}M\eta^\exists_P"']\ar[ru, "\alpha_P", near start]\ar[uu, "M\eta^\exists_P"]
&&M(\exists{}P)\ar[ru, "\alpha_{\exists{}P}"']
\ar[from=lluu, "\ct", crossing over, pos=0.25]
\end{tikzcd}\hspace{0.2cm}
\begin{tikzcd}[row sep=10pt, column sep=2pt]
&M'((\pi^{\vec{A}})^*\forall{}P)\ar[dd, "M'\varepsilon^\forall_P"', near end]\ar[rr, "M'\ct"]&
&M'(\forall{}P)\\[-5pt]
M((\pi^{\vec{A}})*\forall{}P)
\ar[dd, "M\varepsilon^\forall_{P}"']\ar[rr, "M\ct"' near end, crossing over]
\ar[ru, "p" near start]&&
M(\forall{}P)\ar[ru, "\alpha_{\forall{}P}"' near start]&&
\\
&M'P\\
MP\ar[ru, "\alpha_P"]
\end{tikzcd}\hspace{2cm}$$ Next, $\alpha_{\forall{}P}$ is defined to be the unique morphism for which there exists a (necessarily unique) $p$ making the above diagram to the right commute. Since $\alpha_{(\pi^{\vec{A}})^*\forall{}P}$ is the unique morphism $p$ making the top square in the diagram commute, it follows that $p=\alpha_{(\pi^{\vec{A}})^*\forall{}P}$, and hence that we have the desired commuting naturality square for $\varepsilon^\forall_P$.
The argument for naturality with respect to $r^B$ is the same as the one for $\eta^\exists_P$ just given.
*Inductive steps*
We now turn to the inductive steps for the naturality of $\alpha$ with respect to fiber morphisms.
It is immediate that $\alpha$ is natural with respect to $g\circ{}f$ if it is natural with respect to $f$ and $g$.
Next, let $t:\vec{A}\to\vec{B}$ and let $f:P\to{}Q$ be a morphism in $\fib{Pf}^{\vec{A}}$, and suppose $\alpha$ is natural with respect to $f$. Then, in the following cube, the right face commutes by assumption, the front and back squares commute by the naturality of $\alpha$ with respect to the cleavage, shown above, and the top and bottom squares commute by the functoriality of $M$ and $M'$. Hence, by a diagram chase using [@sentai Proposition 2.8], the left face commutes, so $\alpha$ is natural with respect to $t^*f$. $$\begin{tikzcd}
&[-20pt]M(t^*Q)\ar[dd, "\alpha_{t^*Q}", near start]&[10pt]&[-5pt]MQ
\ar[from=ll, "M(\crt{t}Q)"]\\
M(t^*P)\ar[ru, "M(t^*f)"]
&&MP\ar[from=ll, crossing over, "M(\crt{t}{P})", near end]
\ar[ru, "Mf"', near start]&\\
&M'(t^*Q)&&M'Q\car[from=ll, "M'(\crt{t}Q)", near start]\ar[from=uu, "\alpha_Q"]\\
M'(t^*Q)\ar[from=uu, "\alpha_{t^*P}"']\ar[ru, "M'(t^*f)"']&&
M'P\ar[from=ll, "M'(\crt{t}{P})"']
\ar[from=uu, crossing over, "\alpha_P"', near end]
\ar[ru, "M'f"']&
\end{tikzcd}$$
Next, suppose $\alpha$ is natural with respect to $f:P\to{}Q$ and $g:P\to{}R$. We want to prove that it is also natural with respect to $\br{f,g}$. We have the following diagram, where the unlabeled morphisms are $M\br{f,g}$ and $M'\br{f,g}$. Naturality of $\alpha$ with respect to $\br{f,g}$ then follows from the universal property of $M'(Q\wedge{}R)$ by a diagram chase, using the naturality of $\alpha$ with respect $f$ and $g$, which we are assuming, and with respect $\pi_{QR}$ and $\pi'_{QR}$, which we showed above. $$\begin{tikzcd}[row sep=30pt]
&&&&M'P\ar[ldd, "M'f"']\ar[rdd, "M'g"]\ar[d]&\\
&MP\ar[d]\ar[ldd, "Mf"']
\ar[rrru, "\alpha_P"]&&&M'(Q\wedge{}R)
\ar[ld, "M'\pi_{QR}"]\ar[rd, "M'\pi'_{QR}"']&\\
&M(Q\wedge{}R)\ar[ld, "M\pi_{QR}", pos=0.4]
\ar[rrru, "\alpha_{Q\wedge{}R}", crossing over]&&M'Q&&M'R\\
MQ\ar[rrru, "\alpha_Q"', pos=0.8]&&
MR\ar[rrru, "\alpha_R"']\ar[from=luu, "Mg", crossing over, pos=0.6]
\ar[from=lu, "M\pi'_{QR}"', crossing over, pos=0.8]&&&
\end{tikzcd}$$ The proof of naturality with respect to $[f,g]$ is the same.
Next, suppose that $\alpha$ is natural with respect to $f:P\to(\pi^{\vec{A}})^*S$. We then have the following diagram. $$\begin{tikzcd}[column sep=40pt]
&M'((\pi^{\vec{A}})^*S)\ar[rr, "M'\ct"]\ar[from=dd, "M'f", near start]&&M'S\\
M((\pi^{\vec{A}})^*S)\ar[rr, "M\ct", crossing over, near end]
\ar[ru, "\alpha_{\pi^*S}"]\ar[from=dd, "Mf"]&&MS\ar[ru, "\alpha_S"]&\\
&M'(P)\ar[rr, "M'(\ct\cdot\eta^\exists_P)", near start]&&
M'(\exists{}P)\ar[uu, "M'(\mu{}f)"]\\
M(P)\rac[rr, "M(\ct\cdot\eta^\exists_P)"]\ar[ru, "\alpha_P"]
&&M(\exists{}P)\ar[ru, "\alpha_{\exists{}P}"']\ar[uu, "M(\mu{}f)"', crossing over, near start]&
\end{tikzcd}$$ Here, the left face commutes by assumption, the front and back faces commute by the functoriality of $M$ and $M'$, and the top and bottom faces by the naturality of $\alpha$ with respect to cartesian morphisms, shown above. Hence, the right face commutes by the dual of [@sentai Proposition 2.8], and so $\alpha$ is natural with respect to $\mu{}f$.
The proof that $\alpha$ is natural with respect to $\xi{}f$ is the same.
Next, suppose that $\alpha$ is natural with respect to $f:P\wedge{}Q\to{}R$. We want to show that the following square commutes. $$\begin{tikzcd}
MP\ar[r, "Mf^\simm"]\ar[d, "\alpha_P"']&M(Q\To{}R)\ar[d, "\alpha_{Q\To{}R}"]\\
M'P\ar[r, "M'f^\simm"]&M'(Q\To{}R)
\end{tikzcd}$$ By the universal property of $M'(Q\To{}R)$, it suffices to see that the diagram commutes after applying $(-)\wedge{}M'(Q)$ and post-composing with $M'\varepsilon_{QR}:M'(Q\To{}R)\wedge{}M'Q\to{}M'R$. We then have the following diagram, in which each of the sub-diagrams (using the naturality square of $\alpha$ for $\varepsilon_{QR}$, which we have already proved) – except possibly the square on the bottom left – commutes, and the outside of the diagram also commutes (by induction). $$\begin{tikzcd}[column sep=50pt, row sep=30pt]
MP\wedge{}MQ\ar[r, "Mf^\simm\wedge\id"]\ar[d, "\id\wedge\alpha_Q"]
\ar[dd, bend right=60pt, "\alpha_{P\wedge{}Q}"']
\ar[rr, bend left=20pt, "Mf"]
&M(Q\To{}R)\wedge{}MQ\ar[r, "M\varepsilon_{QR}"]\ar[d, "\id\wedge\alpha_Q"']
\ar[dd, bend left=70pt, "\alpha_{(Q\To{}R)\wedge{}Q}"]
&MR\ar[dd, "\alpha_R"]\\
MP\wedge{}M'Q\ar[r, "Mf^\simm\wedge\id"]\ar[d, "\alpha_P\wedge\id"]
&M(Q\To{}R)\wedge{}M'Q\ar[d, "\alpha_{Q\To{}R}\wedge\id"']&\\
M'P\wedge{}M'Q\ar[r, "M'f^\simm\wedge\id"]\ar[rr, bend right=20pt, "M'f"]&
M'(Q\To{}R)\wedge{}M'Q\ar[r, "M'\varepsilon_{QR}"]&M'R
\end{tikzcd}$$ We wish to show that the square on the bottom-left commutes after post-composing with $M'\varepsilon_{QR}$. It suffices to show this after pre-composing with the isomorphism $\id\wedge\alpha_Q:MP\wedge{}MQ\to{}MP\wedge{}M'Q$, and this now follows from a diagram chase.
Finally, suppose $\alpha$ is natural with respect to $f:(\pi^{\vec{A}})^*P\to{}S$. We want to show that the following square commutes. $$\begin{tikzcd}[column sep=40pt]
MP\ar[r, "M(\mu{}f)"]\ar[d, "\alpha_P"']&M(\forall{}S)\ar[d, "\alpha_{\forall{}S}"]\\
M'P\ar[r, "M'(\mu{}f)"]&M'(\forall{}S)
\end{tikzcd}$$ By the universal property of $M'(\forall{}S)$, it suffices to show this after applying $(\pi^{\vec{A}})^*$ and post-composing with $M'\varepsilon^\forall_S:M'(\forall{}S)\to{}S$. This follows from a diagram chase in the following diagram, since the square on the right commutes by the hypothesis and the outside naturality of $\alpha$ with respect $\varepsilon^\forall_{S}$, shown above. $$\begin{tikzcd}[column sep=40pt, baseline=(MpS.base)]
M((\pi^{\vec{A}})^*P)\ar[r, "M(\mu{}f)"]\ar[d, "\alpha_{(\pi^{\vec{A}})^*P}"]&
M((\pi^{\vec{A}})^*\forall{}S)\ar[r, "M\varepsilon^\forall_S"]\ar[d, "\alpha_{(\pi^{\vec{A}})^*\forall{}S}"]&
MS\ar[d, "\alpha_S"]\\
M'((\pi^{\vec{A}})^*P)\ar[r, "M'(\mu{}f)"]&
M'((\pi^{\vec{A}})^*\forall{}S)\ar[r, "M'\varepsilon^\forall_S"]&|[alias=MpS]|M'S
\end{tikzcd}
\tag*{\qed}$$
### {#section-27}
$\Pf_\sigma$ is a free $h^=$-fibration over $\TM_\sigma$. This is precisely the content of Propositions \[prop:syntactic-fib-functor\] and \[prop:syntactic-hfib-nat-trans\].
Lemmas for the proof of freeness {#subsec:freeness-lemmas}
--------------------------------
We now state and prove the Propositions \[prop:longprodf\]-\[prop:longsumf\] which were used in the proof that $\fib{Pf_\sigma}$ is free.
The propositions are all very similar, and there is one for each of the operations of an $h^=$-fibration. In fact, the proofs are so similar that we only give the first one, leaving the remaining ones to the reader.
The statements all have the following form: given any one of the $h^=$-fibration operations – in each case, this is given by a diagram $D$ satisfying some universal property – and a cartesian morphism into each of the objects which are the “inputs” to the operation, we can apply the corresponding operation to the domains of these cartesian morphisms, thus obtaining a diagram $D'$. The claim is then that there is an induced cartesian morphism from the “output” object $P'$ of $D'$ to the “output” object $P$ of $D$. In each case, this is proven by choosing an arbitrary cartesian morphism $\ct$ into $P$, and then using the universal property of $D'$ to conclude that the domain of $\ct$ is isomorphic to $P'$.
Henceforth, let $\fibr{C}CB$ be a fibration.
### {#prop:longprodf}
Suppose we are given – as shown in the solid diagram below – a pullback square in $\B$ with morphisms $f,g,h,k$, a $\prod$-diagram over $k$ based on $Q\in\Ob\fib{C}^C$ which is stable long $g$, a cartesian lift $P\to{}Q$ of $f$, and a $\prod$-diagram over $h$ based on $P$.
There is a unique morphism $q:\prod_hP\to\prod_kQ$ over $g$ for which there exists a (necessarily unique) morphism $r:\widetilde\prod_hP\to\widetilde\prod_kQ$ over $f$ making the whole diagram commute. Moreover, $q$ (and hence also $r$) is cartesian. $$\begin{tikzcd}
\widetilde\prod_hP\ar[d]\ar[r]\car[dr, "r"', dashed]
&\prod_hP\car[dr, "q", dashed]&\\
P\car[rd, "p"']&\widetilde\prod_kQ\ar[d]\ar[r]&\prod_kQ\\
&Q&\\[-15pt]
A\ar[r, "h"]\ar[rd, "f"']&B\ar[rd, "g"]&\\
&C\ar[r, "k"']&D
\end{tikzcd}$$ By the stability along $g$ of the $\prod$-diagram based on $Q$, we obtain a commutative solid diagram $$\begin{tikzcd}[column sep=5pt]
\widetilde\prod_hP\ar[rdd]\ar[rr]
\ar[dr, "v", dashed, shorten >=-6pt, shorten <=-3pt, "\sim"'{sloped, pos=1.4}]&[-5pt]&
\prod_hP\ar[dr, "u", "\sim"'{sloped, near end}, dashed, shorten <=-2pt, shorten >=-2pt]
&[-10pt]&\\[-15pt]
{}&
f^*\widetilde\prod_kQ\ar[rd, "\ct"]\ar[rr, "s"]\ar[d, "t"']&&g^*\prod_kQ\ar[rd, "\ct"]&\\
&P\ar[rd, "\ct"]&\widetilde\prod_kQ\ar[d]\ar[rr]&&\prod_kQ\\
&&Q&
\end{tikzcd}$$ with $(s,t)$ a $\prod_h$-diagram. By the universal property of the $\prod$-diagrams based on $P$, there is a unique isomorphism $u:\prod_hP\to{}g^*\prod_kQ$ over $B$ for which there exists a morphism $v$ over $A$ as shown making the diagram commute (and which is hence an isomorphism). The claim follows.
### {#prop:longprod}
Suppose we are given – as shown in the solid diagram below – a morphism $f:A\to{}B$ in $\C$, objects $P',Q'\in\Ob\fib{C}^B$, a product diagram based on $P'$ and $Q'$ in $\fib{C}^B$ which is stable along $f$, cartesian lifts $p:P\to{}P'$ and $q:Q\to{}Q'$ of $f$, and a product diagram on $P$ and $Q$ in $\fib{C}^A$. $$\begin{tikzcd}
&[-20pt]P\wedge{}Q\ar[ldd]\ar[rd]\car[rrr, dashed, "r"]&[-20pt]&
&[-20pt]P'\wedge{}Q'\ar[rd]&[-20pt]\\
&&Q\ar[rrr, "q", near start, "\carsym"{anchor=center, pos=0.4}]&&&Q'\\
P\car[rrr, "p"]&&&P'\ar[from=uur, crossing over]&&\\
&A\ar[rrr, "f"]&&&B&
\end{tikzcd}$$ There is a unique morphism $r$ making the diagram commute. Moreover, $r$ is cartesian.
### {#prop:longcoprod}
The previous proposition holds with “product” replaced by “coproduct” (and with the direction of all morphisms lying over $A$ and $B$ reversed.
### {#prop:longexp}
Suppose we are given – as shown in the solid diagram below – a morphism $f:A\to{}B$ in $\C$, objects $P',Q'\in\Ob\fib{C}^B$, an exponential diagram in $\fib{C}^B$ based on $P'$ and $Q'$ which is stable along $f$, cartesian lifts $p:P\to{}P'$ and $q:Q\to{}Q'$ of $f$, and an exponential diagram in $\fib{C}^A$ based on $P$ and $Q$. $$\begin{tikzcd}[column sep=0pt]
&[-20pt]&[-15pt]Q\car[rrr, "q"]&&[-23pt]&[-15pt]Q'\\
&(P\To{}Q)\wedge{}P\ar[ldd]\ar[rd]\ar[ru]\car[rrr, "s", dashed]
&&&(P'\To{Q'})\wedge{}P'\ar[rd]\ar[ru]&\\
&&P\To{}Q\car[rrr, "r", near end, dashed]&&&P'\To{}Q'\\
P\car[rrr, "p"]&&&P'\ar[from=ruu, crossing over]&&\\
&A\ar[rrr, "f"]&&&B&
\end{tikzcd}$$ There exists a unique $r$ such that there is a (by Proposition \[prop:longprod\] necessarily unique, and cartesian) $s$ making the whole diagram commute. Moreover, $r$ is cartesian.
### {#prop:longsumf}
Suppose we are given – as shown in the solid diagram below – a pullback square in $\C$ with morphisms $f,g,h,k$, a cocartesian morphism over $k$ with domain $Q$ which is stable along $g$, a cartesian lift $P\to{}Q$ of $f$, and a cocartesian morphism over $h$ with domain $P$. $$\begin{tikzcd}
P\ar[r]\car[dr, "p"']&\sum_hP\car[dr, "q", dashed]&\\
&Q\ar[r]&\sum_kQ\\
A\ar[r, "h"]\ar[rd, "f"']&B\ar[rd, "g"]&\\
&C\ar[r, "k"']&D
\end{tikzcd}$$ There is a unique morphism $q:\sum_hP\to\sum_kQ$ over $g$ making the whole diagram commute. Moreover, $q$ is cartesian.
[^1]: Of course, assuming the law of the excluded middle, $\Omega$ is just $\{\emptyset,\tm\}$, but in general, it needn’t be.
[^2]: We are using “non-standard” in a somewhat non-standard way.
[^3]: I.e., “$X$ is inhabited”. Assuming the law of the excluded middle, this is equivalent to $X\ne\emptyset$.
[^4]: This is a drastic oversimplification. Though we can reproduce all the *logical* operations, we still do not have the set $\Omega$ itself; hence, for example, we cannot express the induction principle as stated in (\[eq:ind-princ-omega\]) above. It is a non-trivial empirical fact that one can still reproduce mathematics in this a priori restricted context – though this does almost immediately require the introduction of *universe* types, which to some extent serve as a substitute for $\Omega$.
[^5]: A point of clarification is needed here. In the type theory from the previous section, one does not have the membership relation $x\in{}X$. However, each term *has an associated type*, determined uniquely by the term’s syntactic structure, and we sometimes use $t:X$ to signify that $t$ is a term of type $X$. It may seem that the situation is the same in dependent type theory. However, in the latter, the same term can have two different types (in case these types are *equal*). What is still true is that the relation $t:X$ is *decidable* – given any two expressions $t$ and $X$, one can automatically determine whether $t:X$ is provable. Hence, given the type $X$, the only creative effort involved in showing $X$ is inhabited is the construction of a term $t$ such that $t:X$.
[^6]: This interpretation of logic in a locally cartesian closed category has also been considered E. Palmgren [@palmgren-bhk].
[^7]: Actually, this will not hold without further (fairly mild) assumptions; for example, that the topological spaces are homotopy-equivalent to CW-complexes, or that the simplicial sets are Kan complexes.
[^8]: More specifically, we use the “first-order” variant of the notion of hyperdoctrine. These are simply called “hyperdoctrines” in [@seely-hyperdocrtines]. The special case of the notion in which the fibers are preorders is called “first-order hyperdoctrines” in [@pitts-note-on-catlog] and “first-order fibrations” in [@jacobscatlogic]. In [@makkai-lauchli1], they are called “$h$-fibrations” (short for “Heyting fibration”), which is also the name we use. We also use the corresponding name $h^=$-fibration for the version with equality, though note that in this case our definition (Definition \[defn:h-fibration\]) differs from that in [@makkai-lauchli1], as we demand only finite products and not all finite limits in the base category.
[^9]: The need to restrict to isomorphisms comes from the fact that not every morphism of structures is an “elementary embedding” – i.e., preserves all logic formulas.
[^10]: We note that, strictly speaking, the semantics given in the fibrational formulation can only be said to agree up to homotopy-equivalence (i.e., isomorphism in $\Ho(\C_\fb)^\to$) with the semantics as described in §\[subsec:htpical-semantics-direct\].
[^11]: This proposition and its converse, Proposition \[prop:frobexp-converse\], are well-known and appear (in a different form) in [@lawvereequality p. 6].
[^12]: This proposition and its converse, Proposition \[prop:beck-chev-otherswap\], are well-known and are mentioned in [@makkai-lauchli1 p. 343] and [@jacobscatlogic Lemma 1.9.7].
[^13]: This notation is somewhat misleading, since the domains of the morphisms in $(\D^\to)_*$ and $\Ho(\D^\to)_*$ are not required to be in $\D$; however, whenever we use this notation (such as in $(\C_\fb^\to)_\fb$ and $(\C_\cfb^\to)_\cfb$), that will happen to be the case.
[^14]: These have also considered by M. Lambert in [@lamberthesis Definition 2.2.15]. We learned the concept from M. Makkai.
[^15]: It would be more natural, perhaps, to demand that this is only an equivalence, and not an isomorphism, but the free f.p. category we construct in the appendix has this stronger property, and it is convenient to assume it.
[^16]: The reader may wonder if this is just a functor $F$ from $\OTM^\op$ to the category of sets and partial functions. It is not, because the domain of the composite $Fg\cdot{}Ff$ is in general smaller than that of $F(g\cdot{}f)$. However – if one is so inclined – it can be described as a *lax pseudo-functor* into the poset-enriched category of sets and partial functions.
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2016 Bills
SB 294 ensures that returning veterans know of their right to state pension credit for their periods of active military service.
SB 563 will increase transparency and accountability within the workers’ compensation review process by ensuring that there are no inappropriate incentives for doctors to modify, delay, or deny requests for medically necessary services.
SB 877 will require the State Department of Public Health to collect data on violent deaths so the state can better understand and prevent fatal violence in our communities.
SB 947 removes barriers for needy families to participate in California Work Opportunity and Responsibility to Kids program (CalWORKs) by allowing them to be interviewed on the phone or through electronic means in lieu of an in-person interview.
SB 1058 ensures that hospital police supervisors receive salary and benefits that are at least generally the economic equivalent to the salary and benefits granted to the employees they supervise—making it easier for state hospitals to recruit and retain qualified employees.
SB 1095 makes sure that all newborn babies are tested for rare diseases that would otherwise go undiagnosed and untreated by making sure that California follows the federal Recommended Uniform Screening Panel’s recommendations. This bill would save lives and allow for cost-effective medical treatment.
SB 1189 would require all autopsies to be conducted by a forensic pathologist. Additionally, if law enforcement personnel was directly involved with the care and custody of an individual who died in their care, then they will not be allowed inside the autopsy room during the performance of the autopsy.
SB 1238 would allow health care providers in state correctional facilities to publish and learn from medical data in order to provide higher quality health care for the more than 127,000 inmates they serve.
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WASHINGTON — Hillary Clinton declined to say Sunday whether she believes in a constitutional right to bear arms, possibly opening the door to a fresh round of attacks from Donald Trump, who has already accused the likely Democratic presidential nominee of wanting to "abolish" the Second Amendment.
In an interview on ABC's "This Week," Clinton deflected twice when asked whether she agrees with the Supreme Court's interpretation of the Second Amendment. The court ruled in 2008 that the Constitution affords private citizens the right to keep firearms in their homes and that such possession need not be connected to military service.
The wording of the Second Amendment has long made the extent of gun ownership rights a point of contention:
"A well regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed."
Questioned by George Stephanopoulos about her view of the amendment, Clinton talked about a "nuanced reading" and emphasized her belief in the rights of local, state and federal governments to regulate gun ownership. Stephanopoulos, formerly a top aide to President Bill Clinton, wasn't satisfied by the response.
"That's not what I asked," he replied.
Clinton then discussed the right to own a gun as a hypothetical. "If it is a constitutional right," she began her next answer, "then it - like every other constitutional right - is subject to reasonable regulations."
Here's the full exchange:
STEPHANOPOULOS: Let's talk about the Second Amendment. As you know, Donald Trump has also been out on the stump talking about the Second Amendment and saying you want to abolish the Second Amendment. I know you reject that. But I want to ask you a specific question: Do you believe that an individual's right to bear arms is a constitutional right - that it's not linked to service in a militia?
CLINTON: I think that for most of our history there was a nuanced reading of the Second Amendment until the decision by the late Justice (Antonin) Scalia. And there was no argument until then that localities and states and the federal government had a right - as we do with every amendment - to impose reasonable regulations. So I believe we can have common-sense gun safety measures consistent with the Second Amendment. And, in fact, what I have proposed is supported by 90 percent of the American people and more than 75 percent of responsible gun owners. So that is exactly what I think is constitutionally permissible and, once again, you have Donald Trump just making outright fabrications, accusing me of something that is absolutely untrue. But I'm going to continue to speak out for comprehensive background checks; closing the gun show loophole; closing the online loophole; closing the so-called Charleston loophole; reversing the bill that Sen. (Bernie) Sanders voted for and I voted against, giving immunity from liability to gun makers and sellers. I think all of that can and should be done, and it is, in my view, consistent with the Constitution.
STEPHANOPOULOS: And, and the Heller decision also says there can be some restrictions. But that's not what I asked. I said, do you believe their conclusion that the right to bear arms is a constitutional right?
CLINTON: If it is a constitutional right, then it - like every other constitutional right - is subject to reasonable regulations. And what people have done with that decision is to take it as far as they possible can and reject what has been our history from the very beginning of the republic, where some of the earliest laws that were passed were about firearms. So I think it's important to recognize that reasonable people can say, as I do, responsible gun owners have a right. I have no objection to that. But the rest of the American public has a right to require certain kinds of regulatory, responsible actions to protect everyone else. | https://www.chicagotribune.com/nation-world/ct-hillary-clinton-second-amendment-20160605-story.html |
From the 2020 NSCA Coaches Conference, Ashley Jones, Strength and Conditioning Coach for the Houston SaberCats Major League Rugby team, presents a high-energy hands-on presentation highlighting fun warm-up games coaches can implement with their athletes.
Tex McQuilkin, Director of Training for Power Athlete HQ, defines athleticism as a trainable performance variable at the 2019 Coaches Conference. McQuilkin illustrates the four phases of the competitive lifecycle for sport athletes and empowers coaches with strategies to best apply progressive ov...
Learn a nine-step plan for opening your own gym. In this session from the NSCA’s 2018 Personal Trainers Conference, David Crump shares his experience owning a facility.
Rob Orr, co-lead of Bond University’s Tactical Research Unit, explains the differences in load carriage requirements and contexts between different tactical forces, and how to prepare tactical personnel for load-carriage tasks. This session from the NSCA’s 2018 Tactical Annual Training also looks... | https://www.nsca.tv/featured-carousel |
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140 Ill. App.3d 230 (1986)
488 N.E.2d 610
AUGUST TRIPI, d/b/a A.J. Products Company, Plaintiff-Appellee,
v.
SIGURD LANDON, Defendant-Appellant.
No. 84-3013.
Illinois Appellate Court First District (5th Division).
Opinion filed January 10, 1986.
*231 Paul E. Peldyak, of Chicago, for appellant.
John H. Anderson, of Oak Lawn, for appellee.
Judgment affirmed.
JUSTICE LORENZ delivered the opinion of the court:
Defendant Sigurd Landon appeals from a judgment order against him for $4,057.22 and costs obtained by plaintiff August Tripi following a bench trial on plaintiff's suit to recover the cost of goods sold to the defendant. On appeal defendant contends (1) the action should have been barred by the Statute of Frauds; (2) defendant's motion for summary judgment should have been granted; (3) the evidence did not support the judgment against him, and (4) the default judgment obtained against a codefendant, Earl Gabrielson, was contradictory.
We affirm.
The following pertinent testimony was adduced at trial.
The plaintiff, August Tripi, testified that in his 20-year-old home-based business, A.J. Products Company, he sold parts to car dealers and trucking companies. For the past nine years he had been doing business with the defendant, known to him only as Mr. Landon. Defendant's business, a cartage company, was known to plaintiff as F. Landon Cartage, and subsequently as F. Landon Leasing. Plaintiff testified that he was never advised that he was dealing with a corporation. The defendant's trucks only bore the name "F. Landon Cartage."
According to the plaintiff the defendant's business was initially at 1000 West Monroe in Chicago. At that location, the defendant had told him he should obtain orders from defendant's head mechanic, known to plaintiff as Gene. Plaintiff followed this practice there. When defendant moved his business to 2114 South May Street, defendant told the plaintiff that orders would be placed by Earl Gabrielson, apparently defendant's new head mechanic. In October and November of 1982 the plaintiff made deliveries of three sets of goods ordered by Earl Gabrielson. Plaintiff personally delivered these goods to defendant's May Street business, where he observed trucks with "F. Landon Cartage" on them. Plaintiff denied that defendant ever instructed him not to accept orders from Gabrielson.
Plaintiff mailed invoices totalling $4,057.22 to the defendant for those three orders, but defendant never paid them. In December 1982 he telephoned the defendant and asked for a meeting so he could obtain payment. Defendant gave him a time and place where they could "work out the money situation" but then never appeared at the time *232 specified. In that telephone conversation the defendant did not deny receiving the goods, nor did he say he would not pay for them.
Louis Bajick testified that between May and November of 1982 he had assisted the plaintiff with deliveries to defendant's business. There the only names he had seen on trucks were "F. Landon" and "F. Landon Cartage."
Defendant testified that he was retired at the time of trial but had been an officer and director of five Landon corporations: F. Landon Cartage Company, F. Landon Trucking Company, Landon Truck Leasing Company (subsequently renamed Landon Interstate Limited), and Landon Truck Leasing Limited, a Wisconsin corporation. Defendant testified that F. Landon was his grandfather. He denied ever hearing of "F. Landon Cartage" or "F. Landon Leasing."
Defendant denied dealing with the plaintiff in anything but his capacity as a corporate agent. He also denied ever operating a trucking company as an individual or a sole proprietor.
Defendant testified that he was unaware of any communications from the plaintiff from May to December of 1982. The Landon businesses ceased operating at the May Street location in April 1982 because of a union strike. They vacated the premises on May 30, 1982, and never returned to that location.
Defendant testified further that Earl Gabrielson, although on the payroll, was not considered an employee for "paperwork purposes." Gabrielson would supply them with tools and would repair their trucks, receiving payment based on invoices he submitted. Defendant stated that Gabrielson had no authority to order parts or equipment on behalf of the Landon businesses. However, he also admitted that when necessary Gabrielson would be given a check and would then buy parts for them. He further conceded that plaintiff might have been paid in the past by the Landon businesses for items purchased by Gabrielson. Defendant also stated that in April 1982 he told the plaintiff that Gabrielson had no authority to place orders, and plaintiff would need written authority from the defendant for any further orders.
Defendant recalled that Earl Gabrielson ceased working for the Landon businesses after May 1982. The defendant's last conversation with the plaintiff had been in April 1982, when they settled a billing dispute for $1,000, paid with a check drawn on Landon Truck Leasing, Ltd., but signed by Sigurd Landon without any corporate title or designation by his name.
Defendant denied that he or the Landon businesses had ever received any invoices from the plaintiff for goods delivered in October *233 or November of 1982. He also denied having a phone conversation with the plaintiff concerning these bills.
Lori Callahan, formerly a dispatcher with F. Landon Cartage Company, testified that the May Street offices were closed in May 1982. She did not recall seeing any invoices from the plaintiff. However, she also stated that it was not her duty to pick up or open the mail and she had only received some of the company mail.
Joseph Powell, an operations manager for Jansen Transfer, formerly located at the same May Street address, testified that the only Landon business he was aware of there was Landon Cartage. This was also the only name he saw on the equipment. Powell testified that after the Landon company moved out he did not see any of their equipment or employees. However Earl Gabrielson continued to work there as a self-employed truck repairer for eight months.
A former Landon truck driver, Phil Santore, testified that he completed moving Landon's equipment out of the May Street address on the last day of May 1982. He did not know if any Landon trucks remained there but he did testify that spare parts, tools, and truck repairing equipment was left there.
The court in its findings clearly indicated that it found the testimony of the defendant and his witnesses to be incredible. It found for the plaintiff, entering judgment in the amount of $4,057.22 plus costs against the defendant, and on the basis of a default judgment against the other named defendants, Sigurd Landon, d/b/a Landon Interstate, Ltd., an Illinois corporation, Sigurd Landon, d/b/a Landon Leasing, and Earl Gabrielson. All those judgments were entered jointly and severally against all the defendants.
OPINION
1 On appeal, defendant first contends that this action should have been barred by the Statute of Frauds. Although defendant initially raised the issue in his answer to plaintiff's amended complaint, he subsequently filed superceding answers to plaintiff's second amended complaint, omitting any mention of the Statute of Frauds. No subsequent attempt was made in the trial court to raise the issue or to specifically adduce evidence concerning it and defendant has clearly waived the issue. O'Hare International Bank v. Feddeler (1973), 16 Ill. App.3d 35, 305 N.E.2d 325.
2 In his briefs before this court, defendant has presented over five pages of argument concerning the propriety of the trial court's denial of his motion for summary judgment. Although the plaintiff has chosen to contest this issue on its merits, we would only note that a *234 denial of summary judgment is not appealable, that determination being deemed to merge into the trial itself. Home Indemnity Co. v. Reynolds & Co. (1963), 38 Ill. App.2d 358, 187 N.E.2d 274; Banwart v. Okesson (1980), 83 Ill. App.3d 222, 403 N.E.2d 1234.
3 We next consider defendant's contentions concerning the trial court's findings at the close of trial. Defendant asserts that the court improperly found him liable based upon an alter ego theory of liability never advanced by the plaintiff in his pleadings. We find that defendant has misconstrued the trial court's finding. The court expressly held that in this cause defendant held himself out and did business as Landon Cartage, no corporate entity. Thus the court found that defendant was doing business as an individual when he ordered goods through his agent from the plaintiff. We therefore need not consider the sufficiency of the evidence to support a corporate alter ego theory of liability. We also need not consider whether the evidence below sufficed to establish the existence of the various corporations of which defendant claimed to be an officer.
The trial court found that the defendant, doing business as Landon Cartage, an unincorporated entity, had, through his agent, Earl Gabrielson, ordered and obtained delivery of three shipments of goods from the plaintiff. We must determine whether these findings were contrary to the manifest weight of the evidence. (Frankenthal v. Grand Trunk Western R.R. Co. (1983), 120 Ill. App.3d 409, 458 N.E.2d 530.) The plaintiff testified that in nine years of doing business with the defendant, known to him only as Mr. Landon, he had never been advised by the defendant or anyone else that he was dealing with a corporation. The only business name he saw on defendant's truck was "F. Landon Cartage." The defendant had instructed the plaintiff that orders would be placed by defendant's head mechanic, Earl Gabrielson. In October and November plaintiff delivered three orders to defendant's place of business pursuant to orders placed by Earl Gabrielson. On all three occasions there were trucks present with the name "F. Landon Cartage" on them. Plaintiff denied that defendant ever told him the business had moved or that he should no longer accept orders from Earl Gabrielson. Although the plaintiff sent invoices to the defendant, he was never paid for the merchandise.
The defendant denied ever hearing of a business called F. Landon Cartage, although his own witness, an employee of another business on the premises, testified that the only name he saw on the equipment was "Landon Cartage." Defendant denied that Earl Gabrielson had authority to order parts for the Landon Corporation. However, defendant also testified that when necessary Gabrielson would be *235 given money to purchase parts. Defendant further testified that in April 1982 he told the plaintiff that Gabrielson had no authority to place orders, thus apparently countermanding authority he claimed never to have given. As we have noted, plaintiff denied ever being told this.
4 The existence of a principal-agent relationship between the defendant and Earl Gabrielson was a question for the court as fact-finder to determine. (Swartzberg v. Dresner (1982), 107 Ill. App.3d 318, 437 N.E.2d 860.) The trial court clearly resolved this and other credibility questions arising from the testimony in favor of the plaintiff. We do not find the court's determination to have been palpably erroneous or wholly unwarranted and we will not disturb that determination on appeal. Kern v. Uregas Service of West Frankfort, Inc. (1979), 90 Ill. App.3d 182, 412 N.E.2d 1037.
5 Defendant's final contention is that there is an inherent contradiction in the court's finding that defendant and Earl Gabrielson were jointly and severally liable to the plaintiff. The judgments against all the named defendants except Sigurd Landon were by default. None of those defaulted parties has appealed from that order, and we find no basis for considering an attack by the defendant Sigurd Landon on the judgments against his codefendants.
The judgment of the trial court is affirmed.
Affirmed.
SULLIVAN, P.J., and PINCHAM, J., concur.
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What we saw while driving northwest along US-90 towards Seminole Canyon State Historical Park was flat wilderness – all the way to the horizon in every direction. I’ve heard a lot about boring drives through Texas, and I believe this is one of those parts of the state that people refer to.
Seminole Canyon State Historical Park is located just off US-90, at the junction of the Pecos and Rio Grande rivers. So of course that puts it right on the border with Mexico. The rugged 2,172-acre park features some impressive deep canyons with rocky terrain and sparse desert vegetation. And hey, there are lots of birds here, too 🙂
With a week to explore under sunny skies, we were both pumped up and ready to go exploring every morning. During our stay we racked up 29 miles of walking and hiking throughout the park, completing every trail – and some of them several times. We finally got our leg muscles back in shape!
The trails here were rocky but not difficult, with only a 210-ft maximum elevation gain. We found that the trail listed as “strenuous” was really only “moderate” by our standards, but perhaps they were considering the condition of parts of the trail more than physical difficulty.
Parts of some trails were a bit boring, but coming upon the rim of the deep canyons and the river that ran through parts of them took our breath away. Fortunately, it was springtime and some sparse wildflowers could be seen blooming along the trails. That stopped me in my tracks for a few shots of the beauties.
We were excited to see a couple of Javelinas walk across a trail in front of us! They were too quick for us to get a picture, so Steve ran through the scrub brush with the camera to pursue them. He finally gave up, with nothing to show for his efforts but a bunch of cactus scratches on his legs that are still healing.
A point of interest along one of the trails was the Panther Cave Overlook. It lies at the confluence of Seminole Canyon and the Rio Grande. Panther Cave is a rock shelter used by the Desert Archaic culture between about 1,300-8,900 years ago. Visible across the canyon from the cave was an immense pictograph panel that spanned the back wall of the rock shelter, and it included a panther image nine feet long. Access to this cave shelter is by private boat only, and tours had been canceled due to shallow waters.
I zoomed into the cave from across the canyon, and the Panther was visible on the right side of the wall. The plaque at the overlook indicated that some archeologists believe the Desert Archaic people were depicting the shaman’s journey to the spirit world. Caves and rock shelters like this one served as sacred portals or passageways for the shamans, and the panther represented an animal tutelary or guardian that protected them.
We were interested in viewing some of the ancient rock art, and this park’s focus is on the Fate Bell Shelter, the largest rock shelter in the region. In many of the canyons, erosion over millions of years has carved massive rock overhangs that were used by prehistoric Indians for shelter. Hiking in the canyons and viewing rock pictographs can only be done on guided tours. We joined two of the tours to see some prehistoric rock art, and to walk on the floor of the canyons.
The Fate Bell Tour involved a fairly rugged walk to the bottom of the canyon, then over to the huge cliff overhang containing many good examples of Pecos river style pictographs.
Radiocarbon dating suggests these pictographs were created between 2,950 and 4,200 years ago. Because they are so old, experts know very little about the people who created them. The walls were once densely painted, but only isolated panels have been able to resist the effects of time and indiscriminant looting of the site before its acquisition by the state.
Our tour guide explained that colors were made with pigments from local stones such as hematite (red ochre) for red, limonite (yellow ochre) for yellow, manganese oxide for black and calcite (or gypsum) for white. These rocks were ground into a fine powder and mixed with a binder (probably animal fat) to make the pigment stick together and to the wall. The soapy juice of the yucca plant root, mixed with water, may have been used to thin the pigment and fat mixture into a smooth paint that has held up for thousands of years. Is that amazing or what?
One of the guided tours we joined was the Upper Canyon backcountry hike, which included visits to a variety of sites that had prehistoric and historic rock art styles. The latter of these were created by folks who built the railroad in the 1880’s that ran through this area.
Did I forget to mention that this park is also another Texas birding trail? The Pyrrhuloxias shown below, along with many Cactus Wrens, alternately sang to us during our walks. And the Canyon Towhees and Black-throated Sparrows helped the beautiful Cardinals empty our feeder every day.
The birdies all knew there was free food at site #1, and we spent many hours sitting by our campfire and watching the excellent variety of birds competing for the goodies.
A sunset with bird silhouettes capped a wonderful week full of outdoor fun. The sun was finally shining on us!
All of the outdoor activities made me forget my strange medical issue. I want to express my appreciation to everyone for sending me your warm thoughts and messages of concern. Just the thought that all of you are thinking of me makes me feel better. I will soon find out what my new doctor has to say!
Next up: Visiting FABULOUS Big Bend National Park! | https://lowestravels.com/2015/03/21/a-journey-back-in-time-seminole-canyon-sp-comstock-tx/ |
[Alzheimer's disease and depression].
Alzheimer's disease is the most frequent cause of dementia (60% of all dementias) and affects nearly 300,000 people in France. Alzheimer's disease is a disease of the elderly which generally begins after 60 years and whose prevalence increases markedly after age 75 years. The elderly population is increasing in all Western countries. Alzheimer's disease thus constitutes a veritable emergent public health problem. The rapid inflation of the epidemiological and etiopathogenetic data have contributed to enhanced nosographic definition and finer semiological characterization of the disease. Thus, the classic concept of senile dementia has been totally abandoned. In contrast, the concept of depressive pseudodementia as defined by Kiloh (1961) remains present in the "psychiatric culture". The concept refers to rare clinical situations in which the controversial concept of "test therapy" with antidepressants retains, in the author's opinion, some utility. Depressive or psychobehavioral signs and symptoms frequently inaugurate Alzheimer's disease giving rise to first-line psychiatric management. The use of multidimensional evaluation instruments such as the neuropsychiatric inventory (NPI) has enabled demonstration of the signs and symptoms and their quantification through the course of the disease. In the dementia stage, the psychobehavioral symptoms are related to the patient's awareness of the degradation in his intellectual functions and the loss of independence and to specific neuropathological lesions responsible for "frontal deafferentation". Certain clinical forms of depression of late onset are also characterized by symptoms reflecting hypofrontal signs (blunted affect, apathy, defective initiative, etc.) and severe cognitive disorders. Those depressions are associated with risk factors shared with Alzheimer's disease (sex, age, vascular function, APOE 4) and constitute a risk factor for progression to dementia, requiring regular clinical and neuropsychological follow-up. Now that we are entering the era of therapy for Alzheimer's disease, the psychiatrist must contribute to the collective effort of early diagnosis and treatment. In close collaboration with all the medical and social professionals involved, the psychiatrist has a fundamental role throughout the disease, towards the patient but also in providing support and psychological assistance for caregivers.
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Dean Stephen Barker presents a stimulating lecture on the citizen artist, what it means, and why it is essential in the current social climate. He will be joined for a roundtable discussion with Dean Miller from humanities and Dean Maurer from social sciences.
Free admission, reservation required
More about the subject matter:
BUILDING THE CITIZEN-ARTIST
Stephen Barker, Dean, UCI Claire Trevor School of the Arts
What does citizenship have to do with being an artist? What, in the end, does it mean to be an “artist?” To be a particular part of—even a central part of, the creation and development of culture? And there’s the key: the artist’s chief role is that, precisely, of “citizen”? Issues like these underlie all of the work we, as faculty and students, do in the Claire Trevor School of the Arts, across various arts disciplines and, more generally, as researchers, through the arts, into the complexities of culture-building. The arts are often (even in a research university) not considered to be part of, let alone central to, this process of citizenship—what I’m calling “culture-building.” But the core task of the Claire Trevor School is precisely “culture-building,” through diligent, systematic inquiry and public activity aimed at the exploration of theories and applications of the creative process to the evolution of the culture—local, national, global—of which we are part. We focus on Building the Citizen-Artist.
The fundamental materials needed for building “the citizen” and “the artist” are actually quite straightforward, and are nearly identical: both require two things: specific skills, and cultural grounding. The cliché is that the ideal artist requires what we call “talent” (which the Greeks called daimon and the Romans genius), but in fact everyone has talent, which is simply aptitude or “natural ability.” The artist becomes an artist when she employs that aptitude to the passion required to create art objects. Ideally the artist manifests special skills, but for the last century or so even skills are optional (as most television clearly shows).
The ideal citizen, on the other hand, requires a very special skill: literacy, that is, learned critical thinking skills developed through reading, writing, and substantive exchange. “Skill” is thus central to both the artist and the citizen—indeed “skill” is endemic to “art” itself: the word “art” derives from the Latin ars, which derives from the Greek tekhnē, which literally means “skill.”
And both build the culture of which they are a part: the citizen “builds” civic structures; the artist “builds” artworks. And both are performative: the arts, in all their culture’s “mirror” (showing culture what it is, for better and for worse) and as its “provoker” (through critique, showing it where it might go), The artist, like the citizen, articulates, evolves, and disseminates cultural values. To be a citizen or an artist is to be engaged; this is both formative and trans-formative, dynamic forces in the process of moving culture to its next configuration.
At the most fundamental level, it is only through this critique that we can develop fully as citizen-artists. This double task is the heart of the citizen-artist’s task. It begins with raw data from the world, in the form of images and language which, when received becomes information; Information, when it is focused, becomes knowledge. And knowledge, when it results in (and from) critical thinking, becomes hypothesis (a Greek word for opinion). Opinion is nothing more nor less than a well-constructed, informed (or ill-informed) world-view. Critical thinking distinguishes between the two—between data and information, on the one hand, and knowledge on the other; informed opinion requires engagement in critical thought. This super-structure is ubiquitous, repeated across all subjects and genres, for both the citizen and the artist. We’ll explore a few of them.
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While Pakistan has experienced rapid economic growth and substantial poverty reduction in recent years, social indicators remain fragile and the country still faces challenges in good governance. Our programs foster greater participation of all citizens, with a focus on women and economic opportunities, to ensure that the benefits of Pakistan’s economic prosperity are expanded and broadly shared. We work with local partners to strengthen the democratic political process, promote inclusive participation, and build the capacity of local institutions to meet the needs of citizens and sustain the gains of economic and social development.
Sofia Shakil,
Country Representative
Contact
The Asia Foundation – Pakistan
P.O. Box 1165
Islamabad, Pakistan
Tel: +92 (51) 235-6007
Email: [email protected]
Strengthening networks to reduce violence against vulnerable groups
This project promotes inclusion of vulnerable communities (like religious minorities) in Pakistan. A preemptive mechanism was developed to ensure an effective government response to include vulnerable populations. Under this initiative, we engaged with local government departments, law enforcement agencies, civil society organizations and community networks. A key element is establishing a Management Information System to document information on exclusion of vulnerable communities in project districts. It acts as a central hub to mobilize emergency response units, including police and rescue services. To cultivate a culture of tolerance and peace among youth, the Foundation also partnered with public sector universities to establish Peace Incubation Centers.
Sustainable Development Investment Portfolio
To support climate resilient livelihoods and inclusive economic growth in South Asia, The Asia Foundation is addressing growing water, food, and energy insecurity. As part of the Australian Government’s Sustainable Development Investment Portfolio we are improving integrated management of water, energy and food while addressing gender and climate change impacts in these ways: establishing a “Water and Environment Research Corner” at Karakoram International University, creating four pilot projects in water and food security in diverse topographies in Pakistan, a water, energy and food nexus framework based on Pakistan’s government policies, and advocacy and bottom-up platforms for citizenry to access policymakers.
NEWS FROM PAKISTAN
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Resilience in the Time of Covid
March 31, 2021
Blog
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The Asia Foundation Releases Study on Covid-19 & Women in the Economy in South Asia
March 19, 2021
News
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Covid-19 & The New Normal for Women in the Economy in Pakistan
March 8, 2021
Publication
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Covid-19 & The New Normal for Women in the Economy in South Asia
March 8, 2021
Publication
Let's Read Asia
Let’s Read, the digital platform of our Books for Asia program, was introduced in Pakistan in 2019, and houses children’s stories in more than 40 international languages. More than 400 have been translated into Urdu for wider outreach in Pakistan. Let’s Read promotes literacy and encourages reading habits among children, and serves as a creative forum where children can learn about different cultures and ideas through books written by authors from different parts of the world. | https://asiafoundation.org/where-we-work/pakistan/ |
Lazarus, I. J.
MetadataShow full item record
Abstract
Linear electrostatic waves in a magnetized four-component, two- temperature electron–positron plasma are investigated, with the hot species having the Boltzmann density distribution and the dynamics of cooler species governed by fluid equations with finite temperatures. A linear dispersion relation for electrostatic waves is derived for the model and analyzed for different wave modes. Analysis of the dispersion relation for perpendicular wave propagation yields a cyclotron mode with contributions from both cooler and hot species, which in the absence of hot species goes over to the upper hybrid mode of cooler species. For parallel propagation, both electron-acoustic and electron plasma modes are obtained, whereas for a single-temperature electron–positron plasma, only electron plasma mode can exist. Dispersion characteristics of these modes at different propagation angles are studied numerically. | https://ir.dut.ac.za/handle/10321/849 |
Blue Geometric Wallpaper
Blue geometric wallpaper with an abstract flower pattern in blue and orange on a white background. Small damage on some parts because the paint sticks to the paper, very minor and not over the total length just some spots.
€75.00 / roll (Excl. VAT ) 40 rolls left
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How Did The Endless Mission Come to Be?>>May 17, 2018
Developed through a collaboration between E-Line Media and Endless Interactive, The Endless Mission is more than a creation-style sandbox game. It’s a hero’s journey into the world of game creation.
You’ve seen the trailer. You’ve read a bit about the game. You might’ve played it at PAX East or EGX Rezzed. But how did The Endless Mission come to be?
The Origin of The Endless Mission
The idea for The Endless Mission stemmed from Endless OS principal Matt Dalio’s desire to leverage the power of video games to support some of the impact goals he had with his business. When asking his engineers how they learned to code, he was fascinated to learn that their answer was almost always the same: “I learned that I could hack my games and it was more fun to hack them than to play them.” The idea of modding games as a driving force in learning to code was an enticing one. Matt recalls obsessing over world-building games like SimCity and Civilization growing up, as well as powerful tools like HyperCard, where he cut his teeth on creating his own technology. At the same time, Matt was also diving into popular fantasy and technology stories such as Ender’s Game, Harry Potter, and The Diamond Age, enamored with the worlds they built and the protagonists inhabiting them. The vision for The Endless Mission was starting to come together.
These past experiences began informing a dream where he imagined that everyone has learned to code – learned to shape the technology around them, instead of being shaped by it. Years later that dream sparked another question: What if creating and manipulating code within the Endless operating system could be gamified?
While Matt’s aim was to use games to inspire people to want to learn to code, he wanted to tell the story of coding as a superpower and send players on an incredible adventure into a vast, unfolding world with plenty of secrets to uncover with their friends. “The beauty of narrative games is that the story you follow can become your own,” says Matt. “Our goal is to let every player manifest their own power to their greatest ability, and discover that they are the protagonists in their story.”
The Evolution of The Endless Mission
With our course set for the emotional reaction we hoped to evoke from players, it was time to explore the game’s experience and design. During the early prototyping process we tried a variety of directions – from Alternate-Reality Games to gamifying coding curriculum within the Endless education products, to turning the entire Endless operating system into a game. While the ideas were interesting, we realized they weren’t hitting the two pillars we’d established as critical: a grand sense of narrative and a visually vast world. We also determined that it would be critical to create a standalone game on a core gaming platform like Steam in order to reach the largest audience possible. From here, the big spark occurred when we asked ourselves one question: what kind of game genre has traditionally been best at inspiring people to overcome great difficulty in non-traditional skills? The answer: creation sandbox games.
A common theme we noticed among creation sandbox games was that players often had to look outside the game environment to learn some, or all, of the skills required to play and create. We decided that our approach would be to build a comprehensive environment that simultaneously offered unlimited creativity and all the tools you need, all without leaving the narrative.
Rather than limit ourselves to any one genre we chose to root ourselves in gaming as a whole with the capability to create any game and any genre. To accomplish this we examined and distilled iconic genres down to their essence, which involved three elements: Avatar (who you play as), Gameplay (the genre and objective of the game), and Scene (where the game takes place). By taking these elements and combining them in different ways, you could build your own completely unique game experiences from the ground up.
In keeping with the spirit of fun, rather than simply selecting these ingredients from a dropdown menu, we opted for something more engaging: grabbing elements in the form of cubes and literally throwing them together to conjure entire game worlds. This mechanic also clearly expressed Matt’s belief in the power technology brings and how it allows you to shape the world around you.
The Future of The Endless Mission
Our goal with The Endless Mission is to provide you with incredibly powerful creation tools and take you on an epic hero’s journey through a rich and mysterious narrative. As we head into the beta and Early Access, we’re excited to hear what you think about how we’re doing. In particular, we’d love to hear your thoughts about what tools we should introduce, what genres we should explore next, and how you think the narrative might evolve. The Endless Mission is a community-driven game and we can’t wait to go on this adventure together! | https://theendlessmission.com/blog/how-did-the-endless-mission-come-to-be/ |
Paul McCartney has shared an excerpt from his forthcoming book The Lyrics: 1956 To The Present, in which he remembers the inspiration for one of his best-known Beatles songs, ‘Eleanor Rigby’.
Writing about his childhood in Liverpool, McCartney recalled doing chores for local residents during the Scouts’ ‘Bob-a-job week’, during which he met an old lady who would go on to inspire the song.
“Eleanor Rigby is based on an old lady that I got on with very well,” McCartney wrote in an extract published by The New Yorker. “I found out that she lived on her own, so I would go around there and just chat, which is sort of crazy if you think about me being some young Liverpool guy.
“Later, I would offer to go and get her shopping. She’d give me a list and I’d bring the stuff back, and we’d sit in her kitchen. I still vividly remember the kitchen, because she had a little crystal-radio set […] So I would visit, and just hearing her stories enriched my soul and influenced the songs I would later write.”
McCartney also recounted the fact that his original name for Eleanor Rigby was Daisy Hawkins. “I can see that “Hawkins” is quite nice, but it wasn’t right. Jack Hawkins had played Quintus Arrius in Ben-Hur. Then, there was Jim Hawkins, from one of my favorite books, Treasure Island. But it wasn’t right.”
Although there is a grave attributed to an Eleanor Rigby in the graveyard of St Peter’s Parish Church in Woolton, Liverpool, where McCartney and John Lennon had spent time sunbathing as teenagers, it is believed to be a coincidence.
“I don’t remember seeing the grave there, but I suppose I might have registered it subliminally,” McCartney wrote.
He has previously said that the name Eleanor was inspired by the actress Eleanor Bron, who starred in the 1965 Beatles film Help!, while Rigby is based on a shop called Rigby & Evens Ltd, Wine & Spirit Shippers that he saw in Bristol.
McCartney’s two-volume book is published on November 2, and will recount the musician’s life through his earliest boyhood compositions, songs by The Beatles and Wings, and from his lengthy solo career. In August, he revealed the names of the 154 songs that are featured.
To accompany the release, the British Library has announced it will host a free display entitled Paul McCartney: The Lyrics between November 5, 2021 and March 13, 2022, while the musician himself will discuss the book live in conversation at the Royal Festival Hall next month. | https://www.nme.com/news/music/paul-mccartney-on-the-woman-who-inspired-eleanor-rigby-hearing-her-stories-enriched-my-soul-3073210 |
1. Field of the Invention
The present invention relates to a two-wavelength diffraction element in which diffraction efficiency can be adjusted arbitrarily, a method of designing a diffraction grating of the diffraction element, an optical head device for recording, reproducing or erasing information on an optical information medium such as an optical disc by using the diffraction element and an optical information apparatus including the optical head device, as well as a computer, an optical information medium player, a car navigation system, an optical information medium recorder and an optical disc server each of which includes the optical information apparatus.
2. Description of the Prior Art
At present, various kinds of recording mediums are available for recording and storing digital audio, images and moving pictures as well as document files and data files produced by computers or the like. An optical disc is used as one of the recording mediums. Especially, digital versatile disks (DVDs) which have higher density and larger capacity than conventional compact discs (CDs) and are coming into wide use also in the field of recorders in place of video tape recorders (VTRs) used predominantly currently. Furthermore, the study of a next-generation optical disc having a higher recording density is being conducted in a number of firms and is expected to appear on the market in the near future.
In order to raise recording density of the optical disc, it is considered that a numerical aperture (NA) of a beam incident upon its information recording face is increased. However, if its optical axis tilts at this time, such a problem as an increase in the amount of aberration arises. In order to solve this problem, it is effective to reduce a thickness of a protective layer or a substrate thickness in the optical disc. In this specification, the “substrate thickness” indicates a thickness from an incident face of the beam to the information recording face in the optical disc.
Referring to the history of the optical disc, the first-generation optical disc is the CD in which an infrared beam having a wavelength of 780 to 820 nm is used as a light source, an objective lens has a numerical aperture of 0.45 and the thickness of the substrate is 1.2 mm. The second-generation optical disc is the DVD in which a red beam having a wavelength of 630 to 680 nm is used as a light source, an objective lens has a numerical aperture of 0.6 and the thickness of the substrate is 0.6 mm. Meanwhile, the third-generation optical disc under development currently is an ultrahigh-density optical disc in which a blue beam having a wavelength of 380 to 420 nm is used as a light source, an objective lens has a numerical aperture of 0.85 and the thickness of the substrate is 0.1 mm.
As is seen from the above, the thickness of the substrate becomes thinner for raising recording density. A single optical information apparatus is expected to be capable of recording and reproducing optical discs having different substrate thicknesses and different recording densities in view of its economical aspect and its occupied space. To this end, it is necessary to provide an optical head device including a condensing optical system which is capable of condensing a beam up to its diffraction limit on the optical discs having the different substrate thicknesses.
Meanwhile, tracking control and focusing control are typically necessary for recording and reproducing the optical disc. In order to detect these control signals by a compact arrangement at low cost, it is advantageous to employ a diffraction grating in the optical head device. In case recording and reproduction should be performed by the single optical information apparatus in a system including two or more light sources having different wavelengths, it is desirable that the diffraction element has an identical diffraction efficiency for the respective wavelengths of the light sources.
An arrangement in which a ratio of a zero-order beam (main beam) to first-order diffraction beams (sub-beams) can be adjusted for a specific wavelength is disclosed in Japanese Patent Laid-Open Publication Nos. 2001-281432, 2002-311219 and 2002-245660 and is described with reference to FIG. 12. FIG. 12 shows a conventional diffraction element 200. A first diffraction grating 200a for diffracting a laser beam of a wavelength λ1 is provided on one face of the conventional diffraction element 200, while a second diffraction grating 200b for diffracting a laser beam of a wavelength λ2 is provided on the other face of the conventional diffraction element 200. Thus, the first diffraction grating 200a diffracts the laser beam of the wavelength λ1 and transmits the laser beam of the wavelength λ2 therethrough as one beam. On the other hand, the second diffraction grating 200b diffracts the laser beam of the wavelength λ2 and transmits the laser beam of the wavelength λ1 therethrough as one beam. Meanwhile, a depth of the first diffraction grating 200a depends on the laser beam of the wavelength λ2 and a width of each of land portions and a width of each of groove portions of the first diffraction grating 200a are formed such that the ratio of the zero-order diffraction beam to the first-order diffraction beams of the laser beam of the wavelength λ1 diffracted by the first diffraction grating 200a falls within a predetermined range. Likewise, a depth of the second diffraction grating 200b depends on the laser beam of the wavelength λ1 and a width of each of land portions and a width of each of groove portions of the second diffraction grating 200b are formed such that the ratio of the zero-order diffraction beam to the first-order diffraction beams of the laser beam of the wavelength λ2 diffracted by the second diffraction grating 200a falls within a predetermined range.
In the above conventional arrangement, the diffraction gratings are, respectively, provided on opposite faces of a light-transmittable substrate, which requires time-consuming and expensive operations.
Meanwhile, since optical loss on the face of the diffraction grating, through which the beam is transmitted totally, is not zero, optical loss of the diffraction element having the two diffraction gratings provided on the opposite faces, respectively becomes large accordingly.
Meanwhile, in the above prior art documents, an infrared beam having a wavelength of 785 to 790 nm for the CD and a red beam having a wavelength of 650 to 658 nm for the DVD are used as the two wavelengths. In the next-generation ultrahigh-density optical disc apparatus, since a blue beam having a wavelength of 380 to 420 nm is used, an element usable for the blue beam should be provided. However, the above prior art documents do not disclose an arrangement including such an element.
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In short, it comes down to questions. I will likely draw critics for this one, but I take a very reductionist view when it comes to research as an idea. Designing research well is hard, performing the appropriate analysis to support your research question is usually even harder. Achieving publication of your completed research is harder still. Yet where great research is not hard is recognizing that it, just as with life, is about asking the right questions. In life we are fraught with such questions as whether we are in the right job, whether we are raising our kids well, and whether we are saving enough for retirement. All legitimate of course, and what continues to drive the market for self-improvement/personal success books (an avid fan myself I must admit) is the continued lesson that both framing and lens selection are among the keys to answering them. These texts, therefore and for a nominal price, offer methods for framing differently, and offer a lens which differs from the one currently employed whenever we seek to do better.
Success in research, just as with success in life, begins with asking the right questions. Since there is not a What Should I Do With My Life volume for research, here are a few questions to consider as you embark on your next research project:
What Keeps Me Up at Night? Palmer and Zajonc (2010) in their text The Heart of Higher Education: A Call to Renewal quote Whitehead who states, “We must be aware of what I will call ‘inert ideas’ – that is to say, ideas that are merely received into the mind without being utilized, or tested, or thrown into fresh combinations… Education with inert ideas is not only useless; it is, above all things, harmful” (p. 58). Research is a process endeavored by few, yet needed by many. Research pushes our society further, answers those important questions, and gives rise to collectively educating the curious. Yet that process is wasted when on questions of low utility, or those meant solely to serve an end such as publication in itself. A dissertation which simply sits on a shelf, an article written only to be quoted by its author, research performed amid an absence of passion indeed generates inert ideas.
What Can I Talk About, Endlessly? Great research takes time, massive amounts of forethought, a healthy dose of metacognition, and elbow grease. We must dig into the existing literature to such an extent that not only do we understand the relative, ongoing theoretical conversation to-date, yet we must also feel comfortable contributing to its furtherance. Where this becomes a problem is when we recognize there is not one correct and finite way to go about this. As Dane (2011) describes in Evaluating Research, “For any particular theory, the number of ways in which a concept may be operationalized is limited only by the imagination of the researcher” (p. 22). This means not only can the very same concept be represented in myriad ways utilizing myriad methods, this also means that our curiosity in a topic cannot be short-lived or our exploration of it will be poorly served. Where inert ideas asks that we identify a source of passion, Dane reminds us we must also be willing to show great amounts of stamina in order to produce equally great research.
What Do Other People Need? As students of research, both at the Master’s and Doctoral level, we are told when first striving to identify a research topic that we must identify something meaningful to us and explore it. This, I feel is only one third of a very critical equation. As mentioned above another facet is to identify the existing conversation in the literature around a topic, and pinpoint where furtherance can be achieved. The final coefficient to this equation, however, has to do with the audience. As Booth, Colomb, and Williams (2008) note in The Craft of Research, “Down the road, you’ll be expected to find (or create) a community of readers who not only share an interest in your topic (or can be convinced to), but also have questions about it you can answer” (p. 19). This gives rise to the consideration for what problems can be solved for others, what questions can be answered for others, and what good your research can do for others. Great research needn’t be solely about a transparent journey into the center of you, it should also serve a purpose outside of the self, and serve a vested audience.
How Can I Help Them? We are not all researchers comfortable with every design available. Some of us prefer quantitative, some qualitative, some mixed methods. When considering the above on operationalization, as well as the furtherance of an existing scholarly exchange, we also have the opportunity to decide from among the designs possible which will ensure the largest captive, receptive audience as appropriate. As noted by the famed methodologist John Creswell (2009) in Research Design, “researchers write for audiences that will accept their research. These audiences may be journal editors, journal readers, graduate committees, conference attendees, or colleagues in the field… The experiences of these audiences with quantitative, qualitative, or mixed methods studies can shape the decision made about this choice” (p. 19). This becomes a task at not only knowing your audience, it also means a task in understanding what design will be most helpful, as it will equally be the design which brings the learning curve down to near non-existence among your readership.
Again, I recognize I am taking a highly reductionist view. I hope those of you who see this as such also recognize this is meant to be a primer alone. These words certainly do not reflect all one should consider when beginning a research project. What this does represent, though, is a list of things to consider to get you on a productive path. A path toward enlightenment, toward understanding, and one well-trodden by those who were just as curious about the world around them. Hopefully, and if you’re lucky, it will also be a path which only leads to more questions.
Booth, W. C., Colomb, G. G., & Williams, J. M. (2008). The craft of research. Chicago, IL: The University of Chicago Press.
Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches. Thousand Oaks, CA: Sage Publications, Inc.
Dane, F. C. (2011). Evaluating research: Methodology for people who need to read research. Thousand Oaks, CA: Sage Publications, Inc.
Palmer, P. J. & Zajonc, A. (2010). The heart of higher education: A call to renewal. San Francisco, CA: Jossey-Bass
About the Author:
Senior decision support analyst for Healthways, and current adjunct faculty member for Grand Canyon University, South University, and Walden University, Dr. Barclay is a multi-method researcher, institutional assessor, and program evaluator. His work seeks to identify those insights from among enterprise data which are critical to sustaining an organization’s ability to complete. That work spans the higher education, government, nonprofit, and corporate sectors. His current research is in the areas of employee engagement, faculty engagement, factors affecting self-efficacy, and teaching in higher education with a focus on online instruction. | https://drjustinbarclay.com/2014/10/05/what-great-research-and-life-have-in-common/ |
Central Thought: Building a strong foundation requires some unappealing dirty work.
Anyone who has been involved in residential or commercial construction knows that many phases and types of work must be performed to transform undeveloped property into a landscaped site with a home or business structure. Every type of construction begins by planning and laying a proper foundation to build on. In fact, it may appear that construction is in reverse when the excavation for a foundation begins.
Some locations feature shallow bedrock or stable compacted soil that requires minimal effort to survey, drive stakes into, and build a perfectly level and square foundation rather quickly. Other locations require an extraordinary amount of effort to drill and dig deep into the earth before anything can rise above the ground.
Building a solid foundation is a rather thankless and dirty task. The foundation is seen in hues of gray. Evidence of the hard and dirty work that has gone into laying the foundation is covered by a structure and landscaping. No one will 'ooh” or 'ahh” over the beauty of a foundation. A structure that is built on a foundation that seems perfect but required very little effort may be the first to fall when a storm hits. A foundation with moorings that sink deeply into the bedrock can withstand a forceful storm. The outward beauty of a structure cannot compensate for a shallow foundation.
Jesus established the hearing and doing of his Word as the only lasting foundation that we can have for our lives. It may not be exciting and may even seem to be boring or dirty work but it does not diminish its importance. Making His Word our highest priority gives us endurance against the storms and trials of our everyday life. No matter what comes against us, we can stand strong without a crack in the foundation.
Devotional Prayer: Lord, help me to not simply know Your Word but to also put it in practice. Help me to establish my life upon You and do the 'dirty work” necessary to ensure that I can stand firm regardless of the storm that may come my way. | https://globalchristiancenter.com/devotions/daily-devotions/10519-january-9-doing-the-dirty-work |
In these pictures one can see the jobsite before we started. The fiberglass pool shell arrives on a flatbed trailer. In this case, you will notice the pool came stacked with another pool. The shell is then lifted off the trailer with our excavator and then taken to the back yard, waiting for the hole to be completed.
STEP 2: EXCAVATION
Before excavation starts, we draw the pool on the ground and mark off segments of the pool area as to follow the proper grades and depths set by the dig plans. Once the hole is complete, the base is formed by lying off string and then screeting in a slab of compacted gravel(we used to use sand but have since changed to gravel). This gravel will provide the level base needed to insure the pool is installed properly and level.
STEP 3: INSTALLATION
Once the base of the pool is properly prepared, the pool shell is lowered in place. At this time, it is checked to insure it’s level and situated correctly. Water (from our water truck) is the next item to be added, and is followed up with gravel(we used to use sand) backfilled on the outside of the shell. These two actions are done simultaneously as to insure proper structural integrity. After the pool has been backfilled about 75%, we then run the plumbing to the filter system, using a special technique of heat bent schedule 40 PVC pipe. Throughout this process, we continue to check with a laser the levelness of the pool.
STEP 4: BACKFILL, FORM & POUR
In this phase, now that the plumbing has been completed, we backfill to the top edge of the pool. The concrete edge is formed in this stage and gravel is spread out and compacted, covering the patio area entirely as to provide a sound base for the concrete. The concrete deck is poured up to the cantilever forms. This provides a very aesthetically pleasing bull-nose concrete coping edge. Once the concrete has been finished and starts to cure, we will then remove the forms. Once removed, we often get into the pool as to make sure the coping face turns out just right.
ALL DONE! | http://www.dynastypoolsri.com/process/ |
The aim of this study was to develop a validated high-performance thin-layer chromatography (HPTLC) procedure for resolution of chemical constituents and identification and quantification of two selected natural anticancer compounds, stigmasterol (PD-1) and cinnamic acid (PD-2), in Pluchea dioscoridis chloroform fraction (PDCF). The chromatographic estimations were conducted on normal HPTLC (20 cm × 10 cm glass-backed silica gel 60 F254) plates with chloroform-methanol-acetic acid (93:5:2, V/V) used as the mobile phase. para-Anisaldehyde was used for the derivatization of the developed plate, and compact spots were scanned at λmax = 513 nm. A well resolved, compact, and intense peaks of PD-1 and PD-2 were recorded at R F = 0.57 and 0.19, respectively. The proposed analytical method for both biomarker compounds was found to be handy, simple, precise, linear (%RSD = 1.03–1.45), accurate (98.91–99.14%), reliable, and sensitive for the analysis of both bio-markers. The LOD/LOQ (ng) for PD-1 and PD-2 were recorded as 38.73/117.37 and 42.58/129.04, respectively, in the linearity range of 200–1400 ng per spot. The obtained result showed maximum quantities of PD-1 and PD-2 (5.36 and 16.98 μg mg−1, respectively). The developed HPTLC was found to be suitable for the routine analysis of these 2 biomarkers in the chloroform fraction of Pluchea dioscoridis and can be further employed in the process quality control of herbal formulations containing the said biomarkers.
SBA-15 materials exhibit important properties including large surface area, highly ordered mesopores, and high stability in a variety of catalytic reactions. In this study, Al and Zr-loaded SBA-15 materials (Si/Zr = 30, Si/Al = 30), 50 molar ratio with Pd (1, 2, and 4% (w/w)) and Pt (0.5 and 1% (w/w)) were tested in the isomerization of C5–C7 linear alkanes. These solids were characterized using thermal gravimetric analysis–differential thermal analysis (TGA–DTA), nitrogen physisorption, and energy dispersive X-ray (EDX). Interestingly, the low porosity SBA-15 materials loaded with 2% (w/w) Pd were most active catalysts for n-heptane isomerization. The mechanochemical addition of tungsten also resulted in the increase of catalytic activity for the reaction. The results herein demonstrate the possibilities of SBA-15 catalysts as a commercial catalyst alternative in the catalytic isomerization of hydrocarbons.
HALLE-WITTENBERG, HALLE(SAALE), GERMANY Received: 13 June, 2001; accepted: 6 August, 2001 Field experiments were conducted at a high latitude site for sunflower (Helianthus annuus L.) production in central Germany (51 o 24' N, 11 o 53' E) in 1996, 1997 and 1998. The responses of sunflower development to various planting patterns differed in the duration from emergence to the middle of the linear growth period as calculated via a tangent hyperbolic model F(t)=(. +ß)×tanh[. ×(t–.)]. Final dry matter accumulation showed few differences among the planting patterns: 12 plants m –2 at 50 cm row spacing at 75 cm row spacing (RS2PD2) and 4 plants m –2 at 100 cm row spacing (RS3PD1). The actual and simulated values for final dry matter were close to 1200 g m –2 . The responses of soil moisture and temperature to planting patterns changed from the upper to the deep soil layers. In a normal year, e.g. 1997, the soil water to 150 cm depth was sufficient for sunflower growth. In a drought year, e.g. 1998, soil water deeper than 150 cm was used by sunflower crops. The soil temperature was mostly lower in RS1PD3 and RS2PD2 than in RS3PD1, particularly in the upper soil, at depths of 5 and 20 cm. The most important factor defining the responses of soil moisture and temperature to planting patterns seems to be the amount of radiation penetrating the ground, which may depend on latitude, wind and row orientation. | https://akjournals.com/search?q=%22PD1%22 |
This week I discovered that DOMS is a thing and it is not fun. After the disappointment of last weeks weigh in, I spent my day on Sunday finally finishing off setting up my garage. This included cleaning, oiling and setting up my weights set up and the cross trainer (elliptical). It took a lot longer than expected, but by mid afternoon all was set up.
I printed out the free weight training regime drawn up for me years ago as well as the records of what I was lifting. The regime is set in 3 workout groups being Triceps & Back, Shoulders & Legs and Chest & Biceps. In between nights i will do sit ups / planks / bridges / other core exercises.
Being a realist and noting that the records from the last time i did any weight training were from 3 years ago, i set the expectation of lifting from the 2nd week of the previous regime as i immediately increased all my weights after the first week last time, and then had steady gains. I have done nothing since so don't want to tear something straight away being a hero.
The plan - 1 cycle of the workout group (each group has approximately 9 exercises across two areas and includes 3 sets and 10 reps) each week. Nominally a day break between each workout depending on my exercise schedule. Before each session i will do 16 minutes on the cross trainer at a medium setting (it works better with 16 as there are 16 tracking lines on the trainer - visual motivation), then stretch then into a workout group, then stretch and cool down. Ideally in the morning, but as it is in my garage i can do it at night also.
So scene set. Monday was a workout morning so i got up to my alarm at 06:00, got dressed and jumped on the cross trainer. 16 minutes later, nice and sweaty i hopped off and started targeted stretching of my triceps and back as well as general stretching. Feeling good, i launched into my weights. To be honest, i just got through the session. Every set i was struggling to finish my last rep of 10, however got there every time. 45 minutes later, repeated the stretching and cool down began.
I had breakfast, jumped in the shower and logged onto work feeling great. Was easily the best morning i have had in a long time. Around lunchtime i was ravenous and had lunch (tuna and salad sandwich) and after that was tired and lethargic. Throughout the day i kept my water intake up and tried to get up and move as much as possible, but no improvement. Fast forward to the night, and i was in bed by 20:30 and asleep not long after.
I woke up very early Tuesday with a drive to Canberra ahead of me. My back was a little stiff and sore, by triceps more-so. While i was making breakfast i made sure to stretch a little and that was it. Through the day my triceps got progressively worse. I had planned to head to the Driving Range that night as i was overnighting, and nearly stopped it. However with a little stretching i came around, and to be honest felt really good swinging the club for 30 minutes. That night, my arms simply stopped working. I could not lift them above shoulder height. It was a horrendous nights sleep.
To wrap up the DOMS story, it wasn't until Thursday lunch-time that i was able to move with freedom. As such, the next workout was postponed until Thursday night. DOMS IS REAL - and it hurts. I can only hope my recovery time improves as i get deeper into this program.
I nailed my diet this week and am back on track as far as weight-loss is concerned. A real surprise was on Wednesday, driving home from Canberra, i realised that i had my belt on the 2nd notch. It is not overly comfortable yet - which is how i noted something was different, however the first notch is a little loose and the 2nd notch a little tight, but very happy with that change in a couple of weeks!
Monthly weigh in and measurements are next week - focussing on staying ahead of the target next week and hopefully getting through the next workout group with as little pain as possible. Any tips on minimising DOMS greatly appreciated. | https://www.murbles.com/post/week-8-44-to-go |
By virtue of his life as a songwriter and couch-surfing vagabond, Tim Schou has found his artistic foothold. He has transferred his ability to be true to himself to his songs, which are autobiographical and deals with adventures and encounters from his journeys.
”What I do, is no longer about breaking through, but about the freedom to follow my dreams. No matter what happens, I need to write my own songs. I used to be driven by a greedy energy: I must to do this in order to achieve this and that. Now I can breathe. I hope you can hear that in my music, which is more sincere, more rough and to the bone.”
Musically speaking, Tim has found his path as a modern singer-songwriter who writes tuneful and melodious pop songs with electronic elements and an indie feel. With his immediate vocal as the unifying element, Tim has created a sound that combines indie, pop, rock and electronica. His songs expresses the entire gamut of feelings, and engage in the emotional states of life for good or worse.
“I always had an understanding of, and a compassion with the people and artists who dedicate their whole being and soul to their art. I’m no exception myself. As an artist I hang inside emotions. Good and bad, and I keep hanging as long as needed. That’s what I do and that’s what my songs are made of”, Tim says.
All the while he has travelled the world with his guitar, he learned from the best, collaborated with some of the most prominent songwriters and producers of the business, and composed and recorded more than 200 songs to himself and others. His debut single Novocaine, produced by the successful producer team Hitimpulse (Alma, Felix Jaehn, Ellie Goulding og Kygo), recently won him IAMA’s (International Acoustic Music Awards) annual awards ‘Best Male Artist’ and ‘Overall Grand Prize’. In August 2017 Tim won the Akademia Music Awards ‘Best Pop Song’ with Novocaine, and the successor Morrison was voted ‘single of the week’ on Scottish radio.
In 2017, Tim received PLATIN as co-writer of the popular Kiss You Goodbye (Anton Hagman) for over 40.000 sold copies, and was appointed ‘Artist Of The Week’ by IAMA and A&R Worldwide Magazine. He is featuring on “Just Another Nobody” by Winterlude as well as “Feel Good” and “Millionaire” by Felix Jaehn, and his cuts includes titles by Danyiom, Jumpa and Yellow Mellow. He worked with producers and songwriters such as Jasper Leak, Nico Farmakalidis, Thomas Troelsen, Mike Green, Miklos Malek, David Botrill, Kiyanu Kim and Martin Gallop, and at the time of writing, he has more than 2.6M plays on Spotify alone.
In 2018, Tim has already had two appearances on Danish TV, including a mini-documentary, a radio interview on Danish P4, on which his previous single “Altar” is still in rotation, and a Danish Tour supporting Barbara Moleko. He just supported Morcheeba on their shows in Munich and Hamburg and has several shows and writing sessions lined up in the n the near future in Denmark, England and Germany, as well as charity shows and his own song writing camp in Ibiza. His compositions are flowing in a steady pace, growing stronger each song, and has cuts and features on newly or soon-to-be released material by Danish, French and Russian artists. In November he will come back to Germany for intimate acoustic shows. | https://www.visionary-collective.de/agency/tim-schou/ |
Today we pick up at Step 6. If you need to review the previous steps please click here.
If you or your friends are parents, this idea would work at home too. So feel free to share this blog with anybody you know that has children at home that may benefit from this as well.
How to Teach Classroom Procedures
- Define what procedures need to be taught
- Brainstorm the steps of the procedure
- Break the process down into as many steps as needed by asking yourself questions
- Answer the questions by thinking through how you want the procedure to go
- Explain the importance of the procedure
- Determine age-appropriate ways to teach the procedure–use the model, practice, review approach
- Practice until you feel comfortable with students’ understanding of the procedure
Determine age-appropriate ways to teach the procedure–use the model, practice, review approach
The sixth step is to make sure that those things are kid friendly in the way that you explain them. This is actually three steps put together, but you’re going to repeat them until you have achieved the mastery level that you expect for this particular procedure. So the three parts are model, practice and review. When I say model, that means you want to make sure that you show them exactly how you want them to do it either by modeling the procedure yourself or having another child do it. Sometimes we also use video or pictures or anchor charts, whatever the procedure may call for. It will vary, of course, depending upon the procedure that you’re teaching and the age of the child.
With the drive-thru example, I have a picture of a drive-thru projected on my Smartboard to help bring up that prior knowledge that they have about it. I would say something like, “You’ve been through a drive-thru, right?” And the most common one at my school was McDonald’s because there was a McDonald’s very close to our school. Next, I would say, “How many of you like French fries?” and most kids would raise their hands. “And how many of you go to McDonald’s for French fries?” And they’d raised their hands again. “How many of you have ever gone through the drive-through to get some?” So I’m building that up and again accessing that prior knowledge of theirs.
And then we talk about, “Okay, well, when you go through the drive-thru, can you just speed around someone if the line isn’t going as fast as you would like or if there is someone who maybe isn’t pulling up? Like if you’re super hungry, do you just get to go to the front of the line?” And immediately, of course, they’re like, no, no, you have to stay in your spot in the line. And I would respond by saying, “You know, that’s right–you have to pay attention to what cars are in front of you when you come in and no, you can’t just speed past them. If you did, you’d get in an accident or make them angry because you cut in line.” And so, right then, I’m explaining to them and modeling to them, giving them a specific picture in their mind of how it is they need to think about lining up. Next, I would tie it into our classroom and explain that lining up is just the same. We have to go from where we are to the line in an orderly way and not just push past or zoom around others.
I would show them exactly where I would want them to walk, how I would want them to walk and what I would want them to do. And then I would have the students practice. And as they say, perfect practice makes perfect. So we would practice until I felt they had achieved the method of lining up I wanted them to do. It’s also important at this stage to be sure if they aren’t quite doing it the way you expect, that you stop right then and in a calm, respectful manner explain what was wrong and what correction is needed. This is not to call anyone out or make them feel bad, but rather, to help students understand each specific part of the process. We want them to understand what the wrong way looks like and what the right way looks like from the start.
The class would work towards the mastery that I wanted them to have, and if they were even slightly off, I would say, “Nope, this is what I saw (and explain where the procedure was executed incorrectly). So let’s go back and try that again.”
Practice until you feel comfortable with students’ understanding of the procedure
For step seven, you’ll do the model, practice and review until you get to the point that you feel like the procedure is completed the way you want it to be. Then you will just need to go back and review as needed. This will likely be each time you line up for the next several days, so plan a little extra time to line up so that you can fix any missteps right on the spot and still get to where you are going on time!
If you have any questions, please feel free to email me at [email protected] or send me a DM on Instagram @yourteachingmentor. | https://yourteachingmentor.com/how-to-teach-classroom-procedures-part-3/ |
Like many religions, Christianity has views on the care of the planet.
Extracts from this document...
Introduction
Care of the Planet Many people have different views on how humans should care for the environment, and religions also usually have some opinion over the way we treat the planet. Despite all the actions we take to try to help, are we doing enough to help? Today we are faced with many different problems, such as resource depletion. People are using too many resources and because of our modern lifestyle, it is predicted that resources such as fossil fuels will run out very soon. Fossil fuels are also polluting, and damage the environment by causing global warming. This leads on other problems, toxic chemicals, acid rain and pesticide use. All these also contribute to pollution. Habitat destruction is also a huge issue, and many animals often become endangered because of this. Destruction of habitats usually means clearing woodland, which is also damaging to the planet as trees are important in the process of turning carbon dioxide to oxygen, which humans need to survive. ...read more.
Middle
Judaism is extremely similar to Christianity, as many teachings are from the same scriptures (e.g. Genesis, which is also in the beginning of the Bible). Under Jewish teachings, God also gave humans the responsibility of stewardship and have a duty to God to help look after the planet. Both Jews and Christians can help by joining environmental groups to help. The responsibility of stewardship is considered important as both religions believe that God will judge humans for what deeds they have done, and to live a good life after death we would want to try to do as many good deeds as possible. Hinduism teaches that we should respect all life, and if we are harming the environment we will also harm life around us. In the Hindu creation story, it is said that Brahma created the world and all living creatures out of his own body, and also gave humans the responsibility to look after the earth and the living creatures. ...read more.
Conclusion
Anyone can easily help protect the planet by just doing a few simple tasks everyday whether it is just turning off the lights when not being used, recycling or reusing things rather than throwing them away. However, many religious teachings are extremely old, and because we have advanced so much in technological terms they often do not link to what the world is like today, such as global warming and new types of transport that did not exist when the teachings were taught. However, teachings such as loving your neighbours can still be linked back to today as the general basis behind the teaching remains the same. My view is that everyone is capable of doing their own bit to help protect the environment, and it should be done as it will help in the future. As a Christian, I believe that as we are given the responsibility to look after the planet we should all do our best to protect and care for it, and that doing something small such as recycling can still help contribute to caring for the environment. ...read more.
This student written piece of work is one of many that can be found in our GCSE Charities, Poverty and Development section.
Found what you're looking for? | https://www.markedbyteachers.com/gcse/religious-studies-philosophy-and-ethics/like-many-religions-christianity-has-views-on-the-care-of-the-planet.html |
!Se habla Español aquí!
We engage students with meaningful and intentional interaction, using Spanish as our primary language in class. Students learn key vocabulary, grammar structures and cultural concepts through storytelling, games, in-class conversation, class presentations and group activities.
This program’s standards and learning outcomes are based on the American Council for Teaching Foreign Language and World Language standards. Students who commit to a two-year program can expect to be fully prepared to test into Spanish II in Grade 9.
This program’s courses include:
Exploratory Spanish
This course, designed for students in Grade 6, emphasizes in-class engagement and inspires students to overcome their fears and believe in their own ability to become bilingual and bi-literate. Students learn through activity, not rote memorization.
Spanish I
During this two-semester course, students in Grades 7 and 8 focus on comprehension. They are not required to “produce” language, but, rather, to focus their full attention on listening to language. Teachers introduce new vocabulary and grammar through activities grouped according to significant themes. As students become more proficient in their ability to comprehend Spanish, they begin to speak the language more organically and authentically. By the end of the second semester, students engage in class regularly by speaking and writing.
Spanish II
While in Spanish I students listen with the intent to understand, in Spanish II, they speak with the intention to be understood.During this two-semester course, students in Grades 7 and 8 significantly increase their conversation and begin to focus on the “how” of the Spanish language. More attention is given to specific grammar principles students have learned but cannot yet explain, such as verb conjugation, subject/modifier agreement, nouns and gender. | https://cherryhillschristian.org/the-spanish-language-program/ |
In the field of tension between changing social, economic and climatic environmental conditions, but also different land use concepts and a growing metropolitan area in the immediate neighborhood, the demands on the agricultural system of the Upper Rhinluch are increasing. With the project VITAL, we want to contribute to the development of an ecologically and economically viable regional agriculture, which takes into account the special land use requirements.
In our workshop, we would like to discuss our research approach with those actors who are various ways involved in land use management and its development. Our aim is to fit our research strategy to their experiences, views and concerns. Finally, Christoph Felgentreu of “Deutsche Saatveredelung AG” and Christian Schröder of Greifswald Mire Centre will give an insight into the innovative developments of locally adapted management of peatlands.
When: Thursday, November 17th, 2016 ; from 1.15 p.m. to 5 p.m.
Where: Leibniz Centre for Agricultural Landscape Research , Gutshof 7, 14641
Paulinenaue
If you are interested in participating in the workshop, please, approach us until November 11th, 2016. For registration or further information, please, contact Ingo Zasada:
We are looking forward to viable discussions. | https://www.zalf.de/en/aktuelles/Pages/SO/Erster_Stakeholderworkshop_VITAL.aspx |
I am extremely excited to be heading into Year 2 of the 3 year project, the Midwest Artist Studios™ (MAS) Project. I will be traveling from July 26 through August 1, 2015 to the following artists/cities/states – Mellissa Redman, Grand Rapids, Michigan; Kate Robertson, Ann Arbor, Michigan; Jenniffer Omaitz, Kent, Ohio; Ellie Honl, Bloomington, Indiana; Jessica Anderson, Jacksonville, Illinois; and Jason Ackman, Rushville, Illinois. In mid-August I will be visiting the John Michael Kohler Arts Center’s Arts/Industry to document Emmy Lingscheit, who is one of our featured 2015 MAS artists and a current artist in resident. In late September, I will finish our documentation/research by visiting Krista Svalbonas, Chicago, Illinois and Emmy Lingscheit, Urbana, Illinois.
The artists selected were based on their responses to an online survey focusing on Art Education, body of work, and a Skype interview.
Throughout our visits I will be introducing you to 8 amazing and talented artists from the Midwest working in printmaking to painting, sculpture to mixed media and collage to installation art.
Click here to read a collaborative reflection from this past school year’s MAS Project.
Join me on this MAS adventure via facebook.com/midwestartiststudios or subscribe to the blog, midwestartiststudios.com.
– Frank Juarez
________________________________________________________________________
Here are two of the questions asked on our survey and the artist’s response.
Please share one positive Art Education experience that you had in middle school, high school or college.
I interned for the West Michigan Center for Arts and Technology from 2013-14 and during that time I saw young teens who were not particularly interested in being in yet another program, let alone an arts program become truly excited about community projects they were involved in. The defining factor for the success of WMCAT is not the facilities or the glamour of the studios, but the genuine interest of the instructors and the hands on approach they have toward their students. These kids were what the school system would have considered not worth the time, but I found them really charming and attentive once given the right amounts of attention and motivation to see a project to it’s completion.
Why is Art Education today?
Art education impacted my decision to pursue art as a career. I was homeschooled from fifth grade until I entered college, but I attended school through fourth grade. The art classes I was involved in helped me to hold on to my creativity through very difficult family circumstances that would have otherwise extinguished my interest in the arts. Once I was homeschooled, my mother recognized that I was artistically inclined she put me in summer programs which fostered education through the arts.
Mellissa Redman
Web: mellissajredman.smoothfolio.com
The precursor to her current body of work was her father’s cancer diagnosis in May of 2011. She took this into her artwork as a way to record her feelings at the time, and it slowly evolved into a series of work on its own. The creative process turned from an escape to a cathartic experience. Coping with life is part of our existence as humans. It is an emotional process, affecting each individual differently. The concept of “masking” the true self is something that is well known by nearly every human being. In many ways, it can be described as an elaborate act, a play of sorts; in others, a survival tactic that maintains order and control. She believes both examples of these methods of coping can have positive outcomes.
The portraits in these pieces are not meant to represent any specific person or people group, rather humanity as a whole. Therefore, the expressions of the faces of these figures are neither threatening nor inviting. They are to be viewed as pensive and introverted; facing the viewer, yet clearly not acknowledging him/her for his or her own thoughts. The patterns she uses throughout the picture plane look may appear to be familiar to a viewer, but only in the way that they simply mimic the human fingerprint or loop/whorl pattern in which human hair grows. In addition to the patterning, she alters the smooth surface of the pieces with resin drips and pouring. Additionally, the patterning represents the complexities within oneself as anxieties multiply and are internalized. When light passes through the translucent screen-printed patterns, the portrait in the layers beneath the resin is interrupted. She begins with washes of watercolor that she builds up to increase color saturation. Over this, she uses colored pencil to flesh out the figure’s skin tone and facial details. The only other part of the body shown in this series is the neck, which she has made uniform in each individual piece to create homogeneity.
Once the portrait is completed, she screen prints a transparent thumbprint pattern directly onto the piece and cover the surface with a coating of clear epoxy resin. More transparent screen-printed patterns are printed in between layers of the resin, before the piece is completed. All of the pieces contain at least three layers of resin to achieve the correct amount of layering. The rest of the body is unimportant to this work as the focus is on the head. Behind the head of the figure is a colored disc. Although in art history, a flat disc behind the head of a figure was regarded as a holy symbol, its additional function is to represent a person’s aura (her reasoning for including it is the latter definition). She has modified the aura to act both as a compositional element to frame the face and head, and also to obscure it. Her goal with this series of work is to make the hidden external, to depict how swallowed fears and anxieties would appear if made tangible and visible. Though it’s well known that there are plenty of destructive, unhealthy, and dangerous coping strategies associated with emotional turmoil, she tends to think that there are an equal amount of positive experiences that can be gathered. It is these experiences that give us growth of character, a will to live. These are the experiences she hopes to convey in her work.
Bio
Mellissa Redman earned her Bachelor’s Degree in Painting and Drawing from the University of Akron. A native of Akron, Ohio, Mellissa volunteered her time or artwork to the local YMCA and YWCA chapters, the University of Akron Ballet Institute, the City of Akron, The Chapel: Akron Campus, and the Akron Children’s Hospital. She now resides in Grand Rapids, Michigan where she recently received a Master’s Degree of Fine Art in Painting at the Kendall College of Art and Design. Though she works with water-based media, her paintings also include drawing, printing and collage.
Gallery
All images copyright of the artist and used with their permission. | https://midwestartiststudios.com/tag/portraits/ |
make a super simple & easy side dish using peels of green pea pods & stems of cauliflower without throwing out
How easily one can take out the peels from the green peas ?
Ingredients used to make the dish :
The whole method of preparation is quite simple and easy. To the tempering, all the veggies are added followed by salt and turmeric powder. Then everything is allowed to cook till well done. That's it. The dish will be ready with few minutes. The awesome and flavorful taste of this quick dish'll never let you to throw away the pea pods after taking out the peas from the pods and also the stems of the cauliflowers. With either hot phulka / roti / paratha or even with rice-dal combo this is a best side dish to enjoy the simple meal. Only the trick is how carefully one can peel the pods, else the dish is ready to make using few ingredients. So do try !
The event related to this post :
Green Peas Peel & Cauliflower Stem Fry
ingredients:
- Green pea peel 1.5 cup
- Cauliflower stem (chopped into 2 inch lengthwise) 1 cup
- Cauliflower floret 1/2 cup
- Spring onion (cut into 2 inch) 1/2 cup
- Green pea 1/2 cup
- Onion (finely sliced) (medium) 1 no.
- Turmeric powder 1/2 tsp
- Salt as needed
- Oil as needed
instructions:
How to cook Green Peas Peel & Cauliflower Stem Fry
- In a pan heat oil.
- Add cumin seeds and allow to splutter.
- Add sliced onions and fry for few seconds.
- Next add chopped cauliflower stem first, fry for 1 minute.
- Add cauliflower, green peas, green pea peels and mix everything.
- Fry for 1-2 minutes.
- Then add salt, turmeric powder and mix well.
- Cover and allow to cook well.
- After that add all the green onions and again give a nice mix.
- Cover for 1 minute more.
- Take out the cover and fry for 2-3 minutes more to completely dry all the moisture from the pan stirring in between.
- Turn off heat and serve as a side dish with hot phulka or paratha or even with rice-dal combo.
NOTES:
Did you make this recipe? | https://www.firsttimercook.com/2020/02/green-peas-peel-cauliflower-stem-fry.html |
Writing new and unique blogs consistently becomes challenging for bloggers. It is practically impossible for a human to develop new and original ideas daily.
There may be moments when you will need to read other blogs and pick up concepts from what they have to say. However, you cannot just copy and reprint their information; this necessitates the use of paraphrasing, which is both necessary and beneficial.
You have the option of either manual paraphrasing or using a paraphrasing tool to rephrase the content.
Throughout this post, we’ve addressed the benefits of paraphrasing as well as some advice for bloggers who want to rewrite material using a rephrase tool. So, let’s get started!
Is it necessary for bloggers to paraphrase?
Although it is not required, it is undoubtedly beneficial for bloggers. It is not a simple effort to come up with fresh ideas and write original material entirely independently.
It will need a significant investment of time and energy, which a blogger may be unable to supply given his commitment to publishing posts virtually every day.
However, when paraphrasing, you do not have to start from the beginning. It will assist you in repurposing someone else’s concept by giving it a new form.
How can bloggers rephrase the content to make it unique?
1. The use of Plagiarism checker Tool
With free plagiarism checker you can quickly determine duplicate content whether or not you are dealing with the real stuff.
Many different types of tools are now accessible on the internet, making the material distinctive and free of repeated words, allowing it to stand out from the crowd.
This program scans the text for instances of plagiarism and identifies passages that have been lifted verbatim from other sources. This allows one to update and improve the information, making it more useful for readers.
2. Take several similar posts and put them all together into a single article
For example, you may be a tax professional who has authored several blog entries on preparing for the hectic tax filing season. Alternatively, you may be a lawyer who has been writing about the developing developments of a significant legal matter.
Take the most crucial point from each post, describe it in a paragraph, and then publish it as a brand-new piece.
3. Read again and again
When you wish to restate or paraphrase anything, you must study the content over and over again with complete focus. This is because, when you carefully read a piece, the advantage of this advice is that your understanding of the text becomes obvious.
As a result, you have a clear sketch of the passage as it originally appeared. In this way, it becomes easy for you to rephrase that paragraph in your wording.
4. Compare the Writings
The next step is to compare your paraphrased section with the original piece after you’ve rephrased your material in your own words, as described above.
Consequently, make sure that your paraphrased text has the significant subject, all necessary elements, and the fundamental meaning of the original section.
5. Use synonyms and comparable phrases
When paraphrasing literature, employ synonyms and similar phrases that reflect a thorough understanding of the topic. Synonyms may be found using Thesaurus. It will significantly assist you in searching for synonyms, but don’t overdo it.
It does not imply eliminating all terms from the original text. Sometimes it is vital to utilize some of the original text’s words, substituting synonyms for certain words to make the writing fresh.
6. Vary the Length of Sentences
Complicated thoughts should be broken down into straightforward content. Alternatively, you may synthesize the content by combining numerous concepts from the original text into one big statement. Try to keep to your writing style so that the paraphrased content complements the rest of your work.
Use of Paraphrasing Tool
Paraphrasers, also known as article rewriters, are artificial intelligence-based systems that scan lengthy texts and modify their wordings in real-time without disrupting the content structure.
These paraphrasers are simple to use, yet they may produce entirely distinct text from the original.
They retain the essential message of the context without affecting the article’s structure by inserting new words into the text.
How Paraphrasing tools are Helpful for Bloggers
1. The grip on a Sentence
Minor errors occur while writing content by hand, which can have a detrimental impact on the reader. You run the danger of boring your readers or giving them the idea that they obtain less information from more words.
An article rewriter will aid you in rewriting the content with better sentence structure and a more visually appealing paragraph grip. In this way, the reader feels more connected to the information.
2. Correct any grammatical errors
Another critical factor for a blogger is to ensure that the blogs he produces are free of grammatical and spelling errors. If you have many grammatical mistakes in your material, it will leave a poor impression on the readers.
To get over this difficulty, you might make use of an artificial intelligence-based paraphrasing tool that will detect and repair any grammatical errors in your written work.
3. Time-Saving
Manual paraphrasing takes a lot of time, work, and comprehension to complete. At the same time, paraphrasing tools are a fantastic method to save time since they are fast to provide outputs.
To use the tool, all you have to do is copy and paste or upload your text into it, click on the Paraphrase button, and it will produce new and original material for you in a matter of minutes.
Conclusion
Readers are always on the lookout for something fresh. Consequently, they will never trust you again if you offer them duplicated data. Make sure that your info is unique to get your reader’s confidence. Bloggers can use these paraphrasing tips to help them create unique content quickly and efficiently. | https://askanyquery.com/how-bloggers-can-rephrase-the-content-to-make-it-unique/ |
The Group's financial instruments comprise cash deposits, bank loans and overdrafts, finance lease obligations, derivatives used for hedging purposes and trade receivables and payables.
The Group reports in Sterling and pays dividends out of Sterling profits. The role of the Group's treasury activities is to manage and monitor the Group's external and internal funding requirements and financial risks in support of the Group's corporate activities.
The Board of Directors has approved a policy which governs all treasury activities.
The Group uses a variety of financial instruments, including derivatives, to finance its operations and to manage market risks from these operations. Derivatives, principally comprising forward foreign currency contracts, foreign currency options and interest rate swaps, are used to hedge against changes in foreign currencies and interest rates. Hedges of net investments in foreign operations are also used in the management of foreign currency risk.
The Group does not hold or issue derivative financial instruments for speculative purposes and the Group's treasury policy specifically prohibits such activity. All transactions in financial instruments are undertaken to manage the risks arising from underlying business activities, not for speculation.
The capital structure of the Group consists of net borrowings and shareholders' equity. At 30 June 2017, net borrowing was £120.0 million (2016: net borrowing was £116.6 million), whilst shareholders' equity was £302.6 million (2016: £276.6 million).
The Group maintains a strong capital base so as to maintain investors', creditors' and market confidence and to sustain future development of the business.
The Group manages its capital structure to maintain a prudent balance between debt and equity that allows sufficient headroom to finance the Group's product development programme and appropriate acquisitions. There were no changes in the Group's approach to capital management during the year.
The Group operates globally, primarily through subsidiary companies established in the markets in which the Group trades. The Group's operating subsidiaries are generally cash generative and none are subject to externally imposed capital requirements.
There are financial covenants associated with the Group's borrowings, which are interest cover, and net debt to underlying EBITDA. The Group complied with these covenants in 2017 and 2016.
Operating cash flow is used to fund investment in the development of new products as well as to make the routine outflows of capital expenditure, tax, dividends and repayment of maturing debt.
The Group's policy is to maintain borrowing facilities centrally which are then used to finance the Group's operating subsidiaries, either by way of equity investments or loans.
The Group has exposure to the following risks from its use of financial instruments:
This note presents information about the Group's exposure to each of the above risks, and the Group's objectives, policies and processes for measuring and managing risk.
Liquidity risk is the risk that the Group will not have sufficient funds to meet liabilities as they fall due. Cash flows and covenants of the Group are monitored quarterly. These are reviewed to ensure sufficient financial headroom exists for at least a 12 month period.
The Group manages its funding requirements through the following lines of credit:
The Group's revised borrowing facilities at 30 June 2017 are detailed in note 21. Refer to note 35 for events after the reporting period for changes in the facility.
Market risk is the risk that changes in market prices, such as foreign exchange rates or interest rates, will affect the Group's income or the value of its holding of financial instruments.
The majority of the Group's borrowings bear interest at floating rates linked to base rate or LIBOR and are consequently exposed to cash flow interest rate risk.
The Group has hedged interest rate risk on a proportion of its revolving credit facility by means of an interest rate swap arrangement whereby the Group's exposure to fluctuations in LIBOR is fixed at a rate of 1.8% on the revolving credit facility. The amount of the revolving credit outstanding at 30 June 2017 was £176.8 million at the year end exchange rates (2016: £151.6 million).
Foreign currency transaction exposure arising on normal trade flows is not hedged. The Group matches receipts and payments in the relevant foreign currencies as far as possible. To this end, bank accounts are maintained for all the major currencies in which the Group trades. Translational exposure in converting the income statements of foreign subsidiaries into the Group's presentational currency of Sterling is not hedged.
The Group hedges selectively expected currency cash flows outside normal trading activities. During the previous year the Group designated a US dollar loan of $120.0 million, as a net investment hedge of US dollar net assets.
Credit risk is the risk of financial loss to the Group if a customer or counterparty to a financial instrument fails to meet its contractual obligations.
The Group considers its maximum credit risk to be £65.2 million (2016: £67.4 million), which is the total carrying value of the Group's financial assets excluding cash and cash equivalents.
Cash is only deposited with highly rated banks in line with our treasury policy.
The Group offers trade credit to customers in the normal course of business. Trade and bank references are obtained prior to extending credit.
Our principal customers are pharmaceutical wholesalers and distributors. The failure of a large wholesaler could have a material adverse impact on the Group's financial results.
The largest customer of the Group sits within the NA Pharmaceuticals segment and accounted for approximately 15.4% of gross trade receivables at 30 June 2017 (2016: 11.8%). This customer accounted for 18.5% (2016: 14.5%) of total Group revenues. One other customers accounted for more than 10% of total Group revenues (2016: none).
Receivables are written off when management considers the debt to be no longer recoverable.
The following table presents the carrying amounts and the fair values of the Group's financial assets and liabilities at 30 June 2017 and 30 June 2016. The following assumptions were used to estimate the fair values:
The financial instruments of the Group are analysed as follows:
In March 2015, the Group made an investment of US$1 million in Jaguar Animal Health Inc. (Jaguar) to potentially gain access to the EU marketing rights for a companion animal product. At 30 June 2017, the Company holds 178,571 shares in Jaguar following its IPO. The Company also holds 89,286 warrants, which are valid for three years. The shares and warrants have been fully impaired during the period.
The table below analyses the Group's financial instruments carried at fair value, by valuation method. Where possible, quoted prices in active markets are used (Level 1). Where such prices are not available, the asset or liability is classified as Level 2, provided all significant inputs to the valuation model used are based on observable market data. If one or more of the significant inputs to the valuation model is not based on observable market data, the instrument is classified as Level 3. There were no transfers between Level 1 and Level 2 during the year.
Deferred and contingent consideration is recorded at fair value based on risk-adjusted future cash flows discounted using appropriate interest rates, which are reviewed annually. The inputs relating to future cash flows will include cash flows relating to the relevant contractual arrangements. There would be no material effect on the amounts stated from any reasonably probable change in such inputs at 30 June 2017. Refer to note 4 for amounts recognised in the Consolidated Income Statement in the year.
At 30 June 2017, the deferred and contingent consideration balance is made up of £0.5 million in relation to the Brovel acquisition, £3.0 million in relation to the Phycox acquisition, £3.6 million in relation to the Kane licensing agreement and £27.9 million in relation to the Animal Ethics licensing agreement. Movements in deferred and contingent consideration consist of: £0.8 million decrease due to foreign exchange differences, £28.8 million and £3.7 million in relation to the acquisition of the Medical Ethics Pty Ltd and Kane Biotech Inc licensing agreements respectively; payments of £0.7 million and £0.3 million of unwinding of discount.
The following table shows financial assets which are overdue and for which no impairment provision has been made:
The movement in the impairment provision was as follows:
The following table shows the cash flow commitments of the Group in respect of financial liabilities at 30 June 2017 and 30 June 2016. Where interest is at floating rates, the future interest payments have been estimated using current interest rates:
The contractual undiscounted cash flows in respect of derivative financial instruments are as follows:
The Group has a contractual obligation to pay £nil (2016: £50,000), as its interest rate swap arrangement ended on 31 October 2016.
There are no other assets (2016: none) that have been impaired during the year.
The Sterling equivalents of financial assets and liabilities denominated in foreign currencies at 30 June 2017 and 30 June 2016 were:
A 2.0% increase in annual interest rates compared to those ruling at 30 June 2017 would reduce Group profit before taxation and equity by £3,620,000 (2016: £3,120,000).
The Group has significant cash flows and net financial assets and liabilities in Danish Krone, US Dollar and Euro. The Group does not hedge either economic exposure or the translation exposure arising from the profits of non-Sterling businesses. The Group is hedging certain foreign currency translations through the designation of a US dollar loan as a net investment hedge of US dollar net assets.
The following table shows the impact on the Group's profit after taxation of a 10% appreciation of Sterling against each of these currencies compared to the rates prevailing at the year end date. In this analysis, only financial assets and liabilities held on the balances sheet at the year end are assessed and are only considered sensitive to foreign exchange rates where they are not in the functional currency of the entity that holds them. There is no impact on other equity reserves.
The sensitivities above represent the Directors' view of reasonably possible changes in each risk variable, not worst case scenarios or stress tests. The outputs from the sensitivity analysis are estimates of the impact of the effect of changes in market risks assuming that the specified changes occur at the year end and are applied to the risk exposures at that date. Accordingly, they show the impact on profitability and the balance sheet from such movements.
Actual results in the future may differ materially from these estimates due to commercial actions taken to mitigate any potential losses from such rate movements, to the interaction of more than one sensitivity occurring and to further developments in global financial markets. As such, this table should not be considered as a projection of likely future gains and losses. | http://dechra.annualreport2017.com/financial-statements/notes-to-the-consolidated-financial-statements/24-financial-instruments-and-related-disclosures |
As a part of the U.S. Geological Survey's Volcano Hazards Program, the California Volcano Observatory aims to advance scientific understanding of volcanic processes and lessen the harmful impacts of volcanic activity in the volcanically active areas of California and Nevada.
Selected Volcano
Change Map Base Layer
News
Understanding Your Community’s Volcano Hazard Risk is One Way to Plan During National Preparedness Month
Disasters and emergencies can happen at any time, often without warning. Natural hazards threaten thousands of lives and cause billions of dollars in damage every year throughout the nation.
Unpacking CalVO's new seismic monitoring boxes
If you've noticed any changes to the earthquake counts released in CalVO's weekly updates, don't worry - some behind-the-scenes improvements to our monitoring system have been implemented which allow us to focus on unrest specifically related to our volcanoes and volcanic regions.
Clearing up the volcanic history at Clear Lake
A group of scientists at CalVO have recently begun a new investigation of the Clear Lake Volcanic Field (CLVF), a seismically and geothermally active volcanic system located just ~150 km (93 miles) north of the San Francisco Bay Area. | https://www.usgs.gov/observatories/california-volcano-observatory |
The main cause and effect essay is another typical essay kind, either being an essay kind on its own, or as an element of a bigger essay including a number of paragraphs examining factors and impacts. This site provides info on just what an underlying cause and effect essay is, just how to format this kind of essay, and exactly how to make use of cause and impact framework terms (change signals) because of this variety of essay. There is an illustration cause and effect essay on the main topics females at the job, in addition to some exercises that will help you exercise this area.
For the next look at the exact same content, read the infographic for cause and impact essays ».
What exactly are cause & effect essays?
A cause and essay that is effect at the reason why (or causes) for one thing, then discusses the outcome (or results). Because of this good explanation, cause and impact essays are occasionally named reason and result essays. These are typically the most typical kinds of organisation in scholastic writing. Often the entire essay is supposed to be cause and effect, though sometimes this can be just an element of the essay that is whole. It’s also possible, specifically for quick exam essays, that just the causes or the results, maybe maybe perhaps not both, are talked about. Begin to see the examples below.
- Talk about the factors and outcomes of worldwide warming ’cause and impact’ essay
- Explain the death that is high in Chernobyl ’causes’ only essay
- Talk about the WTO and its particular impacts regarding the Chinese economy economy that is chinese’effects’ only essay
There’s two main how to shape an underlying cause and essay that is effect. They are like the real how to structure problem-solution essays, specifically utilizing a block or even a string framework. For the block framework, most of the reasons are listed first, and all sorts of regarding the impacts are listed a while later. For the string structure, each cause is followed instantly by the effect. Often that impact will then trigger the effect that is next which is the reason why this framework is named ‘chain’. Both kinds of structure have actually their merits. The previous is typically clearer, specifically for smaller essays, while the ensures that are latter any results you provide relate straight to the reasons you have got provided.
The 2 forms of framework, block and chain, are shown into the diagram below.
Cause and Effect Construction Words
Cause and effect words that are structure change signals which reveal the main cause and impact relationships. It is essential to be clear which can be the reason (or explanation) and which will be the end result (or result), and also to make use of the transition that is correct or expression. understand that a cause happens very very first, and also the impact occurs later on.
Here are some cause that is common effect framework words. X is employed to point a reason, while Y can be best place to buy essay used to point the consequence.
- The very first cause of (Y) is (X)
- The next reason is (X)
- Due to (X), (Y)
- As outcome of (X), (Y)
- As a result of (X), (Y)
- because/since/as (X)
- to derive from (X)
- (X) outcomes in (Y)
- to function as the consequence of (X)
- (Y) is born to (X)
- Because of (X), (Y)
- (Y) is basically because of (X)
- (Y) is the end result of (X)
- (Y) is the result of (X)
- Worsening pollution amounts in towns are because of the increased use of cars.
- Due to the increased utilization of vehicles, air pollution amounts in urban centers are worsening.
- Being a total outcome of this increased utilization of vehicles, air pollution amounts in towns and cities are worsening.
- The consequence associated with the increased utilization of automobiles is really a worsening of air pollution amounts in urban centers.
- 1st effectation of (X) is (Y)
- Another consequence of (X) is (Y)
- As being a total result, (Y)
- As a result, (Y)
- Consequently (Y)
- Therefore, (Y)
- Therefore (Y)
- Thus (Y)
- (X) outcomes in (Y)
- (X) causes y that is(
- (X) has an impact on (Y)
- (X) impacts (Y)
- (X) is just one of the factors behind (Y)
- (X) ‘s the reason for (Y)
- Vehicles are employed increasingly for metropolitan transportation. As a result, air pollution amounts in urban centers are worsening.
- Increased utilization of automobiles for metropolitan transport adversely affects air pollution amounts in towns.
- Increased usage of vehicles for metropolitan transportation is just one of the reasons for worsening air air pollution amounts in metropolitan areas.
Sample essay
Below is an underlying cause and essay that is effect. The block is used by this essay framework. Click the areas that are differentwithin the shaded bins into the right) to emphasize the various structural aspects in this essay, in other words. Reasons, Results, and framework terms. This can emphasize not only the paragraphs, but additionally the thesis declaration and summary, since these perform the complexities and results included in the body that is main.
Title: progressively women can be now venturing out be effective plus some ladies are now the salary that is major into the family members. Do you know the reasons for this, and exactly what impact is this wearing families and culture?
In past times, the majority of women remained in the home to manage domestic chores such as for instance cooking or cleaning. Ladies’ liberation and feminism have actually meant that this case happens to be changed plus in modern culture women can be playing a nearly equal part to males with regards to of work. This has already established significant effects , in both regards to your family , for instance by increasing total well being and increasing youngsters’ feeling of self-reliance , as well as for culture it self with greater sex equality .
The reasons that are main the rise of females on the job are ladies’ liberation and feminism. The ladies’s liberation motion originated from the 1960s and had been popularised by writers such as Simone de Beauvoir. Because of this, new legislation emerged, giving females equal liberties to males in a lot of areas, in specific work. A result of this, women have more time to pursue their own careers and interests because of feminist >As.
These have generated some effects that are significant both to household life also to culture all together.
Even though making ability of a female in her own life time is normally not as than compared to a guy, she can nonetheless produce a significant share to your family earnings. Probably the most crucial result of this might be a better standard of living. The pressure on the husband is cons >hence improving both the husband’s and the wife’s emotional wellbeing by helping to maintain a steady income for the family. Also, the power that is purchasing of family may also be raised. Which means the household are able to afford more luxuries such as for example international travel and a household automobile.
A further influence on the household may be the advertising of independency into the kiddies. Some might argue that having both moms and dads working could be damaging to your kiddies as a result of too little parental attention. Nonetheless, such kids need to figure out how to look on them to help with the housework after themselves at an earlier age, and their parents often rely. This consequently shows them crucial life abilities.
The most significant impact of women going to work is greater gender equality as regards society. You can find a number that is increasing of that are becoming politicians, solicitors, as well as CEOs and business supervisors. As a result has generated greater equality for ladies in most aspects of life, not only work. For instance, women now have much more resilient rights to safeguard on their own against domestic physical physical violence and intimate discrimination in the workplace.
To conclude, the increasing quantity of females at work has had about some essential modifications to family members life, including enhanced total well being and increased self-reliance for young ones, in addition to impacting culture it self. Its clear that the sexes are nevertheless a way that is long being equal in every aspects of life, nevertheless, and maybe the task when it comes to current century would be to make sure this occurs. | https://brentwoodshutters.com/academic-language-for-essay-writing-how-to-build-2/ |
Spinal trauma is a serious condition, which must be identified readily as it has critical implications on patient outcome. The etiologies are vast ranging from blunt to penetrating, with a myriad of imaging presentations. Often if the patient has suffered a recent trauma, CT imaging maybe used for a rapid diagnosis of fractures. However, magnetic resonance imaging is also in the forefront in the diagnosis of spinal trauma, specifically when identifying pathology involving the spinal cord, and its associated complications. The purpose of our electronic exhibit is to present multiple cases to highlight the range of conditions as well as the mechanism of injury that lead to serious spinal trauma and present key features of their CT, MRI, and angiographic examinations.
Mutliple CT as well as 1.5 and 3T MRI images were reviewed over the past five years from multiple institutions.
We will present cases including bullet injury to the spinal cord causing gross transection, bilateral facet dislocation, Jefferson fracture from axial loading injury, acute cord compression related to an acutely herniated disc status post MVC, and additional cases highlighting imaging features of spinal trauma subsequent to blunt and penetrating trauma.
It is critically important to be familiar with the imaging features of spinal trauma. Additionally, a thorough understanding of mechanism of injury also provides insight into correlative imaging features. The literature clearly demonstrates marked improvement in patient morbidity and mortality when spinal trauma is accurately diagnosed. While there are classic imaging signs, it is also important to understand and identify potential imaging pitfalls as there may be dire implications. We hope that by showing the various entities leading to spinal trauma, a greater insight and understanding of this disease process will be obtained. | http://theassr.org/abstract/level-1-trauma-at-an-inner-city-hospital-imaging-spectrum-of-spinal-trauma/ |
Those in the in-group enjoy relationships with the leader that is marked by trust and mutual respect. Preliminary efficacy of a web-based family intervention for children with traumatic brain injury.
Situational leadership theory has been criticized on both theoretical and methodological grounds. Regardless of the state of evidence in the research literature for specific behavioral procedures, the selection of such procedures in the case of a specific student should be made on the basis of a functional behavior analysis.
The number of points at the end of the day or period dictates the nature or magnitude of a reward at that time. Brief History of the Systems Approach Principle General systems theory, incorporating the systems approach principle, was first proposed formally in with the publication of Ludwig von Bertalanffy's "General System Theory: Contingency system approach theories have much in common.
Journal of Substance Abuse Treatment. They tend to be involved in important activities and decisions.
Meta-analysis of day treatment and contingency-management dismantling research: Like trait research, leader behavior research did not consider situational influences that might moderate the relationship between leader behaviors and leader effectiveness.
For this reason, a consistent and well implemented behavior management system, including careful management of consequences, is particularly important when the student returns to school and resumes a normal school schedule. Behavior specialists who rely heavily on contingency management do not neglect the antecedents of behavior.
In addition, the student may become accustomed to short work sessions and to controlling activities more than is allowed in school. Preliminary efficacy of a family problem-solving intervention for children with traumatic brain injury.
The situation is most favorable when followers respect and trust the leader, the task is highly structured, and the leader has control over rewards and punishments. Furthermore, behavior management systems that rely on punishment are dangerous for many reasons.
In general, these studies simply looked for significant associations between individual traits and measures of leadership effectiveness. Potential Advantages Ignoring undesirable behavior can have the effect of reducing the likelihood of that behavior, assuming the behavior was intended consciously or unconsciously to have an effect on others.
This neurological problem makes behavior management a difficult issue. The theory continues to generate empirical research. Substitutes foe Leadership s Characteristics of the organization, task, and subordinates may substitute for or negate the effects of leadership behaviors. A cumulative and up to date review of EHR implementations.
Fiedler's research indicated that task-oriented leaders were more effective when the situation was either highly favorable or highly unfavorable, but that person-oriented leaders were more effective in the moderately favorable or unfavorable situations.
Nov 22, · Contingency Approach Main contributors–John Woodward, Fiedler, Lorsch &Main contributors–John Woodward, Fiedler, Lorsch & degisiktatlar.comce. The latest approach to management which interactThe.
WHAT IS CONTINGENCY MANAGEMENT? Contingency management is based on the principle that behavior is a function of its consequences. That is, what people do – how they behave – is related in a predictable way to the consequences of their behavior. Abstract.
Contingency, an amount of funds added to the base cost estimate to cover estimate uncertainty and risk exposure, is a topic of interest for both project managers and sponsors alike. HISTORICAL DEVELOPMENT Three main theoretical frameworks have dominated leadership research at different points in time.
These included the trait approach (s and s), the behavioral approach (s and s), and the contingency or situational approach (s and s).
The Gateway to Up-To-Date Information on Integrated 'Whole Building' Design Techniques and Technologies. The goal of 'Whole Building' Design is to create a successful high-performance building by applying an integrated design and team approach to the project during the planning and programming phases.
WBDG Updates. Contingency planning aims to prepare an organization to respond well to an emergency and its potential humanitarian impact. | https://tokejulunecysak.degisiktatlar.com/contingency-system-approach-44739kq.html |
Fossil Focus: Pterosaurs
Introduction:
Pterosaurs are often mistakenly called flying dinosaurs, but they are a distinct, although related, lineage. They are an extinct group of reptiles from the Mesozoic era (251 million to 66 million years ago) and were the first vertebrates to evolve powered flight (Figs 1 and 2). Pterosaurs were first described as early as 1783 and recognized as flying reptiles shortly afterwards, and more than 150 species are now known. Fossil pterosaurs have been found around the world, with every continent yielding specimens.
Figure 1 — The holotype specimen of Pterodactylus, the first pterosaur known, described in 1783. Permission to use this photo was kindly granted by the Bavarian State Collection of Munich, Germany. Photograph taken by Georg Janssen.
Adult pterosaurs ranged in size from around 1 metre in wingspan to more than 10 metres; the largest species were the biggest flying animals of all time. They occupied the skies for much of the Mesozoic era and had the air to themselves until the birds first appeared in the middle to late Jurassic period (176 million to 146 million years ago). Pterosaurs died out along with the non-avian dinosaurs and many other groups 65 million years ago, in the great extinction at the end of the Cretaceous period.
Figure 2 — The ‘dark wing’ specimen of Rhamphorhynchus. This beautiful specimen is partially preserved in three dimensions and also has spectacularly detailed wing membranes. Top: the specimen under natural light. Bottom: under ultraviolet light, where extra details can be seen. Natural-light image by D. Hone, ultraviolet photograph kindly sent by Helmut Tischlinger.
The fossil record for pterosaurs is poor compared to that for many Mesozoic reptile groups, because their bones were fragile and so were not readily preserved. Until the past few years, there was little research dedicated to pterosaurs, and as a result many things relating to their biology are still either contentious or poorly understood. However, a recent resurgence in interest in this group and a raft of new finds are helping palaeontologists to get to grips with this important group, or clade.
Phylogeny:
The origins and the relationships of the pterosaurs have long been contentious, although a consensus is forming on both issues. Often confused with dinosaurs, pterosaurs are members of their own clade, but are close relatives of their more famous cousins.
Over the years, palaeontologists have hypothesized that pterosaurs originated from various parts of the reptile evolutionary tree. Very early researchers considered them to be the ancestors of birds or even bats, and for a long time it seemed that they were probably basal archosaurs (the clade that contains dinosaurs, birds, crocodilians and some other groups). More recently evidence has begun to stack up that they are a separate group to the dinosauromorphs (dinosaurs and their closest relatives) but that the two groups evolved from a common ancestor. Most researchers now support this position. This makes pterosaurs reasonably close relatives to birds, but they are not bird ancestors as is sometimes wrongly reported.
Pterosaurs are divided into two broad groups. The basal pterosaurs are called the rhamphorhynchoids and are characterized by a number of features of the skeleton, including: relatively small heads with a separate nostril and antorbital fenestra (an opening in the front of the skull between the eye and the nostril, also present in dinosaurs); short necks and large bodies; a short first bone in the fourth finger; a short pteroid bone (see below); a long fifth toe; and a long tail. The more derived pterosaurs have been grouped into the pterodactyloids and had the opposite set of characters: a long head with a combined (and often very large) nostril and antorbital fenestra forming one large opening in the skull; a long neck and short body; long fourth finger and pteroid bones; a short fifth toe; and a short tail (Fig. 3). (As an aside, the name pterodactyloid obviously derives from Pterodactylus, the genus of a type of pterosaur, although neither of these really means the same as the term pterodactyl, which is often misused in place of ‘pterosaur’).
Figure 3 — Skeletal outlines of Rhamphorhynchus (left) and Pterodactylus (right) showing off the basic body plans of the rhamphorhynchoids and pterodactyloids respectively. Note the different sizes of the heads and bodies, and the different proportions of the wings. Image by Edina Prondvai, based on an original by Peter Wellnhofer.
The rhamphorhynchoids and pterodactyloids remained really rather separate with a large anatomical gap between them, until the discovery of Darwinopterus in 2010. This animal is from the Middle Jurassic of China and has a mixture of traits: the large head, combined nasoantorbital fenestra and long neck of the pterodactyloids, but the long tail, short fourth finger bone, long fifth toe and other features otherwise seen only in the basal forms (Fig. 4). Darwinopterus (and several close relatives that have since been discovered) is a wonderful example of a transitional fossil showing in part how one group of animals evolved into another.
Figure 4 — Skeleton of Darwinopterus. Note the pterodactyloid-like head and long neck, but the rhamphorhynchoid body, wings and tail (compare with Fig. 3). Image kindly provided by Lü Junchang.
Anatomy:
Pterosaurs can be identified instantly by their highly modified arms. The first three fingers of the hand are small and would be used to move around when not in flight. The fifth finger of the hand is absent, but the fourth is both robust and massively elongated, and would have provided the main support for the wing membrane. Many of the bones of pterosaurs were thin-walled and hollow like those of birds and some dinosaurs, making the skeleton light overall.
Running from the tip of each wing finger to each ankle was the main wing membrane. This was not leathery, as is often stated, but in fact was a skin-like structure with layers of stiffening fibres, blood vessels and a sheet of muscle (Fig. 2). A smaller membrane sat in the crook of the elbow, supported by a modified wrist bone called the pteroid that was unique to pterosaurs. Finally, a membrane spanned the space between the legs. In rhamphorhynchoids, this was a single broad sheet and was anchored to the long fifth toe on each foot. In pterodactyloids, it was split into two smaller parts, with each half running from the ankle to the base of the tail. This arrangement freed the legs and allowed the reptiles to walk more easily on the ground. In addition to all this flight apparatus, the rhamphorhynchoids also had a vane, on the end of their long tails.
Pterosaurs were also ‘furry’. Their bodies were covered in thin, hair-like fibres termed pycnofibres. This was neither true fur as in mammals nor the simple feathers seen in early dinosaurs and baby birds, but probably evolved independently. It may have been linked to their ability to fly and there is a strong suggestion that pterosaurs were homeothermic (‘warm blooded’).
As might be expected of flying animals, in general the pterosaurs had rather conservative anatomy; that is to say that the restrictions on body shape imposed by flight meant that their overall shape was relatively similar between taxa. Over time there was a general trend for increasing size, with the earliest pterosaurs being rather small and the later ones being especially large. Early forms had lots of — often large — teeth, whereas the most derived forms from the late Cretaceous period were toothless. The most obvious deviation from conservatism was in the remarkable array of head crests that many members of the group sported. These had many different sizes and shapes and could be made of bone, soft tissues or a combination of both (Fig. 5).
Lifestyle:
Despite occasional reports, there is currently no evidence that any pterosaurs were flightless. Pterosaurs were not clumsy flappers or gliders as they have occasionally been portrayed, but were excellent fliers. It is likely that most pterosaurs hunted on the wing, and many lineages seem to have been well adapted to catching fish: some specimens have fish preserved in the stomach. However, other lineages were filter feeders, insect eaters, shellfish specialists or predators who hunted on land. Some species have been suggested to have fed mostly on fruits or seeds.
Pterosaurs laid thin-shelled eggs, which were probably buried in soil with vegetation to keep them moist. Several fossil eggs are known, including some preserved with intact embryos. Both the embryos and very young pterosaurs have remarkably well-formed bones, and it seems likely that even very young pterosaurs could fly.
Rhamphorhynchoids are generally thought to have had difficulty walking on the ground: no footprints have been found for them, and they would probably have stuck to the trees when not flying. The pterodactyloids were better adapted for life on the ground and numerous tracks are known for them (Fig. 6).
Figure 6 — Drawing of a pterodactyloid fossil trackway showing the large four-toed feet and the splayed three-fingered hands (the wing finger does not normally leave a mark; it is held up out of the way, as seen in Fig. 7). Drawing by Mark Witton.
At least some species lived in large colonies, and many may have been social animals. The head crests were probably some form of sexual adornment or signalling structure.
Fossil Record:
Pterosaur specimens are found spanning most of the Mesozoic era. Their fossil record is rather mixed — they are generally rare and often known only from fragments, but areas of exceptional preservation can produce superb specimens and some species are known from large numbers of fossils. The famous Pteranodon is known from more than 1,000 individuals, although most are fragmentary and in poor condition. Rhamphorhynchus is known from more than 100 specimens, most of which are more or less complete. Pterosaurs from areas of exceptional preservation are often preserved with soft tissues including wing membranes and head crests, but the bones are typically crushed flat.
The rhamphorhynchoids arose in the late Triassic period (around 200 million years ago) and go at end of Jurassic period. There are records of some in the early Cretaceous of China, but more recent studies suggest that these are the result of errors in fossil dating and the specimens are in fact older. Intermediate forms such as Darwinopterus date from the middle Jurassic; shortly afterwards, in the late Jurassic, the first pterodactyloids appear. Pterosaur footprints first appear in the late Jurassic alongside the origin of the pterodactyloids, and are found in many locations around the world.
Summary:
Pterosaurs were an important component of Mesozoic land and sea ecosystems. This group lived for more than 150 million years alongside the dinosaurs; they filled numerous ecological niches and included the largest flying animals of all time (Fig. 7). Well adapted for flight, these were not clumsy gliders as they are often unfairly portrayed, but were probably every bit as good as birds in the sky. In some ways they even may have been more agile. Pterosaur research and discoveries are currently booming, and palaeontologists are rapidly gaining a better understanding of the evolution and biology of these fascinating creatures.
Figure 7 — A full-sized azhdarchid pterosaur of around 10-metre in wingspan, standing next to a modern giraffe for scale. The giant azhdarchoids were the largest flying animals of all time. Image kindly provided by Mark Witton.
*This article has been modified from it’s original form. The article originally stated “Fossil pterosaurs have been found around the world, with every continent except Antarctica (so far) yielding specimens.” This has been corrected to “Fossil pterosaurs have been found around the world, with every continent yielding specimens.”
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In Assistive Technology: Matching Device and Consumer for Successful Rehabilitation, contributing authors explore ways psychologists and other helping professionals can collaborate with users of assistive technology to help them get the most out of these devices. Thanks in large part to the past century's advances in technology, people with disabilities can live independent lives, contribute to their communities, attend regular schools, and work in professional careers as a result of assistive technology. This technological evolution has fomented a shift from a medical model to a social model of technology delivery, an approach that puts as much emphasis on the user's community reintegration as it does on his or her physical capabilities. This change means that those in the field can no longer focus on the delivery of technology as an end in itself, but must go one step further and partner with consumers and communities to ensure that assistive devices are put to their best possible use. This forward-looking, interdisciplinary book provides research-based guidance for finding the perfect match between device and consumer, including key information on personality assessment, the influence of pain, coping skills, and the power of new technology and social programs. This volume will be of interest to rehabilitation psychologists, researchers, and anyone working with or using assistive technology.
Product Details
- ISBN-13:
- 9781557988409
- Publisher:
- American Psychological Association
- Publication date:
- 12/28/2002
- Edition description:
- 1 ED
- Pages:
- 325
- Product dimensions:
- 7.30(w) x 10.00(h) x 1.10(d)
Table of Contents
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Small noncoding HIV-1 leader exon 3 is defined by its splice sites A2 and D3. While 3′ splice site (3′ss) A2 needs to be activated for vpr mRNA formation, the location of the vpr start codon within downstream intron 3 requires silencing of splicing at 5′ss D3. Here we show that the inclusion of both HIV-1 exon 3 and vpr mRNA processing is promoted by an exonic splicing enhancer (ESEvpr) localized between exonic splicing silencer ESSV and 5′ss D3. The ESEvpr sequence was found to be bound by members of the Transformer 2 (Tra2) protein family. Coexpression of these proteins in provirus-transfected cells led to an increase in the levels of exon 3 inclusion, confirming that they act through ESEvpr. Further analyses revealed that ESEvpr supports the binding of U1 snRNA at 5′ss D3, allowing bridging interactions across the upstream exon with 3′ss A2. In line with this, an increase or decrease in the complementarity of 5′ss D3 to the 5′ end of U1 snRNA was accompanied by a higher or lower vpr expression level. Activation of 3′ss A2 through the proposed bridging interactions, however, was not dependent on the splicing competence of 5′ss D3 because rendering it splicing defective but still competent for efficient U1 snRNA binding maintained the enhancing function of D3. Therefore, we propose that splicing at 3′ss A2 occurs temporally between the binding of U1 snRNA and splicing at D3.
During human immunodeficiency virus type 1 (HIV-1) long terminal repeat (LTR)-driven transcription, RNA polymerase II generates a pre-mRNA that encodes at least 15 viral proteins (1). The preformed 43S ribosomal subunit recognizes the CAP structure moving along its template until it encounters a translational start codon, defined by its surrounding sequence (2). Thus, the position of the Gag and Gag/Pol open reading frames (ORFs) proximal to the 5′ end of the unspliced viral mRNA ensured their efficient recognition. However, proper HIV-1 replication is intimately connected to the expression of seven other ORFs located distal to CAP that encode the viral proteins Vif, Vpr, Tat, Rev, Nef, Vpu, and Env. Alternative splicing removes inhibitory upstream AUGs, thereby placing downstream ORFs near the CAP structure and allowing their efficient translation by the scanning ribosome. The particular HIV-1 protein encoded by a spliced mRNA is almost always specified by the ORF that is immediately downstream of the 3′ splice site (3′ss) used to create the mRNA. The sole exception is the env ORF within the bicistronic vpu-env mRNAs, whose translation is dependent on a minimal upstream ORF within the HIV-1 vpu leader (3, 4).
On the basis of their intron contents, three different-sized viral mRNA classes can be defined: the unspliced (9-kb), intron-containing (4-kb), and intronless (1.8-kb) viral RNAs (Fig. 1A) (5; for a recent review, see reference 6). The accumulation of these viral mRNA classes occurs in a temporal order (7, 8). In the early phase of viral gene expression, the HIV-1 pre-mRNA is extensively spliced, leading to intronless 1.8-kb mRNA species such as the tat, rev, and nef mRNAs. Rev is necessary for the onset of the late phase of viral gene expression that is characterized by a shift within the cytoplasmic mRNA pool toward isoforms with increased intron content and in which the normal nuclear retention mechanisms are bypassed (9). Rev recognizes an RNA secondary structure, called the Rev-responsive element (RRE), within the env coding sequence. Rev-RRE interactions target the intron-containing (4-kb) and unspliced (9-kb) viral mRNAs for CRM1 export receptor pathway-mediated transport into the cytoplasm, which essentially relies on the multimerization capacity of Rev (10). During the late phase of viral gene expression, the accessory and structural proteins Vif, Vpr, Vpu, and Env are translated from the respective intron-containing viral mRNAs (4 kb). In addition, the unspliced viral mRNA (9 kb) is used for translation of the structural and enzymatic components or enclosed as genomic RNA in progeny virions. Viral mRNA diversity is further increased by the alternative inclusion of either one or both of the two noncoding leader exons, 2 and 3. Noncoding leader exon 3 is flanked by 3′ss A2 and 5′ss D3. The formation of intron-containing vpr mRNA, however, requires the activation of 3′ss A2 but silencing of 5′ss D3, since the ORF of Vpr starts within the downstream intron of exon 3. Thus, vpr mRNA processing and exon 3 inclusion are mutually exclusive. Nevertheless, both splicing patterns are negatively regulated by an exonic splicing silencer (ESSV) within exon 3 (Fig. 1B) (11–13). ESSV contains three (pyrimidine)UAG motifs which promote the binding of members of the hnRNP A/B protein family to the viral mRNA, inhibiting splicing at the upstream 3′ss A2. In the absence of functional ESSV, the levels of exon 3 and vpr mRNA splicing are excessively increased, This leads to a severe perturbation of the balance between spliced and unspliced viral mRNAs that is detrimental to virus particle production (11, 13).
Alternative splicing of the HIV-1 pre-mRNA. (A) Schematic of the HIV-1 genome. The ORFs are indicated by open boxes. The LTRs are located at both ends of the provirus. All HIV-1 proteins are encoded in a single primary RNA. Alternative splicing allows all viral proteins to be efficiently translated within the host cell. The 5′ (SD) and 3′ (SA) splice sites are depicted. Alternatively spliced noncoding exons 2 and 3 within Rev-independent (1.8-kb size class) and Rev-dependent (4-kb size class) spliced mRNAs are shown as boxes (exon 2, dark gray; exon 3, light gray). The positions of the primers used in RT-PCRs for the analyses of viral mRNA splicing are indicated by arrows (E1 [fwd], exon1; E4 [rev], exon 4; I4 [rev], intron 4; E7 [rev], exon 7). (B) Intrinsic strength of the 5′ss (D1 to D4) and 3′ss (A1 to A7) distributed along the HIV-1 pre-mRNA. Each value in parentheses reflects the predicted intrinsic strength (5′ss, HBond score [www.uni-duesseldorf.de/rna]; 3′ss, MaxEnt score [http://genes.mit.edu/burgelab/maxent/Xmaxentscan_scoreseq_acc.html]). The nomenclature of the viral splice sites is from reference 9. Positions of known enhancer (white) and silencer (black) sequences within the HIV-1 pre-mRNA are shown. Exon 3 is flanked by 3′ss A2 and 5′ss D3. The positions of the Ld-2 (35), ESE-Vif (36), ESEM (14), G4 (36), GI2-1 (M. Widera, and H. Schaal, submitted for publication), ESSV (11–13), ESS2p (37), ESE2 (38, 39), ESS2 (40–42), GAR, guanosine-adenosine rich (GAR) ESE (16, 17, 27), E42 (27), ISS (15), ESE3 (43), and ESS3 (43–45; adapted from references 27 and 46) sequences are shown.
In this work, we identified an exonic splicing enhancer (termed ESEvpr) between the repressing ESSV and the 5′ss D3 that acts positively on its recognition by the U1 snRNP. We show that the defect in virus particle production seen in the context of ESSV-negative provirus was efficiently rescued by additional inactivation of this ESE but that vpr mRNA processing critically depended on the presence of intact ESEvpr. Furthermore, we identified Tra2-alpha and Tra2-beta as the splicing regulatory proteins by mass spectrometry and overexpression analyses. Finally, replacement of 5′ss D3 with a splicing-incompetent but U1 binding-competent 5′ss-like sequence revealed that the ESEvpr-mediated U1 snRNP stabilization to HIV-1 exon 3 is essential for vpr mRNA expression. This argues for a function for U1 snRNP binding to 5′ss D3, unrelated to splicing, that resulted in the activation of 3′ss A2, potentially via the formation of an exon definition complex.
Oligonucleotides.The oligonucleotides used in this study were obtained from Metabion GmbH (Martinsried, Germany).
Primers for site-directed mutagenesis.The oligonucleotide primers used for site-directed mutagenesis are described in Table 1.
Primers used for semiquantitative and quantitative real-time RT-PCR.The oligonucleotide primers used for semiquantitative and quantitative real-time reverse transcription (RT)-PCR are described in Table 2.
HIV-1-based subgenomic splicing reporter.The HIV-1 NL4-3 (GenBank accession no. M19921)-derived parental plasmid LTR ex2 ex3 contains the two small noncoding leader exons 2 and 3 and the 5′ part of tat exon 1 interspersed with their authentic intronic sequences.
LTR ex2 ex3 was constructed as follows. First, the EcoRI/PstI fragment of the previously described LTR SD SA tatCAT minigene (14) was replaced with a PCR product obtained with primer pair 1814/1817 and pNLA1 (15)—a cDNA derivate of pNL4-3—as the template, leading to LTR SD ex2 ex3 SA. Subsequently, viral splice site D1 was inserted via BssHII/EcoRI restriction sites by using an amplicon obtained by PCR with primer pair 2346/2347 and SV-1-env (16) as the template, generating LTR D1 ex2 ex3 SA. In the next step, the NdeI/SalI fragment of LTR D1 ex2 ex3 SA was replaced with an NdeI/XhoI-digested PCR product amplified with primer pair 2386/2381 by using pNLA-1 as the template. This fragment contained viral splice site A3, an SpeI restriction site inserted via reverse primer 2381 and duplicated sequences downstream of 3′ss A3, generating LTR D1 ex2 ex3 A3 dupl. To remove the duplicated sequences obtained during this cloning step, the SpeI/XmaI fragment of LTR D1 ex2 ex3 A3 dubl was replaced with a PCR product obtained with primer pair 2384/2385 and LTR SD SAtatCAT as the template, leading to LTR ex2 ex3/pNLA1. Finally, the LTR ex2 ex3 splicing reporter, whose viral nucleotide sequences are identical to those of pNL4-3, was cloned by insertion of the NdeI/EcoRI fragment from pNL4-3 into LTR ex2 ex3/pNLA1.
LTR ex2 ex3 ESSV and ESEvpr mutants were constructed by PCR mutagenesis. For construction, the AlwNI/SpeI fragment of LTR ex2 ex3 was replaced with the respective PCR products using an appropriate forward mutagenesis primer and 2588 as the reverse PCR primer containing AlwNI and SpeI restriction sites. All plasmid sequences can be obtained on request.
Proviral HIV-1 plasmids.pNL4-3 mutants were constructed by replacing the region between PflMI and EcoRI of proviral clone pNL4-3 with the respective mutated LTR ex2 ex3 minigene fragments, which were generated as described above.
U1 snRNA expression plasmids.pUCBU1αD3 and pUCBU1αD3(+1G>C) were constructed by the insertion of a PCR product amplified with primer pairs 3924/3926 and 3925/3926, respectively, containing BglII and XhoI restriction sites into the template pUCBU1 (kindly provided by Alan M. Weiner) into pUCBΔU1 (17).
Cell culture and RT-PCR analysis.HeLa and HEK 293T cells were maintained in Dulbecco's high-glucose modified Eagle's medium (Invitrogen) supplemented with 10% fetal calf serum and 50 μg/ml each penicillin and streptomycin (Invitrogen). Transfections were done in six-well plates with 2.5 × 105 cells per plate using FuGENE6 reagent (Roche) according to the manufacturer's instructions. Total-RNA samples were collected 30 h after transfection from either HeLa or HEK 293T cells transfected with subgenomic or proviral constructs and pXGH1 as a control. For RT, 4 μg of RNA was subjected to DNA digestion with 10 U of DNase I (Roche). DNase I was heat inactivated at 70°C for 5 min, and cDNA synthesis was allowed to occur for 1 h at 50°C and 15 min at 72°C by using 200 U Superscript III RNase H− reverse transcriptase (Invitrogen), 7.5 pmol oligo(dT)12-18 (Invitrogen) as the primer, 20 U of RNasin (Promega), and 10 mM each deoxynucleoside triphosphate (Qiagen). For semiquantitative analysis of minigene mRNAs, cDNA was used as the template for a PCR with forward primer 1544 and reverse primer 3632. For a transfection control, a PCR was performed with primers 1224 and 1225 to specifically detect GH1 mRNA. For analysis of exon 3 inclusion in viral tat mRNAs and vpr mRNA splicing, a PCR was performed using primers 1544 (E1) and 3632 (E4). For analysis of 1.8-kb HIV-1 mRNAs, a PCR was carried out with forward primer 1544 (E1) and reverse primer 3392 (E7). Partially spliced 4.0-kb HIV-1 mRNAs were detected with primers 1544 (E1) and 640 (I4). PCR products were separated on 8% nondenaturing polyacrylamide gels, stained with ethidium bromide, and visualized with a Lumi-Imager (Roche).
Quantitative real-time PCR assays for the detection of single viral mRNA species were done with primer pairs 3389/3390 for unspliced mRNA, 3391/3392 for multiple spliced mRNA, 3395/3396 for vif mRNA, 3397/3398 for vpr mRNA, 3397/3636 for exon 3 inclusion, and 3629/3637 for exon exclusion. For normalization, primers 3387 and 3388 were used and the level of overall viral mRNAs present in each sample was monitored. Fluorescence emission was read by a LightCycler 1.5 (Roche). Data are presented as the average of three independent RT-PCR experiments.
Antibodies.The following primary antibodies were used for immunoblot analysis. A mouse antibody against α-actin (A2228) was obtained from Sigma-Aldrich. A mouse antibody against hnRNP A1 (9H10) was purchased from Santa Cruz Biotechnology. A rabbit antibody against Tra2-beta (ab50846) was obtained from Abcam. A sheep antibody against HIV-1 p24 was purchased from Biochrom AG. Rabbit antiserum against Vif and rabbit antiserum against Vpr were obtained through the NIH AIDS Research and Reference Reagent Program from Dana Gabuzda (18) and Jeffrey Kopp, respectively. For detection, we used a horseradish peroxidase (HRP)-conjugated anti-rabbit antibody (A6154) from Sigma-Aldrich, an HRP-conjugated anti-mouse antibody (NA931) from GE Healthcare (Munich, Germany), and an HRP-conjugated anti-sheep antibody from Jackson ImmunoResearch Laboratories Inc. (West Grove, PA).
Protein analysis.Transfected cells were lysed in radioimmunoprecipitation assay buffer (25 mM Tris HCl [pH 7.6], 150 mM NaCl, 1% NP-40, 1% sodium deoxycholate, 0.1% SDS, protease inhibitor cocktail [Roche]). Proteins were separated by SDS-polyacrylamide gel electrophoresis (SDS-PAGE), transferred to nitrocellulose membranes, and subjected to an immunoblotting procedure. The membranes were probed with the respective primary and secondary antibodies and developed with ECL chemiluminescence reagents (GE Healthcare).
Protein isolation by RNA affinity chromatography.Short RNA oligonucleotides were obtained from Metabion. The RNA oligonucleotides were covalently coupled to agarose beads (Sigma). Immobilized RNAs were incubated in HeLa nuclear extract (Cilbiotech) diluted to a concentration of 40% with buffer D (20 mM HEPES-KOH [pH 7.9], 5% [vol/vol] glycerol, 0.1 M KCl, 0.2 mM EDTA, 0.5 mM dithiothreitol). To remove unspecific bound proteins, samples were washed five times with 1 ml buffer D containing 4 mM magnesium chloride (800 rpm, 2 to 3 min, Eppendorf microcentrifuge). Precipitated proteins were eluted from the RNA by heating to 95°C for 10 min in protein sample buffer. Protein samples were subjected to mass spectrometry or loaded onto an SDS-polyacrylamide gel for Western blot analysis.
Mass spectrometry and mass spectrometric data analysis.Protein samples from RNA affinity purification experiments were loaded onto an SDS-polyacrylamide gel, concentrated in the stacking gel, stained with silver, reduced, alkylated, and digested with trypsin. Peptides were extracted from the gel and subjected to liquid chromatography in 0.1% trifluoroacetic acid. For peptide separation over a 140-min gradient, an Ultimate 3000 Rapid Separation liquid chromatography system (Dionex/Thermo Scientific, Idstein, Germany) equipped with an Acclaim PepMap 100 C18 column (75-μm inside diameter, 50-cm length, 2-μm particle size; Dionex/Thermo Scientific, Idstein, Germany) was used. Mass spectrometry was carried out with an Orbitrap Elite high-resolution instrument (Thermo Scientific, Bremen, Germany) operated in positive mode and equipped with a nanoelectrospray ionization source. The capillary temperature was set to 275°C, and the source voltage was set to 1.5 kV. Survey scans were carried out with the Orbitrap analyzer over a mass range of 350 to 1,700 m/z at a resolution of 60,000 (at 400 m/z). The target value for the automatic gain control was 1,000,000, and the maximum fill time was 200 ms. The 20 most intense doubly and triply charged peptide ions (minimal signal intensity, 500) were isolated, transferred to the linear ion trap (LTQ) part of the instrument, and fragmented by collision-induced dissociation. Peptide fragments were analyzed by using a maximal fill time of 200 ms and an automatic gain control target value of 100,000. The available mass range was 200 to 2,000 m/z at a resolution of 5,400 (at 400 m/z). Two fragment spectra were summed, and already fragmented ions were excluded from fragmentation for 45 s.
Raw files were further processed for protein and peptide identification and quantification using MaxQuant software suite version 1.3.0.5 (Max Planck Institute of Biochemistry, Planegg, Germany). Within the software suite, database searches were carried out by using 86,875 human sequences from the UniProtKB/SwissProt database, including the Trembl part (release 06.2012), with the following parameters: mass tolerance Fourier-transformed mass spectra (Orbitrap) first/second search, 20 ppm/6 ppm; mass tolerance fragment spectra (linear ion trap), 0.4 Da; fixed modification, carbamidomethyl; variable modification, methionine oxidation and acetylation at protein N termini. Label-free quantification was done by using the “match between runs” option with a 2-min time window. Peptides and proteins were accepted at a false-discovery rate (FDR) of 1%, and proteins identified with a minimum of two peptides and quantitative information available for all 10 measured samples were subjected to subsequent statistical analysis. To discriminate selective from nonselective binding protein besides calculating conventional Student t tests on log-transformed data, the significance analysis of microarrays (SAM) algorithm (19) implemented in Perseus version 1.2.7.4 (Max Planck Institute of Biochemistry, Planegg, Germany) was used (FDR threshold, 0.05; constant S0, 1.2). The algorithm accounts both for the change in protein abundance and standard deviation of measurements.
ESEvpr is necessary for vpr mRNA processing.Using an enhancer-dependent splicing reporter (16), we systematically screened exon 3 for splicing regulatory elements and found the 25-nucleotide-long fragment between ESSV and D3 to contain an enhancer sequence (20). On the basis of hexamer score changes, we singled out two point mutations suspected to impair ESEvpr enhancer function (Fig. 2A) (S. Theiss, S. Erkelenz, and H. Schaal, unpublished data) (21).
ESEvpr is necessary for exon 3 inclusion and vpr mRNA processing. (A) The wild-type ESEvpr sequence and the amino acid sequence encoded by the overlapping vif ORF are shown below exon 3. Mutated ESEvpr nucleotide residues are denoted by their positions relative to the GT dinucleotide of viral 5′ss D3. The black box represents the upstream ESSV. Uppercase letters represent exonic positions, and lowercase letters represent intronic positions. (B) HEK 293T cells (2.5 × 105) were transiently transfected with 1 μg of each of the proviral plasmids. At 30 h after transfection, total-RNA samples were collected and used for RT-PCR analyses with different sets of primer pairs. HIV-1 mRNA species are indicated to the right of the gels in accordance with the nomenclature published previously (5). (C) cDNA samples were prepared as described for panel B and used in real-time PCR assays to specifically quantitate the relative abundances of unspliced (a), multiply spliced (b), Vif (c), and Vpr (d) mRNA species and exon 3 inclusion ratios (e). For normalization, primers 3387 and 3388 were used to detect the total viral mRNA content of each sample. Data represent expression ratios relative to that of wild-type pNL4-3 (bar 1), which was set to 100%. Values and error bars show the average ± standard deviation of three independent transfection experiments. Bars correspond to lanes in panel B. (D) HEK 293T cells (2.5 × 105) were transiently transfected with 1 μg of each of the proviral plasmids. At 48 h posttransfection, viral supernatants were collected, layered onto 20% sucrose solution, and centrifuged at 28,000 rpm for 90 min at 4°C to pellet the released virions. In addition, cells were harvested and resuspended in lysis buffer. Supernatants and cellular lysates were resolved by 12% SDS-PAGE and electroblotted onto nitrocellulose membranes. To determine virus particle production and the expression of viral proteins, samples were probed with primary antibodies specifically detecting structural p24gag (CA) and the viral infectivity factors Vif and Vpr. Equal amounts of cell lysates were controlled for by the detection of α-actin. E, extended exon; dm, double mutation; sn, supernatant; ly, lysate.
In order to confirm their relevance for exon 3 splice site activation and vpr mRNA formation, we used proviral clone pNL4-3 (GenBank accession no. M19921) and mutant forms thereof to transfect HEK 293T cells. Semiquantitative RT-PCRs were set up with different primer pairs to detect exon 3 inclusion and vpr mRNA processing within intron-containing and intronless viral RNAs (Fig. 2B). In the presence of the repressing ESSV, the ESEvpr single mutation −16A>G, as well as the −25T>C −16A>G double mutation, led to nearly undetectable levels of exon 3 inclusion in the tat, nef, and env mRNAs (Tat3, Env8, Nef4, Rev7+8) (Fig. 2B, lanes 1 to 4). Additionally, it was impossible to detect vpr mRNAs (Fig. 2B, lanes 1 to 4), indicating that ESEvpr is also required for the activation of 3′ss A2. In line with previous observations (13), inactivation of ESSV resulted in a shift from exon 3-less to exon 3-containing nef, rev, tat, and env mRNAs. Additionally, we observed a considerable increase in the expression of vpr mRNAs (Fig. 2B, cf. lanes 1 and 5). However, the inclusion of exon 3 in the nef, rev, tat, and env mRNA species could be gradually reduced to near-wild-type levels, starting from −25T>C, followed by −16A>G and then the double mutation (Fig. 2B, lanes 6 to 8). Taken together, these results demonstrated that ESEvpr contributed to the regulation of exon 3 inclusion in each of the viral mRNA species. To thoroughly examine ESEvpr for its impact on the regulation of HIV-1 exon 3 splicing, quantitative RT-PCR analyses were performed. Different primer pairs were used to specifically quantitate the relative levels of viral unspliced, spliced, vpr, vif, and exon 3-containing mRNAs. Quantitative RT-PCR assays showed that ESEvpr mutations did not significantly alter the levels of unspliced, spliced, and vif mRNAs in the context of the ESSV-positive virus (Fig. 2C, parts a to c, bars 1 to 4). However, the single point mutation −16A>G alone was able to downmodulate the relative amount of vpr mRNA, indicating that ESEvpr was necessary for the activation of 3′ss A2 even in the presence of ESSV (Fig. 2C, part d, bars 1 to 4). Furthermore, the levels of exon 3-containing mRNA species were greatly reduced (Fig. 2C, part e, bars 3 and 4). Consistent with previous work (13), disruption of ESSV resulted in a large reduction (∼10- to 20-fold) in the level of unspliced mRNA (Fig. 2C, part a, cf. bars 1 and 5). Moreover, the relative amount of multiply spliced mRNAs was upregulated approximately 10-fold (Fig. 2C, part b, cf. bars 1 and 5). In addition, loss of ESSV function because of mutagenesis induced a strong decrease in vif mRNA levels of up to 20-fold (Fig. 2C, part c, cf. bars 1 and 5). In contrast, expression of vpr and exon 3-containing viral mRNAs was detected at highly elevated levels (Fig. 2C, parts d and e, cf. bars 1 and 5). These results were consistent with recent studies showing that disruption of ESSV causes a dramatic deregulation of viral splicing. However, second-site mutations within the ESEvpr element could compensate for the lack of ESSV activity (Fig. 2C, parts a to e, bars 6 to 8). ESEvpr double mutations restored at least normal levels of unspliced, spliced, and vif mRNAs (Fig. 2C, parts a to c, cf. bars 1 and 8). vpr mRNA levels were also decreased in the case of the ESEvpr double mutation both with and without ESSV (Fig. 2C, part d, cf. bars 1 and 4, and 5 and 8), albeit they did not completely return to wild-type levels (Fig. 2C, part d, cf. bars 1 and 8), possibly because of residual enhancer activity. This notion was supported by the finding that the expression of exon 3-including mRNAs also did not entirely return to normal levels (Fig. 2C, part e, cf. bars 1 and 8).
Furthermore, we performed Western blot analyses to evaluate the levels of both intracellular viral proteins and virus particles released into the supernatant (Fig. 2D). In agreement with the data obtained from real-time PCR assays, the levels of Gag and Vif proteins were mostly unaffected by the ESEvpr mutants in the context of the intact ESSV (Fig. 2D, lanes 2 to 5). Moreover, similar viral capsid (CA, p24gag) levels within the supernatant samples indicated that virus particle production was not significantly changed (Fig. 2D, lanes 2 to 5). However, in the absence of ESSV, the levels of the Gag and Vif proteins were strongly reduced (Fig. 2D, lane 6). Moreover, we observed a striking defect in Gag processing, characterized by loss of the Gag precursor p55 cleavage products p41 and p24 (Fig. 2D, lane 6). As expected from the RT-PCR results, Vpr protein expression was drastically increased in the ESSV mutant (Fig. 2D, lane 6). As anticipated on the basis of earlier studies (13), mutation of ESSV led to a defect in the liberation of virus particles into the cell supernatant, as indicated by the detection of only small amounts of p24gag (Fig. 2D, lane 6). This has been hypothesized to result from insufficient amounts of intracellular Gag, which is needed to drive virus assembly at the cellular plasma membrane. The expression and normal processing of structural proteins and Vif within the cells were reinstated following the double mutation of ESEvpr (Fig. 2D, lane 9). Furthermore, the failure to efficiently produce virus particles of ESSV-negative clones could be rescued by the −16A>G mutation and the double mutation (Fig. 2D, lanes 8 and 9). Finally, Vpr protein amounts were strongly reduced following the insertion of the −25T>C −16A>G double mutation in the absence of ESSV, demonstrating again that ESEvpr is required for 3′ss A2 activation.
In summary, ESEvpr appears to promote the use of splice sites A2 and D3. Additionally, these observations demonstrate that a functional enhancer is critical for the expression of vpr mRNA, noteworthy even when ESSV is active, indicating a delicate interplay between ESSV and ESEvpr in the regulation of viral HIV-1 exon 3 splicing.
The splicing factors Tra2-alpha and Tra2-beta bind to the ESEvpr sequence.To identify cellular factors that bind to the ESEvpr sequence, RNA affinity purification experiments were performed. Therefore, we incubated short, in vitro-synthesized RNA substrates of either the wild-type or the double-mutated (−25T>C −16A>G) ESEvpr sequence (each n = 5) in HeLa cell nuclear extracts (Fig. 3A). After SDS-PAGE purification, proteins were in-gel digested with trypsin. The peptides obtained were separated by liquid chromatography and mass spectrometry for label-free quantitative analysis. This allowed us to quantify 602 RNA affinity-purified proteins in each of the 10 samples analyzed. To discriminate unspecific binding proteins from proteins bound to the ESEvpr sequence specifically affected by the double mutation, FDR-controlled statistical analysis based on the SAM method (19) was used. This algorithm assigns a score based on the change in protein abundance relative to the standard deviation of repeated measurements and estimates the FDR by using permutations. This approach revealed 6 proteins in the double-mutated ESEvpr group, as well as 12 proteins in the wild-type group, to be significantly enriched (Table 3; Fig. 3B). Eight of those 12 proteins could be assigned to the gene ontology biological process term mRNA processing, representing a significant enrichment of this biological process (FDR-adjusted P value, 0.01). Besides binding to several members of the cleavage stimulation factor complexes, as well as the cleavage polyadenylation-stimulating factor complexes, we found a significant increase in the proteins Tra2-alpha, Tra2-beta, and CUGBP1 in the wild-type ESEvpr sequence. Because of their known role in regulating alternative splicing, we chose them for further validation (22–25).
ESEvpr is bound by the splicing factors Tra2-alpha and Tra2-beta. (A) In vitro-transcribed RNA substrates used for RNA pulldown experiments (dm, double mutation). (B) Volcano plot of RNA binding proteins purified by RNA pulldown with a nonmutated or a mutated ESEvpr sequence with HeLa cell nuclear extract. The precipitated proteins were digested with trypsin and subjected to quantitative mass spectrometry analysis. The x axis of the volcano plot shows the relative difference in protein abundance as calculated by the SAM method, whereas the y axis shows the −log t-test P value of the groupwise comparison of protein abundances. Besides the majority of probably unspecifically binding proteins (circles), some proteins preferably bound to the wild-type ESEvpr sequence (triangles) or the mutated ESEvpr variant (squares). The proteinsTra2-alpha and Tra2-beta were selected for validation experiments. (C) Immunoblot analysis with an antibody specific for Tra2-beta and hnRNPA1 confirmed significantly smaller amounts of Tra2-beta for the double mutant. (D) HeLa cells (2.5 × 105) were transiently cotransfected with 1 μg of each of the HIV-1-based LTR ex2 ex3 splicing reporters, 0.2 μg of SVctat (47); 1 μg of pXGH5 (GH1) as a transfection control, and 1 μg of pcDNA3.1(+), an expression plasmid for Tra2-alpha, Tra2-beta, CUGBP1, and SRSF7. At 30 h posttransfection, total RNA was isolated and subjected to semiquantitative RT-PCR analyses with primers 1544 and 3632. For measurement of equal transfection efficiencies, a separate PCR was carried out with a primer pair (1224/1225) specific for human growth hormone 1 (GH1). (E) RT-PCR analyses of intronless (2-kb) and intron-containing (4-kb) viral mRNA species following the transient transfection of HEK 293T cells with 1 μg of the respective proviral construct and 1 μg of pcDNA3.1(+), an expression plasmid for either Tra2-alpha or Tra2-beta. E, extended exon; dm, double mutation.
Western blot analyses confirmed that while the levels of hnRNP A1 were not changed by the double mutation, Tra2-beta was precipitated with significantly reduced efficiency by the mutated ESEvpr sequence (Fig. 3C, cf. lanes 3 and 4 and lanes 5 and 6). To unravel whether the splicing factors identified are functionally involved in exon 3 splice site activation, we performed coexpression experiments and analyzed their effects on HIV-1 exon 3 and vpr mRNA splicing. In the context of an HIV-1-based minigene (Fig. 3D), coexpression of Tra2-alpha, -beta, and both increased exon 3 splice site activation in the presence of the wild-type ESEvpr sequence (Fig. 3D, cf. lanes 1 to 4), while it failed to promote exon 3 inclusion following inactivation of the enhancer (Fig. 3D, cf. lanes 7 to 10). Coexpression of CUGBP1 or SRSF7, however, did not increase exon 3 splice site activation in either in the context of wild-type ESEvpr (Fig. 3D, cf. lanes 1, 5, and 6) or that of double mutant ESEvpr (Fig. 3D, cf. lanes 7, 11, and 12). Tra2 proteins were also coexpressed together with pNL4-3 and the derived mutants (Fig. 3E), reiterating their role in ESEvpr-controlled exon 3 splice site activation. Once again, it was found that Tra2-alpha and -beta overexpression increased the inclusion of exon 3 in the case of wild-type ESEvpr but not that of double mutant ESEvpr (Fig. 3E, cf. lanes 1 to 6). Therefore, we concluded that Tra2 proteins bind to newly found ESEvpr and are thereby involved in the activation of exon 3 inclusion and vpr mRNA splicing.
A modified U1 snRNA fully complementary to 5′ss D3 strongly activates exon 3 inclusion and vpr mRNA expression.To determine the importance of ESEvpr for exon 3 splice site use under conditions of optimal 5′ss D3 recognition, we generated a 5′-end-mutated U1 snRNA matching all 11 nucleotides of 5′ss D3 (Fig. 4A). HEK 293T cells were transiently cotransfected with this modified U1 snRNA expression vector and proviral DNA containing either wild-type or mutant exon 3 sequences. In general, RT-PCR analysis of RNA isolated from the transfected cells revealed a dramatic shift toward vpr and/or exon 3 spliced mRNAs upon the coexpression of the mutated U1 snRNA, indicated by larger amounts of vpr mRNA species (Fig. 4B, e.g., Vpr3, cf. lanes 1 to 8 and 9 to 16) and the increased levels of exon 3-containing mRNA species (Fig. 4B, e.g., Tat3 [cf. lanes 1 to 8 and 9 to 16] or Nef4 [cf. lanes 1 to 4 and 9 to 12]). On the basis of these results, we concluded that the coexpressed U1 snRNA seemed to assemble correctly into mature snRNPs and that U1—as expected and anticipated by a recent publication (26)—increased 5′ss D3 recognition. Interestingly, the strong increase in exon 3 splice site activation appeared to suppress the inclusion of exon 2 in the viral mRNA species (Fig. 4B, Tat2, cf. lanes 1 and 9), indicating that a balanced exon 3 splicing activity is also necessary to permit the use of exon 2 splice sites A1 and D2. However, in the absence of functional ESEvpr, U1 snRNA coexpression predominantly activated vpr mRNA splicing, while only a minor influence on exon 3 inclusion was observed (Fig. 4B, Vpr and to a lesser extent Tat3, cf. lanes 3 to 4 and 11 to 12), indicating not only that ESEvpr may enhance the early recognition of 5′ss D3 but also supports its use later in the splicing reaction. It is worth noting that inactivation of both ESSV and ESEvpr allowed efficient vpr mRNA splicing and exon 3 inclusion upon the coexpression of modified U1 snRNA (Fig. 4B, Vpr3 and Tat3, lanes 8 and 16), rather arguing for a repressive activity of ESSV that addresses splicing after initial 5′ss recognition and that is counteracted by active ESEvpr. Western blot analysis of the p24 levels within the supernatant suggested that the coexpression of the U1 snRNA could induce excessive exon 3 splicing even in the presence of ESSV, thereby dramatically reducing virus particle production (Fig. 4B, cf. lanes 1 and 2 and lanes 9 and 10). However, binding of the coexpressed U1 snRNA relied on the presence of ESEvpr to elicit excessive activation of the exon 3 splice sites, causing severely reduced viral particle production (Fig. 4B, cf. lanes 9 and 12). When ESSV was disrupted, U1 coexpression efficiently inhibited viral particle production independent of ESEvpr activity (Fig. 4B, cf. lanes 5 to 8 and 13 to 16). These findings emphasize that the complementarity between the 5′ss D3 and the U1 snRNA is not the sole determinant of exon 3 inclusion. However, the absence of functional ESEvpr could be (at least partially) bypassed by increasing base pairing between U1 snRNA and 5′ss D3.
Coexpression of a modified U1 snRNP with full complementarity to 5′ss D3 induces HIV-1 exon 3 splicing and vpr mRNA expression. (A) Schematic drawing of a 5′-end-modified U1 snRNA (right) perfectly matching the 5′ss D3 sequence. Mutated nucleotides are indicated by gray capital letters. Additional base pairing interactions between 5′ss D3 and the optimized 5′ end of the U1 snRNA are indicated by vertical gray lines. (B) HEK 293T cells (2.5 × 105) were transiently cotransfected with 1 μg of both a proviral plasmid and a U1 snRNA expression plasmid. Total RNA was isolated and subjected to RT-PCR analyses. PCR products were resolved by PAGE and stained with ethidium bromide. RT-PCR samples are shown at the top. The main viral mRNA species are indicated on the right. Viral supernatants were collected as well and analyzed for viral p24gag concentrations by immunoblotting (bottom). E, extended exon; dm, double mutation; sn, supernatant.
vpr mRNA expression can be modulated by up and down mutations of 5′ss D3.The use of 3′ss A2 results in the formation of vpr-mRNA but only when splicing at the downstream 5′ss D3 is suppressed because this would remove the Vpr translational initiation codon within intron 3 from the mature transcripts. Remarkably, ESEvpr was shown to be critical for vpr mRNA expression, although it is located close to 5′ss D3 and separated from 3′ss A2 by the repressor ESSV. It was found previously that efficient recognition of a 5′ss by the U1 snRNP exerts positive feedback on the assembly of splicing factors at the upstream 3′ss—most likely via interactions across the exon (27–29). To analyze the interdependence of ESEvpr, 5′ss D3, and vpr mRNA expression, mutations predicted to either decrease (D3 down) or increase (D3 up) the intrinsic strength of the viral 5′ss D3 were tested in the context of a replication-competent provirus (Fig. 5A). Mutations were chosen so that the overlapping Vif ORF was not changed. Following transient transfection of HEK 293T cells with proviral DNA, the exon 3 abundance within the viral mRNA species was determined for each of the mutant proviruses by RT-PCR analysis (Fig. 5B). As expected, the extent of the complementarity between U1 snRNA and the 5′ss basically correlated with the amounts of exon 3 present in the viral transcripts. In the presence of ESEvpr activity, weakening of 5′ss D3 caused a decrease in the levels of exon 3-containing isoforms within both major viral mRNA classes, whereas an increase in the complementarity of 5′ss D3 partially overcame the general repression of exon 3 splicing by dominant negative ESSV (Fig. 5B, lanes 1 to 3). This was in line with the hypothesis that the stability of U1 snRNP binding to a 5′ss plays a pivotal role in the recognition of the entire exon. However, mutant forms of ESEvpr showed no detectable exon 3 inclusion, regardless of their intrinsic 5′ss strength (Fig. 5B, lanes 4 to 6), indicating a strict requirement for functional ESEvpr to enable stable binding of the U1 snRNP to 5′ss D3. Mutant ESSV was also associated with a poor response of the vpr3 mRNA to the distinct 5′ss variants in the experiment shown (Fig. 5B, lanes 7 to 9) and lacked any response in parallel experiments (data not shown). Efficient exon 3 inclusion was detected in each case, irrespective of up or down mutations within D3 (Fig. 5B, e.g., Nef4, lanes 7 to 9), suggesting that in the absence of ESSV activity, recognition of a weaker 5′ss can be compensated for by stronger activation of 3′ss A2. Finally, when both ESSV and ESEvpr were mutated, the intrinsic 5′ss strength again up- or downmodulated the frequency of exon 3 inclusion in the viral mRNAs (Fig. 5B, e.g., Nef4, lanes 10 to 12). This reinforces the notion that in a less favorable environment with regard to enhancer strength, the efficiency of exon inclusion exhibits a higher dependency on the ability of a splice site to bind the U1 snRNP on its own. Taken together, the data show that the overall efficiency of HIV-1 exon 3 splicing is adjusted by the individual strength of the preceding exonic splicing regulatory elements and the intrinsic strength of 5′ss D3. Western blot analyses were consistent with these results and revealed that Vpr expression was under the combined control of ESSV, ESEvpr, and 5′ss D3 (Fig. 5C). Correspondingly, increasing base pairing between the 5′ end of U1 snRNA and 5′ss D3 was accompanied by a higher abundance of Vpr protein within the transfected cells, whereas a reduction of the intrinsic strength showed the opposite effect on Vpr expression (Fig. 5C, lanes 2 to 4 and 11 to 13). However, in the presence of only one intact splicing regulatory element, either ESEvpr or ESSV, exon 3 splicing efficiency was either too low or too high to allow tuning by alterations of 5′ss strength (Fig. 5C, Vpr, lanes 5 to 7 and 8 to 10). These results substantiate the observations that U1 snRNP binding to 5′ss D3 enhances the use of upstream 3′ss A2 and that splicing of exon 3 can be considered the integrated outcome of exonic elements (ESEvpr and ESSV) and the intrinsic strength of 5′ss D3.
5′ss D3 up and down mutations modulate HIV-1 exon 3 splicing and vpr mRNA formation. (A) Silent mutations predicted to decrease or increase the complementarity to the 5′ end of the endogenous U1 snRNA were introduced into viral 5′ss D3. Exonic nucleotides are denoted in uppercase letters, and intronic nucleotides are denoted in lowercase letters. Complementarity and predicted intrinsic strength by HBond score (HBS) and MaxEnt score algorithms are both shown next to the 5′ss sequence. Nucleotides complementary to the U1 snRNA are in capital letters, while mismatches to the U1 snRNA are in lowercase letters. (B) HEK 293T cells (2.5 × 105) were transiently transfected with 1 μg of each of the different infectious clones. RNA was isolated from the cells, DNase I digested, and reverse transcribed. The resultant cDNA served as the DNA template in semiquantitative PCRs using primer pairs E1/I4 and E1/E7 to specifically detect viral 4.0- and 1.8-kb viral mRNAs, respectively. Proviral mutants are shown above the panels. The main HIV-1 mRNA species are indicated at the right. (C) Protein lysates and viral supernatants were collected from HEK 293T cells transfected with 1 μg of pNL4-3 or mutant derivatives. Samples were loaded on 12% SDS-polyacrylamide gels and, after separation, transferred to nitrocellulose membranes. Viral proteins and α-actin (as a loading control) were determined by probing with specific primary antibodies. For detection, appropriate HRP-conjugated antibodies and ECL detection reagent were applied. HBS, HBond score; MaxEnt, MaxEnt score; dm, double mutation; E, extended exon; sn, supernatant; ly, lysate.
Binding of the U1 snRNP to a nonfunctional 5′ss is sufficient to augment splicing at upstream 3′ss A2.The presented results suggest that U1 snRNP binding to the 5′ss fulfills two functions during pre-mRNA splicing; i.e., (i) it enhances the formation of exon definition complexes and therefore promotes recognition of the upstream 3′ss (27–29), and (ii) it commits the bound 5′ss to splice site pairing with a 3′ss across the downstream intron into the prespliceosome (30). It is hypothesized here that vpr mRNA splicing requires exon definition, so that 5′ss D3 is recognized by U1 snRNA but splicing at D3 must occur with lower efficiency. To gain a broader understanding of how vpr mRNA expression is regulated by U1 snRNP binding, 5′ss D3 was replaced with GTV (17), a U1 binding-competent but splicing-incompetent sequence with nonetheless nearly full complementarity to U1 snRNA (Fig. 6A). As a control, 5′ss D3 was also inactivated by a G-to-C mutation at position +1. The resulting set of variants ranged from a functional 5′ss, which supported efficient binding of the U1 snRNP, as well as splicing (D3), to sequences allowing either efficient binding (GTV) or neither efficient binding nor splicing (D3+1G>C). These variants were tested in the context of ESSV-negative or ESSV/ESEvpr double-negative proviruses. Splicing efficiency at 3′ss A2 was determined by semiquantitative RT-PCR analyses.
U1 snRNP binding to a splicing-incompetent 5′ss enhances vpr mRNA expression. (A) 5′ss D3 was replaced with a splicing-incompetent sequence that perfectly matches the free 5′ end of the cellular U1 snRNA except for position +1 (GTV). As a control, 5′ss D3 was disabled for splicing by a G-to-C mutation at position +1, decreasing its complementarity to the U1 snRNA (D3+1G>C). Complementarity patterns are shown next to the 5′ss sequences. Matches to the U1 snRNA are indicated by uppercase letters, and residues not complementary are in lowercase letters. (B) HEK 293T cells (2.5 × 105) were transiently transfected with 1 μg of each of the proviral constructs and analyzed by semiquantitative RT-PCR. RT-PCR products were resolved by PAGE, followed by ethidium bromide staining. Mutants are depicted at the top. Main viral mRNAs are indicated on the right. (C) Cellular lysates and viral supernatants were obtained from transfected HEK 293T cells and loaded onto 12% SDS-polyacrylamide gels. After transfer to nitrocellulose membranes, viral proteins were determined with specific antibodies for p24gag and Vpr. To ensure the loading of equal protein amounts, the membrane was also probed with an antibody to cellular α-actin. (D) Schematic drawing of a 5′-end-modified U1 snRNA perfectly matching the 5′ss D3+1G>C sequence (left). Mutated nucleotides are indicated by gray capital letters. Additional base pairing interactions between 5′ss D3 and the optimized 5′ end of the U1 snRNA are indicated by vertical gray lines. HEK 293T cells (2.5 × 105) were transiently transfected with 1 μg of both proviral pNL4-3 DNA and U1 snRNA expression plasmid. Total RNA and cellular lysates were isolated and subjected to RT-PCR or Western blot analysis (right). E, extended exon; dm, double mutation; sn, supernatant; ly, lysate.
While 5′ss D3 was efficiently used in HEK 293T cells transfected with an ESSV mutant, neither GTV nor D3+1G>C was spliced and thus did not allow the accumulation of exon 3-containing mRNAs (Fig. 6B, Tat3, Env8, and Nef4, lanes 1 to 3). In the presence of ESEvpr, replacement of D3 with GTV or D3+1G>C monotonically increased the tat1-to-vpr mRNA expression ratio (Fig. 6B, Vpr and Tat1, lanes 1 to 3), which was consistent with greater complementarity between GTV and U1 snRNA than between GTV and D3+1G>C. Even in the absence of ESEvpr, wild-type D3 still retained some vpr mRNA expression and exon 3 inclusion, although at a higher level of tat1 mRNA expression (Fig. 6B, Vpr and Tat1, lane 4), while the disruption of both ESEvpr and wild-type D3 abolished all vpr mRNA expression (lanes 5 and 6). These results were in agreement with the hypothesis that U1 snRNP binding to splicing-incompetent U1 snRNA binding sites alone suffices to augment cross-exon interactions and splicing at 3′ss A2.
The change in mRNA expression ratios induced by these mutations was entirely consistent with the decreasing amount of Vpr protein detected by Western blot analysis (Fig. 6C, Vpr). Moreover, it was observed that decreasing vpr mRNA splicing by both reducing U1 snRNA complementarity at D3 and mutating ESEvpr could rescue virus particle production of ESSV-negative provirus (Fig. 6C, p24gag [sn/ly]).
The finding that ESEvpr-dependent binding of U1 snRNA promotes the use of 3′ss A2 was further confirmed by the coexpression of a U1 snRNA fully complementary to splicing-inactive 5′ss D3 (+1G>C) (Fig. 6D), which largely activated vpr mRNA splicing (Fig. 6D, top), as well as Vpr protein expression (Fig. 6D, bottom).
In summary, these findings recapitulate earlier studies showing that the two functions of U1 snRNP binding to the 5′ss can be dissected, supporting (i) the formation of exon definition complexes and (ii) the assembly of a prespliceosome across a downstream intron (27–29).
Splice site recognition is commonly found to be under the combined control of multiple nearby splicing regulatory elements that can either compete or cooperate to regulate splicing activation. It was previously shown that the HIV-1 noncoding leader exon 3 harbors a negative splicing regulatory element—termed ESSV—within its central portion that selectively represses upstream 3′ss A2 (12) and concomitantly inhibits exon 3 inclusion in the different viral mRNA species (13). ESSV disruption results in the strong accumulation of vpr mRNA and exon 3-containing isoforms, which leads to viral replication incompetence. However, ESSV does not act alone; using an in silico-based mutagenesis strategy, we identified an enhancer sequence—termed ESEvpr—upstream of 5′ss D3 that was essential for exon 3 splicing in the context of wild-type ESSV and that provided excess exon 3 splice site activation when ESSV was inactive. This notion was strengthened by the finding that ESSV/ESEvpr double-negative provirus retrieved the capability to efficiently replicate, probably because of reduced exon 3 splicing and a recovery of the ratio of unspliced to spliced viral mRNAs. Therefore, it was concluded that balanced exon 3 splicing is under the combined control of ESSV and ESEvpr (Fig. 7). In addition, we identified the splicing factors Tra2-alpha and Tra2-beta as proteins that bind to the ESEvpr sequence, thereby promoting the inclusion of exon 3 in the viral mRNAs.
HIV-1 exon 3 splicing is under the combined control of ESSV and ESEvpr. Splice site recognition at HIV-1 exon 3 is regulated by ESSV and a novel exonic splicing enhancer (ESEvpr) embedded in the region upstream of 5′ss D3. ESSV is associated with hnRNP A/B proteins, which may multimerize along the 5′ end of exon 3, occluding 3′ss A2. Tra2-alpha and Tra2-beta bind to the ESEvpr sequence, potentially enhancing recruitment of the U1 snRNP to 5′ss D3, which in turn may promote interactions across the upstream exon and activation of 3′ss A2. U1, U1 snRNP.
Interestingly, we further demonstrated that ESEvpr is crucial for vpr mRNA expression. While 3′ss A2 needs to be activated, splicing at 5′ss D3 removes the vpr-coding intron located downstream and therefore needs to be repressed. However, whereas the use of 5′ss D3 in the splicing reaction excludes vpr mRNA expression, we showed that its early recognition by the U1 snRNP is required. This is in line with the exon definition hypothesis in which binding of the U1 snRNP to the 5′ss promotes interactions across the exon and thereby activation of the upstream 3′ss (27–29). By use of 5′ss D3 variations with either decreased or increased complementarity to the U1 snRNA, we demonstrated that the splicing enhancer promotes U1 snRNP recruitment and thereby exon definition. Accordingly, up and down mutations of 5′ss D3 increased or decreased vpr mRNA expression levels, underlining the importance of 5′ss D3 recognition for regulated levels of Vpr. These results recapitulated previous findings demonstrating that reduced recognition of 5′ss D2 leads to lower levels of Vif expression (29). Notably, vif mRNA formation requires the activation of upstream 3′ss A1, while the downstream intron needs to be retained, which is comparable to the mechanism providing vpr mRNA, as shown in the accompanying paper by Widera et al. (31). The use of splice sites D2 and D3 in the splicing reaction counteracts the expression of vif and vpr mRNAs, respectively. However, the aforementioned observations indicate that their initial recognition by the U1 snRNA is essential for obtaining Vif and Vpr expression. This idea was highlighted by the finding that ESEvpr-dependent recognition of splicing-incompetent U1 snRNP binding sites sufficed to enhance exon definition and activation of 3′ss A2 for vpr mRNA formation. Therefore, our results—in agreement with earlier findings (29)—indicate that recognition of 5′ss D2 and D3 must occur with higher efficiency than their use in the splicing reaction in order to permit the expression of Vif and Vpr. It was shown previously that exon definition complexes can readily progress into intron definition complexes, finally performing the splicing reaction across an upstream intron (32). However, it was also found that splicing regulatory elements can negatively control the conversion of exon-to-intron definition complexes (33). Therefore, a model seems plausible in which ESEvpr promotes the formation of an exon definition complex that then can be efficiently changed into an upstream intron definition complex but in which the same process may occur less efficiently across the downstream intron, ultimately allowing the formation of vpr mRNAs. Herein, splicing factors binding to the vpr intron might play a particular role through interference with the assembly of the spliceosome, as described for the polypyrimidine tract binding protein (33).
Alternatively, it has been suggested that high-mobility group A protein 1a renders the U1 snRNP inactive at 5′ss D3 for progression with the splicing reaction, thereby mediating vpr mRNA expression (34). Although disabled from carrying out the splicing reaction, this “dead-end” U1 snRNP should maintain its capability to enhance the assembly of exon definition complexes and thus activation of 3′ss A2. However, since we showed here that ESEvpr-mediated early recognition of 5′ss D3 is critical for Vpr expression, interference with the removal of the downstream vpr intron appears to occur at a later step during spliceosome assembly. Whether this is achieved by either one of the alternative suggested models needs to be further clarified.
We thank Joshua Madsen and Martin C. Stoltzfus for critical readings of the manuscript and helpful comments. We are grateful to Alain Cochrane and Kinji Ohno for providing expression plasmids for Tra2-alpha, -beta, and CUGBP1. We thank Björn Wefers for excellent technical assistance.
The following reagents were obtained through the NIH AIDS Research and Reference Reagent Program, Division of AIDS, NIAID, NIH: HIV-1HXB2 Vif antiserum from Dana Gabuzda and HIV-1NL4-3 Vpr antiserum (1-46) from Jeffrey Kopp. This work was supported by the DFG (SCHA 909/3-1); the Heinz Ansmann Foundation for AIDS Research, Düsseldorf (H.S.); and the Jürgen Manchot Stiftung (H.S).
Accepted manuscript posted online 19 December 2012.
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. 2010. Exon definition complexes contain the tri-snRNP and can be directly converted into B-like precatalytic splicing complexes. Mol. Cell 38:223–235.
. 2008. Polypyrimidine tract binding protein controls the transition from exon definition to an intron defined spliceosome. Nat. Struct. Mol. Biol. 15:183–191.
. 2011. HMGA1a is involved in specific splice site regulation of human immunodeficiency virus type 1. Biochem. Biophys. Res. Commun. 406:512–517.
. 2012. The HIV-1 major splice donor D1 is activated by splicing enhancer elements within the leader region and the p17-inhibitory sequence. Virology 432:133–145.
. 2008. Negative and positive mRNA splicing elements act competitively to regulate human immunodeficiency virus type 1 vif gene expression. J. Virol. 82:3921–3931.
. 2001. A second exon splicing silencer within human immunodeficiency virus type 1 tat exon 2 represses splicing of Tat mRNA and binds protein hnRNP H. J. Biol. Chem. 276:40464–40475.
. 2006. Biochemical and NMR study on the competition between proteins SC35, SRp40, and heterogeneous nuclear ribonucleoprotein A1 at the HIV-1 Tat exon 2 splicing site. J. Biol. Chem. 281:37159–37174.
. 2004. SC35 and heterogeneous nuclear ribonucleoprotein A/B proteins bind to a juxtaposed exonic splicing enhancer/exonic splicing silencer element to regulate HIV-1 tat exon 2 splicing. J. Biol. Chem. 279:10077–10084.
. 1994. Presence of negative and positive cis-acting RNA splicing elements within and flanking the first tat coding exon of human immunodeficiency virus type 1. Mol. Cell. Biol. 14:3960–3970.
. 1999. hnRNP A/B proteins are required for inhibition of HIV-1 pre-mRNA splicing. EMBO J. 18:4060–4067.
. 1997. Splicing efficiency of human immunodeficiency virus type 1 tat RNA is determined by both a suboptimal 3′ splice site and a 10 nucleotide exon splicing silencer element located within tat exon 2. Nucleic Acids Res. 25:861–867.
. 1995. Identification of positive and negative splicing regulatory elements within the terminal tat-rev exon of human immunodeficiency virus type 1. Mol. Cell. Biol. 15:4597–4605.
. 1995. Presence of exon splicing silencers within human immunodeficiency virus type 1 tat exon 2 and tat-rev exon 3: evidence for inhibition mediated by cellular factors. Mol. Cell. Biol. 15:4606–4615.
. 1998. The exon splicing silencer in human immunodeficiency virus type 1 Tat exon 3 is bipartite and acts early in spliceosome assembly. Mol. Cell. Biol. 18:5404–5413.
. 2006. Role of viral splicing elements and cellular RNA binding proteins in regulation of HIV-1 alternative RNA splicing. Curr. HIV Res. 4:43–55.
. 1993. Use of DNA end joining activity of a Xenopus laevis egg extract for construction of deletions and expression vectors for HIV-1 Tat and Rev proteins. Gene 124:275–280. | https://jvi.asm.org/content/87/5/2721?ijkey=000b45c48026c15277b65316c2334e24f08c7269&keytype2=tf_ipsecsha |
The Statue of Liberty is a colossal neoclassical sculpture on Liberty Island in New York Harbor, designed by Frédéric Bartholdi and dedicated on October 28, 1886.
Running of the bulls
Spain
The Running of the Bulls is a practice that involves running in front of a small group (typically a dozen) of bulls that have been let loose, on a course of a sectioned-off subset of a town's streets. The most famous one is in Pamplona.
The Alhambra
Spain
The Alhambra is a palace and fortress complex constructed during the mid 14th century by the Moorish rulers of the Emirate of Granada in Al-Andalus, occupying the top of the hill of the Assabica on the southeastern border of the city of Granada in the Autonomous Community of Andalusia.
The Pyramids
Egypt
The Egyptian pyramids are ancient pyramid-shaped masonry structures located in Egypt. The most famous Egyptian pyramids are those found at Giza, on the outskirts of Cairo.
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The Sagrada Familia
Spain
The Sagrada Família is a large Roman Catholic church in Barcelona, Catalonia, Spain, designed by Catalan architect Antoni Gaudí (1852–1926). Although incomplete, the church is a UNESCO World Heritage Site and in November 2010 was consecrated and proclaimed a minor basilica by Pope Benedict XVI. | https://www.sullivanslist.com/user/131 |
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The IRS informed the National Association of College and University Business Officers (NACUBO) that no more extensions are planned for colleges and universities to comply with a statutory change for filing Form 1098-T, Tuition Statement (reporting amounts billed for tuition and related expenses). NACUBO had requested that the IRS again delay implementation of the rules, which require that schools filing the 2018 form, used by students claiming the American opportunity tax credit, complete Box 1, “Payments received for qualified tuition and related expenses.” The “amounts billed” option in Box 2 will no longer be available, although schools can still choose Box 2 for 2017. See our prior coverage here.
AICPA and NYSBA Suggest Changes to Proposed Partnership Audit Regulations
The American Institute of Certified Public Accountants has suggested changes to the proposed partnership audit regulations and has identified areas that need clarification, including the partner-level penalty defenses, the option to elect out for some partnerships, and the procedures for appointing and replacing a partnership representative. The New York State Bar Association’s suggestions can be found here.
Information contained in this publication should not be construed as legal advice or opinion or as a substitute for the advice of counsel. The articles by these authors may have first appeared in other publications. The content provided is for educational and informational purposes for the use of clients and others who may be interested in the subject matter. We recommend that readers seek specific advice from counsel about particular matters of interest.
Copyright © 2017 Stradley Ronon Stevens & Young, LLP. All rights reserved.
Stradley Ronon served as counsel in the structured sale of Agilis Med Holdings, LLC, the parent company of Agile Medical, one of the largest e-commerce retailers of sleep apnea and respiratory equipment in the U.S., to AdaptHealth LLC (NasaqCM: AHCO), a full-service home medical equipment company.
Stradley Ronon is representing Atmos, a Japan-based global streetwear and sneaker brand with 49 stores across Asia and the U.S., in its sale to Foot Locker for $360 million.
Stradley Ronon represented Nuveen-sponsored closed-end fund Nuveen Core Plus Impact Fund (NYSE: NPCT) in connection with its initial public offering, which raised $575 million.
Stradley Ronon attorneys Karl Myers and Melissa Perry won a decisive appellate victory in the Pennsylvania Supreme Court, which upheld the use of comfort dogs for vulnerable testifying witnesses on behalf of several pro bono clients, the Animal Legal Defense Fund (ALDF), Association of Prosecuting Attorneys (APA) and Lutheran Church Charities (LCC).
Stradley Ronon represented Anna Holdings LLC, the owner of a dispensary permittee operating under the name Keystone Shops, in the sale of three Philadelphia-area medical cannabis dispensaries to Trulieve Cannabis Corp.
Private equity firm and management team acquires the business of the global investment manager
Stradley Ronon advised Customers Bancorp, Inc. (Customers) the parent company of Customers Bank, a full-service bank with $19.6 billion in assets, in the closing of its underwritten public offering of $100 million aggregate principal amount of its 2.875% Fixed-to-Floating Rate Senior Notes due 2031.
Stradley Ronon’s investment management group represented Dimensional Funds and its Independent Directors in the launch of Dimensional ETF (exchange-traded funds) Trust, three new activity managed, tax-efficient ETFs.
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Our size – 200 lawyers – is a distinct advantage. Clients receive concierge service because every client is important in a firm our size. No one gets lost in the shuffle. Our decision-making is nimble; we can tailor fee arrangements, staffing and technology to your needs quickly and efficiently. We punch above our weight; our lawyer talent is prodigious. Finally, we have a global reach without the overhead that often gets passed along to clients. We are the only firm in Philadelphia affiliated with Meritas, an international network of independent litigation and transactional law firms that provides you with access to first-rate lawyers and law firms in more than 230 markets, including 97 countries abroad. Read more...
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Diversity
Stradley Ronon might sound to you like every other big law firm out there: sophisticated clients, high-rise offices and exceptional attorneys. So what sets Stradley apart? It could be our commitment to providing young attorneys with immediate real-world experience. It could be that Stradley services some of the biggest clients in the country but still maintains a friendly, small-firm atmosphere. Or it could be that so many of our associates and partners are Stradley “lifers” – attorneys who joined the firm as summer associates and have built their careers with us. We invite you to get to know us better.
D&I Education and Training Resources List
Stradley is committed to diversity for many reasons: because it provides a voice to the unique perspectives of all our attorneys; because it reflects the evolving face of the legal profession; and because it allows us to provide the highest-quality services to our clients, who are as diverse as we are. Most of all, though, Stradley is committed to diversity because we believe there is strength in the differences among our experiences and world views. Read More...
Copyright © 2022 Stradley Ronon Stevens & Young, LLP. All rights reserved. Review our | https://www.stradley.com/insights/publications/2017/08/tax-insights-august-23-2017 |
The incumbent serves as a Readjustment Counselor at a Vet Center in the Readjustment Counseling Service (RCS) providing direct counseling services, outreach, referral, and follow-up care coordination to eligible individuals. In this capacity, the incumbent functions as a member of a small multi-disciplinary team of 4 or more members, including social workers, other mental health professionals, outreach workers and office managers.
Learn more about this agency
Responsibilities
This is an OPEN CONTINUOUS announcement
The incumbent will be responsible for providing services to eligible individuals who are experiencing a variety of military related psychological and psychosocial problems. Incumbents must possess the knowledge and experience to independently implement psychotherapeutic modalities in working with individuals, families, and groups to enhance the readjustment of Veterans, active duty Service members, and their families. Functions and responsibilities include, but are not limited to:
Intake screening and assessment: Complete all readjustment counseling intake procedures, including an assessment of risk for self-harm, military history and military related psychosocial stressors, and assess family readjustment stressors and document identified behaviors or symptoms.
Readjustment counseling goal setting and service planning: Develop and periodically update an individualized readjustment counseling service plan that reflects a course of therapeutic and psychosocial interventions, including identification of achievable goals and measurable outcomes.
Direct readjustment counseling service provision: Incorporate complex multiple causation in differential psychosocial assessment and direct service delivery to eligible Veterans, including making psychosocial and psychiatric diagnostic inferences within the clinical scope of practice.
Risk assessment and crisis intervention: Conduct timely assessment of Veterans in crisis to identify immediate needs, evaluate risk, and initiate safety planning as appropriate.
Care coordination: Links the Veteran with other community services, resources, and opportunities to maximize the Veteran's independence, health, and well-being.
Coordinating readjustment counseling and outreach: Maintain active coordination with Vet Center outreach workers to ensure seamless referral to Vet Center services for those individuals engaged in the community at outreach events that may require follow-up readjustment counseling.
Team cohesion and coordination: Actively participate in staff meetings designed to promote team building and staff development.
Documentation/administrative responsibilities/consultation: Document all clinical interactions with eligible individuals and episodes of care coordination on their behalf as required by policy.
Other duties as assigned
Work Schedule: Monday
Financial Disclosure Report: Not required
NOTE: Readjustment Counselors are multi-disciplinary and can be filled by: Social Worker, Licensed Professional Mental Health Counselor, Marriage & Family Therapist, and Psychologist
Travel Required
Occasional travel
Supervisory status
No
Promotion Potential
11
0185 Social Work
Similar jobs
Community Mental Health Social Workers
Family Social Workers
Mental Health Social Workers
Psychiatric Social Workers
Social Workers
Social Workers, All Other
Requirements
Help
Requirements
Conditions of Employment
You must be a U.S. Citizen to apply for this job
Selective Service Registration is required for males born after 12/31/1959
You may be required to serve a probationary period
Subject to a background/security investigation
Must be proficient in written and spoken English
Selected applicants will be required to complete an online onboarding process
Qualifications
Basic Requirements:
United States Citizenship: Non-citizens may only be appointed when it is not possible to recruit qualified citizens in accordance with VA Policy.
Education: Must have a master's degree in social work from a school of social work fully accredited by the Council on Social Work Education (CSWE). Graduates of schools of social work that are in candidacy status do not meet this requirement until the school of social work is fully accredited. A doctoral degree in social work may not be substituted for the master's degree in social work. Verification of the degree can be made by going to the CSWE website to verify if that social work degree meets the accreditation standards for a Master of Social Work.
Licensure: Persons hired or reassigned to social worker positions in the GS-185 series in VHA must be licensed or certified by a state to independently practice social work at the master's degree level.
Exception: VHA may waive the licensure or certification requirement for persons who are otherwise qualified, pending completion of state prerequisites for licensure/certification examinations. This exception only applies at the GS-9 grade level. For the GS-11 grade level, the candidate must be licensed or certified.
May qualify based on being covered by the Grandfathering Provision as described in the VA Qualification Standard for this occupation (only applicable to current VHA employees who are in this occupation and meet the criteria).
Grade Determinations:
GS-9: Experience, Education, and Licensure: None beyond the basic requirements.In addition to the experience above, the candidate must demonstrate all of the following Knowledge, Skills, and Abilities (KSAs):
Ability to work with Veterans and family members from various socioeconomic, cultural, ethnic, educational, and other diversified backgrounds utilizing counseling skills.
Ability to assess the psychosocial functioning and needs of Veterans and their family members, and to formulate and implement a treatment plan, identifying the Veterans problems, strengths, weaknesses, coping skills, and assistance needed.
Ability to implement treatment modalities in working with individuals, families, and groups to achieve treatment goals. This requires judgment and skill in utilizing supportive, problem solving, or crisis intervention techniques.
Ability to establish and maintain effective working relationships and communicate with clients, staff, and representatives of community agencies.
Fundamental knowledge of medical and mental health diagnoses, disabilities, and treatment procedures. This includes acute, chronic, and traumatic illnesses/injuries; common medications and their effects/side effects; and medical terminology.
GS-11 Experience and Licensure: Completion of a minimum of one year of post-MSW experience equivalent to the GS-9 grade level in the field of health care or other social work-related settings (VA or non-VA experience) and licensure or certification in a state at the independent practice level. OR
Education: In addition to the basic requirements, a doctoral degree in social work from a school of social work may be substituted for the required one year of professional social work experience in a clinical setting.
In addition to the experience above, candidates must demonstrate all of the following Knowledge, Skills, and Abilities (KSAs):
Knowledge of community resources, how to make appropriate referrals to community and other governmental agencies for services, and ability to coordinate services.
Skill in independently conducting psychosocial assessments and treatment interventions to a wide variety of individuals from various socio-economic, cultural, ethnic, educational and other diversified backgrounds.
Knowledge of medical and mental health diagnoses, disabilities and treatment procedures (i.e. acute, chronic and traumatic illnesses/injuries, common medications and their effects/side effects, and medical terminology) to formulate a treatment plan.
Skill in independently implementing different treatment modalities in working with individuals, families, and groups who are experiencing a variety of psychiatric, medical, and social problems to achieve treatment goals.
Ability to provide consultation services to new social workers, social work graduate students, and other staff about the psychosocial needs of patients and the impact of psychosocial problems on health care and compliance with treatment.
References: VA Handbook 5005/120, Part II, Appendix G39, dated September 10, 2019.
Physical Requirements:Moderate lifting 15-44 lbs, walking and standing up to 4 hours, kneeling 1-2 hours, Operation of motor vehicle, both eyes required, depth perception, ability to distinguish colors, hearing (aid permitted), glasses and prosthesis allowed, able to hear whispered voices, and emotional and mental stability; Environmental factors: outside and inside, electrical energy, working closely with others, protracted or irregular hours of work, and repetitive computer use.
Education
IMPORTANT: A transcript must be submitted with your application if you are basing all or part of your qualifications on education.
Note: Only education or degrees recognized by the U.S. Department of Education from accredited colleges, universities, schools, or institutions may be used to qualify for Federal employment. You can verify your education here: http://ope.ed.gov/accreditation/. If you are using foreign education to meet qualification requirements, you must send a Certificate of Foreign Equivalency with your transcript in order to receive credit for that education. For further information, visit: http://www.ed.gov/about/offices/list/ous/international/usnei/us/edlite-visitus-forrecog.html. | https://jobs.livecareer.com/l/social-worker-community-care-network-case-manager-department-of-veterans-affairs-8339ec7181343635fa8fc3bba02502b4?sid=86a73b70-273f-4f6c-a5b0-7faaec192c9c&q=Psychiatrics&l=Seffner%2C+Florida&fid=&score=&resumenm=&isresumesearch=false&resumeguid=&bgclr=3 |
Engulf yourself in the world of Jordan Berman and take a journey through the awkward phase of what it means to “settle down.” Come along for the ride as he experiences heartbreak, a rift with his BFF, the rollercoaster of one’s late-twenties, and the ever-relatable feeling of worry about never finding someone to love. Come for the drama, stay for the tart comedy, and leave with poignant memories of a lovely story about humanity and human foibles.
February 19 and 20 at 7:30 p.m.
Thanksgiving Play
Native American playwright Larissa FastHorse has written a searing comedy about four white people charged with creating a politically correct and respectful elementary school pageant about the first Thanksgiving. Through biting humor, THANKSGIVING PLAY takes on the mythology of people simply finding peace over a bountiful meal in what FastHorse herself calls “performative wokeness.” American Theatre Magazine named this satirical powerhouse of a script as one of the ten most produced plays of the 2019–20 season. Don’t miss this wickedly funny play by an important voice in the American Theatre.
March 5 & 6 @ 7:30 pm
Dog Sees God
March 19 & 20 at 7:30 pm
Cry It Out
April 9 & 10 at 7:30 pm
Henry IV Part One
April 15 & 16 at 7:30 pm
Red Bike
April 30 & May 1 at 7:30 pm
PLAY ON! Shakespeare Series
Play On! Shakespeare’s mission is to enhance the understanding of Shakespeare’s plays in performance for theatre professionals, students, and audiences by engaging with contemporary translations and adaptations. University of Evansville Theatre is thrilled to partner with Play On! Shakespeare to present three projects during the Spring 2021 semester.
Pericles
Invited Reading Only
February 7
Edward III
Invited Reading Only
February 14
Richard II
Open to the public! Check back here for link to tickets.
March 26 and 27 at 7:30 p.m.
All three projects were commissioned by Oregon Shakespeare Festival, Artistic Director Bill Rauch, Executive Director Cynthia Rider, as part of Play On! Shakespeare. To learn more about this organization, visit https://playonfestival.org.
Office Phone:
812-488-2744
Office Email:
[email protected]
Office Location: | https://www.evansville.edu/majors/theatre/current.cfm |
System variables are, for the purposes of this guide, variables that certain commands will use or modify without asking (i.e. without supplying them in the command's arguments). This is a somewhat ill-defined category, and in fact the system variables we'll discuss are of a somewhat miscellaneous nature. They include equation and plot variables, window and table parameters, statistical variables, and finance variables.
Contents
Equation variables and plot variables
The equation variables include Y,,0,, through Y,,9,, (function variables), X,,1T,, and Y,,1T,, through X,,6T,, and Y,,6T,, (parametric variables), r,,1,, through r,,6,, (polar variables), and u, v, w (sequential variables). They are used when graphing an equation on the graph screen. From the point of view of a programmer, it's important to realize that these variables are actually Strings in disguise. That is to say, they are stored internally as strings, though the string operations and commands won't work on them. Instead, the following operations are valid for equations:
- Evaluating them, by using the equation variable in an expression.
- Evaluating them at a point, using the syntax EqVar(value) - for example, Y,,1,,(5). This will plug value in for the independent variable - X, T, θ, or n for function, parametric, polar, and sequential variables respectively.
- Storing a string to them.
- Using the Equ►String( or String►Equ( command to go back and forth from string to equation form.
In addition to their string contents, an equation variable contains two other pieces of information - the graphing style, set with the GraphStyle( command, and the state (enabled or disabled), set with the FnOn and FnOff commands. There is unfortunately, no way to find out which graphing style or state a variable is in, you can only set these to a value.
Plot variables are used for displaying data graphically, in one of six styles: scatter plot, line plot, histogram, box plot, modified box plot, and normal plot. What the variables actually store varies with the plot type, but includes one or more data lists, and settings. All of this can be set with the PlotN( command. All or some plots can be turned on and off with the PlotsOn and PlotsOff commands. The Select( command is also useful for dealing with plots.
Window and table parameters
These store auxiliary information about the graphing window and include:
- Xmin, Xmax, Ymin, Ymax -- determine the lower and upper X and Y bounds of the graph screen itself.
- ΔX, ΔY -- defined to be (Xmax-Xmin)/94 and (Ymax-Ymin)/62 respectively; storing to them will change Xmax and Ymax to arrange this.
- Xscl, Yscl -- determine how wide the marks on the axes (and, with GridOn, the grid marks) are spaced.
- Xres -- determines the quality of Function graphing (how many pixels apart points are calculated), and must be an integer 1-8.
- Tmin, Tmax, Tstep -- determine the values at which T is evaluated in Parametric graphing (as though it were a For( loop).
- θmin, θmax, θstep -- the same idea, but for the θ variable in Polar graphing.
- nMin, nMax -- bounds for the n variable to evaluate (the step is always 1; also, these must be integers).
- u(nMin), v(nMin), w(nMin) - override the value of u, v, and w at nMin, for the purposes of recursively defining them.
- PlotStart, PlotStep -- actually have nothing to do with plots, but determine which values of n are graphed.
- all of the above have a Z equivalent (ZXmin, Zθstep, etc.) that's used by ZoomSto and ZoomRcl.
- XFact, YFact -- the scaling factors for Zoom In and Zoom Out.
- TblStart, ΔTbl -- the starting value and step for the independent variable in the table screen.
- TblInput -- a 7-element list of the values of a variable in the table.
When transferring variables between calculators, or grouping variables, these variables are grouped in Window, rclWindow, and TblSet.
You can store to each of the window variables directly, or use the Zoom commands, which affect multiple variables at once to achieve some effect.
Statistical Variables
These cannot be stored to, although their values can be used - instead they are modified by statistical commands, which use them as output. See the Statistics pages for more information.
Finance Variables
The seven finance variables are N, I%, PV, PMT, FV, P/Y, and C/Y. They are used by the finance solver tool; to use the solver in a program, set these variables to the values you want, choose one of the options Pmt_End or Pmt_Bgn, and then use the value tmv_VAR (where VAR is the variable you want to solve for).
Another use of the finance variables is in speed-critical sections of code: they are faster to access even than Ans. Somewhat off-setting this advantage is the fact that they are two byte tokens, and that they don't work as the variables of a seq( command or a For( loop (they will actually throw a ERR:SYNTAX error) or [[|is|IS>(]] and DS>(. In addition the value of C/Y is altered when P/Y is altered. | https://learn.cemetech.net/index.php?title=TI-BASIC:System_Variables&printable=yes |
Your child will have a number of tests to investigate their symptoms and confirm a diagnosis of Hodgkin disease, including:
- medical history and physical examination
- blood tests
- medical imaging, which may include
- chest X-ray
- computed tomography (CT) scan
- magnetic resonance imaging (MRI)
- positron emission tomography (PET) scan
- lymph node biopsy – where a small sample of a lymph node is removed to be examined under a microscope
- bone marrow aspiration and biopsy – where a sample of bone marrow and a small piece of bone are taken to be examined under a microscope.
These tests are explained in more detail in How is cancer diagnosed?.
Staging
If your child is diagnosed with Hodgkin disease, some of the diagnostic tests will also help to stage the tumour. Staging determines where the tumour is, how big it is, which nearby organs are involved and whether it has spread to other parts of the body. This is important to determine the outlook (prognosis) for your child, and to decide on the best options for treatment.
There are different ways to assess the stage or extent of disease. One of the most common ways of describing stages for Hodgkin disease is as follows:
- Stage I – the cancer is found only in 1 lymph node area or lymphoid organ (such as the thymus), or only in 1 organ outside the lymphatic system.
- Stage II – the cancer is found in 2 or more lymph node areas on the same side of the body (either both above or both below the diaphragm, which separates the chest and the abdomen), or the cancer has spread from 1 lymph node area into 1 nearby organ.
- Stage III – either the cancer is found in more than 1 lymph node area on both sides of the diaphragm (both above and below), or it is found in lymph node areas both above and below the diaphragm and has spread to a nearby organ and/or the spleen.
- Stage IV – the cancer has spread to 1 or more organs outside the lymphatic system; or it is found in 2 organs in distant parts of the body (but not in the nearby lymph nodes); or it is in the liver, bone marrow, lungs or cerebrospinal fluid. | https://childrenscancer.canceraustralia.gov.au/types-childrens-cancers/hodgkin-disease/diagnosis |
During that period he undertook an intense study of the human anatomy, filling his sketchbook with drawings that would later help to inspire the anatomical collection titled "Ode To Anatomy."
Through his uncanny mastery of detail and precision of the human form one cannot help but notice the influence of Old Masters such as Leonardo Da Vinci and Marco d’Agrate.
In 1999, he and his brother Matt founded their mural painting company, Lmc Murals & Fine Art. Their work can be found in corporate and private collections around the United States, Europe, and the Middle East. In 2009, he opened a studio and gallery in Johnson City, Texas where he continues to create murals and fine art, as well as work on art restoration.
He is a member of the Oil Painters of America and his work has been featured in the Texas Country Reporter show and magazines like American Art Collector, Country Accents, Texas Monthly, U.S. News and World Report. He is represented by RoGallery in NYC, Ny, the Adobe Western Art Gallery in Fort Worth, Tx, and The Fredericksburg Good Art Co. in Fredericksburg, Tx. | https://www.leecasbeer.com/theartist |
Dear Writing Huntress,
This is my 3rd year afield. My 1st year, I started hunting deer in September and went all the way until the end of turkey season in May. I essentially took the summer off, reading books and lazing about, silently pleading for fall to arrive. Unfortunately, by late summer, I had to play catch up for the upcoming season; my arms were weak and I reduced the poundage on my bow, my hunting felt sluggish and it even felt like my senses were off. Now that I’m into my second pre-season summer, I want to spend it productively in order to make the best of my upcoming hunts. What should I be doing during this time? What have you done in the past that has helped your hunting game come September 1st?
Sincerely,
Off-Season in Oregon
Dear Off-Season,
Your plight is a common one, especially for huntresses quasi-new to the sport, with no prior pre-season summer experiences.
During middle and high school, I played hockey from early September to late April. We would generally have a few weeks off, until summer training began, where we’d endure hours of workouts, off-ice drills and absolutely dreadful dry-land practices. I didn’t understand the importance of these practices, until the end of my first summer, when I realized I had learned new skills, sharpened my stick handling, kept my muscles in check and grew closer with my teammates. Hockey, like hunting, doesn’t bode well with time off.
As much as you yearn to take your book outside, get some sun and relax, it’s important, as you’ve already learned, to hone your skills during summer. Without further adieu, allow me to introduce the Writing Huntress’s summer activity list.
Bowfishing
I’ve said it time and time again — bowfishing rocks. It’s awesome for huntresses who want to become better and stronger bow hunters. Bowfishing forces shooters to become more accurate and more comfortable with their bows. Also, it’s super fun. I’ll be writing more about how to bowfish in my next column so be sure to check it out!
Fishing
Fishing is a great way to get out in the elements and enjoy nature, when air conditioning and your couch are beckoning. Fishing can be relaxing, as in flinging cast after cast in the direction of sunfish and perch, but it can also be high-energy and exhausting, especially in the case of deep-sea fishing. Either way, get out and cast a line, you’ll be out in nature and also able to get a nice tan.
Go at it alone
Use this summer to accrue new life skills via friends, experts in the area, or classes a la “Becoming an Outdoors-Woman.” Amass a new skill each week, from foraging to fire making and everything in between. Cap your summer of learning off with a solo camping trip, where you can put your newfound skills to good use, thus preparing for surprise nights afield or pack-in hunting situations.
Challenge Yourself
It’s the age of cold-water, push-up, and cinnamon challenges. While I don’t suggest the latter (Believe me, I did it, and it wasn’t pretty. Lots of cinnamon sneezes.), I am advocating for a summer workout challenge geared toward whatever hunting you plan on doing during season. Personally, now that I’ve moved to Texas, and shall be facing hikes up canyons and across vast tracks of desert, I’ve begun a running regime, not only to acclimate myself to the heat (more on that on a future Ask Writing Huntress column) but also to keep in hunting shape.
Shoot!
I don’t think I can ever stress the importance of keeping your shotgun clean and well fired during the off-season. The best way to ensure your shooting skills thrive during the summer is to visit your local skeet range. My personal favorite shotgun workout is at the 5-Stand range. 5-Stand helps prepare for the erratic flight of oncoming ducks or doves and teaches shooters to keep their heads on a swivel.
Continuous firing of your bow is necessary, too. Either set up a target range in your backyard, or join one in your local community and get shooting! Increase your yardage as the summer wears on, and by September 1st, you’ll be shooting farther than you ever have.
Off-Season, I hope you spend this summer sharpening your game for the upcoming hunting season. Have fun, be safe and remember to relax a little, too!
Happy Hunting,
WH
How could you forget competitions? It’s not about winning, it’s about the fellowship, getting more woods time, and shooting. Find the 3D shoots in your area & gather up a couple friends. They’re so much fun, and they’ll help you gauge distances and shoot in “real life” type scenarios, with trees and uneven ground. Did I mention they’re fun? | https://www.womensoutdoornews.com/2014/06/ask-writing-huntress-activities-hunting-season/ |
This record is currently in review state, the data hasn’t been validated yet.
Trajectory Optimization for Wheeled-Legged Quadrupedal Robots Driving in Challenging Terrain
AuthorShow all
Date2020-07-01
Type
- Journal Article
Abstract
Wheeled-legged robots are an attractive solution for versatile locomotion in challenging terrain. They combine the speed and efficiency of wheels with the ability of legs to traverse challenging terrain. In this letter, we present a trajectory optimization formulation for wheeled-legged robots that optimizes over the base and wheels' positions and forces and takes into account the terrain information while computing the plans. This enables us to find optimal driving motions over challenging terrain. The robot is modeled as a single rigid-body, which allows us to plan complex motions and still keep a low computational complexity to solve the optimization quickly. The terrain map, together with the use of a stability constraint, allows the optimizer to generate feasible motions that cannot be discovered without taking the terrain information into account. The optimization is formulated as a Nonlinear Programming (NLP) problem and the reference motions are tracked by a hierarchical whole-body controller that computes the torque actuation commands for the robot. The trajectories have been experimentally verified on quadrupedal robot ANYmal equipped with non-steerable torque-controlled wheels. Our trajectory optimization framework enables wheeled quadrupedal robots to drive over challenging terrain, e.g., steps, slopes, stairs, while negotiating these obstacles with dynamic motions. Show more
External links
Journal / seriesIEEE Robotics and Automation Letters
Volume
Pages / Article No.
PublisherIEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
SubjectLegged robots; wheeled robots; motion planning; optimization and optimal control
Organisational unit03737 - Siegwart, Roland Y. / Siegwart, Roland Y. | https://www.research-collection.ethz.ch/handle/20.500.11850/437203 |
Designing and building complex systems, such as aircraft, space vehicles, marine vessels, marine platforms such as oil rigs, land vehicles such as automobiles and trucks, and the like, is a complex process that involves several disciplines. For example, typically several years of design, testing, analysis, and systems integration are performed before a complex system is put into operation. Furthermore, before a component, subassembly, or assembly is built, a design for the component, subassembly, or assembly is analyzed.
Such an analysis typically entails generating a mathematical model, such as a finite element model, of the component, subassembly, or assembly. The finite element model is a three dimensional, mathematical definition of a component. The model includes surfaces and exhibits geometric properties, material properties, mass, stiffness, and the like. The finite element model can be subjected to static and dynamic testing. Thus, use of mathematical models such as finite element models greatly reduces time and labor to analyze components over building, testing, and analyzing physical models.
However, generating finite element models of components or subassemblies in complex systems, using currently known methods, is a time-consuming and labor-intensive process. Further, generating finite element models of components in complex systems entails engineering efforts across several disciplines. For example, developing a finite element model for all of the major components for mounting an engine under a wing of a commercial airplane, involves a cross-disciplinary team of loads engineers, stress engineers, designers, and weights engineers.
Typically, engineers from each discipline will develop, from a set of requirements, a preliminary design document. From the preliminary design document, a designer configures a two-dimensional centerline preliminary design drawing. The preliminary design drawing represents definition of lines of a component, but the preliminary design drawing does not represent structure of the component. A designer takes the line definition from the preliminary design drawing and develops structural definition for the component. Structural definition includes assigning properties and materials, and gages. Next, a designer generates surfaces for the component based on the structural definition. Surface generation is a very detailed, time-consuming process.
In a series of manual operations, a modeler takes required information off the structural definition to generate a finite element model of the component. Generating the finite element model includes generating surfaces, structural breaks, and properties and materials for the component.
The surface geometry is transferred from a CAD computing environment to a modeling-computing environment such as UNIX. Because manually generated surfaces typically include flaws, the surfaces are cleaned up. For example, meshing operations in commercially-available modeling software may introduce surface flaws. In most cases, a surface is so flawed that the surface must be re-created.
Each surface is mesh-seeded. If the surface is not corrupted, grid and nodal generation is completed as desired. A limited number assignment to the mesh, that is grid and element numbers, is created.
Property and materials are assigned to the created elements. Mass is evaluated and changed, if desired. Finally, numbering errors are manually modified to allow proper interfacing with other finite element models.
The above process results in just one iteration of each component being modeled. Each model can then be subject to static and dynamic testing, as desired or required. Finally, all the finite element models are integrated into a model of a subassembly or assembly. Integration of the component models involves determining connection points and interface connections. When the component models are integrated into an integrated finite element model, documentation of the model is generated, and the model is released. The above process can take thousands of labor hours and hundreds of manufacturing days, and results in just one iteration of an integrated finite element model.
As a result, a first iteration of an integrated finite element model may not be released until well after a 25% design review and may not be released until a 90% design review. Such a long analysis cycle time introduces program risk and is unresponsive to unanticipated growth of work statements in complex system integration projects. Further, such a process is unresponsive to design changes.
The foregoing examples of related art and limitations associated therewith are intended to be illustrative and not exclusive. Other limitations of the related art will become apparent to those of skill in the art upon a reading of the specification and a study of the drawings.
| |
Heavy stellar traffic, deflected comets, and a closer look at the triggers of cosmic disaster
As stars pass close by our solar system, they can nudge comets from the distant Oort cloud into the inner regions around the Sun. Thus, stellar encounters are an important factor in determining the risk of large cosmic impacts on Earth. Now, Coryn Bailer-Jones from the Max Planck Institute for Astronomy has used data from the ESA satellite Gaia to give the first systematic estimate of the rate of such close stellar encounters. Every million years, up to two dozen stars pass within a few light-years of the Sun, making for a near-constant state of perturbation. The results have been published in the journal Astronomy & Astrophysics.
In-depth description: Heavy stellar traffic, deflected comets, and a closer look at the triggers of cosmic disaster
Strikes by large asteroids or comets are a global danger to be taken seriously – they have occurred in Earth's past, and they will occur again. The first piece of good news is that impacts with regional or even global consequences are exceedingly rare, and occur at a rate of no more than one per million years. A random person's risk of dying in a plane crash is an estimated 25 times as great as the risk of perishing due to a cosmic impact event. The second piece of good news is that current asteroid monitoring systems yield a fairly complete picture of the larger asteroids and comets – those more than a few hundred meters in diameter – that can be found in our solar system neighborhood, and indicate reliably that none of these currently known ``Near-Earth Objects'' pose a concrete threat to the Earth.
Understanding cosmic collisions
Still, the threat is so fundamental that scientists are eager to understand the underlying mechanisms. When it comes to impacts by comets, the chain of events leads even further into the depths of space, far beyond our solar system. Our Sun is only one of an estimated 200 to 300 billion stars in our home galaxy, the Milky Way. Viewed from afar, you would see the Milky Way as a stately disk, so large that it takes light 100,000 years to travel from one side to the other. Within this disk, the stars orbit the Milky Way's center; our Sun completes one orbit in about 225 to 250 million years. Look more closely, and stellar motion becomes more complicated, with the stars following individual orbits that can cross each other, bringing certain stars into close proximity now and then in (cosmically speaking) brief encounters.
These close encounters play an important role when it comes to cometary impacts on Earth. The solar system is believed to be surrounded by a gigantic cloud of small, icy bodies, namely comets. This "Oort cloud" is a roughly spherical shell extending from 2000 to 50,000 times the Earth-Sun distance from the Sun, so its outer edge is about one light-year from the Sun. There are likely to be billions of these comets with sizes up to a few or even a few dozen kilometers.
Given their great distance, these comets feel only a very slight pull of the Sun's gravity; only just enough to keep them in an orbit about the Sun. Thus, the gravity of a star that passes within a few light-years of the Sun can be strong enough to deflect them markedly from their original paths. The amount of deflection depends not only on how close the star passes, but also on how massive it is and how fast it is moving.
From encounters to collisions
Some of the comets can be deflected in a way that carries them into the inner solar system. As they approach the inner regions, the Sun's light as well as its particle winds will strip material from the icy object, creating the distinctive long tail of a comet. After its closest approach to the Sun, the comet will head back out towards the Oort cloud, ready to repeat its orbit as long as it remains intact. In a few, rare cases, a comet could instead collide with a body in the inner solar system.
The existence of the Oort cloud and occasional disturbances are thought to be the explanation for long-periodic comets, whose journey around the Sun takes between 200 and thousands of years per orbit. In fact, the gigantic size of these comets' orbits, as inferred from observations of their passage through the inner solar system, was the motivation for postulating the existence of the Oort cloud in the first place. What we know so far about long-period comets supports this hypothesis, even though the Oort cloud has not yet been observed directly.
Thus, there is a direct connection between close encounters with stars and comet impacts on Earth, and to understand the latter, you need to research the former: How often do close stellar encounters occur? What encounters have there been in the past how have they influenced the frequency of impact events, and do they have a bearing on our estimates for the present?
Reconstructing stellar motions and close encounters
Answers to these questions depend crucially on the available data for stellar motion in our direct cosmic neighborhood, and in particular on how precise those data are. Now, Coryn Bailer-Jones, a staff scientist at the Max Planck Institute for Astronomy, has published the first systematic estimate of the probability for such near stellar encounters. Bailer-Jones' calculations are based on the first data release (DR1) from the ESA astrometry satellite Gaia, which was published on 14 September 2016.
Gaia's mission is to measure the position, distance, and velocity of over one billion stars in our Galaxy, with an accuracy which has never before been achieved for so many stars. The final results will contain precisely what is needed to describe the orbits of stars in our galactic neighborhood: where these stars are in the surrounding space, and in which direction they are moving.
While the full reduction and analysis of the Gaia data is still in progress, DR1 published preliminary results on a special data subset that goes at least part of the way. This is the so-called TGAS catalogue of 2,057,050 stars, which makes use of both the first Gaia data and the data of the ESA-satellite Hipparcos, twenty years earlier, to yield the best stellar distances and motions to date.
Using this catalogue, Bailer-Jones identified 468 stars that, at their current rate of movement, would seem to come within 32.6 light-years (10 parsec) of the Sun, either in the past or in the future. For these stars he performed a computer simulation of their orbits – taking into account the gravity of our home galaxy – to determine more precisely their closest approach to the Sun. He found that 16 stars will pass, or have already passed within less than 6.5 light-years (2 parsec) of the Sun. (The light-year values are not round numbers since Bailer-Jones chose his sample values to be round in another distance unit used by astronomers, 1 parsec = 3.26 light-years.)
The closest future encounter
The closest encounter found is for the star Gl 710 ('Gliese 710'), which has been known for some time to be heading for a close encounter with the Sun. The new data and calculations show that this encounter, which will take place in 1.3 million years, comes much closer than was thought before DR1: just a quarter of a light-year (or 16,000 times the Earth-Sun distance), well within the Oort cloud. This confirms similar calculations by two Polish astronomers, Filip Berski and Piotr Dybczyński, in a 2016 article, also based on the first Gaia data release . Although Gl 710 has a comparatively low mass, just 60% the mass of the Sun, its velocity is also low, giving it plenty of time for exerting its gravitational influence on the Oort cloud. Given that Gl 710 may bring its own Oort cloud, this raises the intriguing possibility that our Sun could even swap comets with passing stars!
Deriving the rate of encounters
But beyond identifying the closest encounters, Bailer-Jones went an important step further. Astronomical surveys are never complete; they will only detect their targets down to some minimal brightness, and miss light sources dimmer than that. Bailer-Jones modelled the encounters that DR1 detected, compared them to the survey's limitation, and used statistical tools to estimate how many stellar encounters his DR1-based evaluation was likely to have missed. In addition, Bailer-Jones took into account the uncertainties of the Gaia data, which are known from systematic studies by the Gaia team. For each star, he calculated not only the orbit corresponding to the nominal Gaia values for distance, position and motion, but the orbits for a whole swarm of virtual stars. The swarm represents the (sometimes large) uncertainties in the data – and hence the fact that, with a certain probability, the derived parameters for the encounter could be different from the nominal estimates. This gives a more reliable estimate than relying on the nominal data alone.
As a result, Bailer-Jones obtained the first systematic estimate of the average stellar encounter rate for the past and future 5 million years. (The model reconstruction is not accurate enough to extrapolate to encounters further in the past or future with the DR1 data.)
The result, which meshes with earlier, less systematic estimates, is that within each period of a million years, between 490 and 600 stars will pass the Sun within a distance of 16.3 light-years (5 parsec) or less. This covers stars of all masses, although the most common ones are low mass stars, like Gl 710. Within a smaller distance of 3.26 light-years (1 parsec), some 19 to 24 encounters are expected per million years. Given that it takes several million years for disturbances to abate, our Oort cloud seems to be in fairly constant upheaval, with no extended periods of calm in between.
Not quite to the dinosaurs
This is valuable information for the scientist attempting to calculate the rate of cometary impacts on Earth. Gaia's next data release – DR2, in April 2018 - should allow an extension of these reconstructions to 25 million years from the present. The release after that, DR3, will contain estimates of the masses and radii of the observed stars, based on the data from Gaia's on-board spectrometer. Bailer-Jones is, in fact, in charge of the group of Gaia data analysts tasked with deriving these and other astrophysical quantities from Gaia's huge treasure trove of measurements. Information about masses and radii will allow the astronomers to estimate how large the disturbances in the Oort cloud will be on average, allowing for more precise estimates of their consequences for the impact rate.
Extending the reconstructions further will be difficult; as the simulations reconstruct longer orbits, uncertainty about the mass distribution in the Milky Way becomes a major limiting factor (although the Gaia data themselves will help us improve knowledge of this). Astronomers in search of stars that might be responsible for sending a comet to Earth that, 66 million years ago, caused or at least hastened the demise of the dinosaurs, will need to know our home galaxy much better than we currently do. | https://www.mpia.de/news/science/2017-09-stellar-encounters?seite=2 |
All relevant data are within the paper and its Supporting Information files.
Introduction {#sec001}
============
Estimations of abundance are fundamental in ecology and conservation to answer a wide range of questions. Often scientists and decision makers need to compare abundance through space or time, and for this purpose, it is important to estimate possible sources of variation in the probability of detecting an individual \[[@pone.0128133.ref001]\]. Several methods to estimate abundance that cope with imperfect detection have been developed, but these methods make several assumptions. Mark-recapture methods require live-trapping and marking of animals or naturally recognizable individuals for camera-trap applications of the models \[[@pone.0128133.ref002]\]. Distance sampling and Random Encounter Models require a minimum number of encounters and random design \[[@pone.0128133.ref003], [@pone.0128133.ref004]\]. These conditions are difficult or impossible to meet in rare or elusive species that are also difficult to catch or when individuals are not consistently recognizable.
Random designs yield extremely low encounter rates are observed in species that are found associated with a particular feature on the landscape, such as rocky outcrops for reef fishes (*Pseudopercis semifasciata* \[[@pone.0128133.ref005]\]), cliffs for mountain vizcachas (*Lagidium viscacia* \[[@pone.0128133.ref006]\]) and the Andean condor (*Vultur gryphus* \[[@pone.0128133.ref007]\]), or conspicuous warrens for maras (*Dolichotis patagonum* \[[@pone.0128133.ref008], [@pone.0128133.ref009]\]) and plains vizcachas (*Lagostomus maximus* \[[@pone.0128133.ref010]\]). In these cases, counts are commonly made by making focal observations close to the feature where animals concentrate (e.g. *Suricata suricatta \[[@pone.0128133.ref011]\]*, *Vultur gryphus* \[[@pone.0128133.ref012]\], reef fishes \[[@pone.0128133.ref005], [@pone.0128133.ref013]\], *D*. *patagonum* \[[@pone.0128133.ref014], [@pone.0128133.ref015]\]). Focal observations provide an index that is restricted to specific points in space so additional information is needed to obtain density estimates for a given area. As an alternative, density can be estimated by searching for sign and estimating occupancy rates of suitable sites combined with independent estimates of the average number of individuals aggregated around each sign \[[@pone.0128133.ref016]\]. We propose a two-step protocol in which random sampling designs are used to search for signs and continuously recording video cameras are used to perform abundance counts at the points where animals are concentrated. However, to avoid confounding effects, we incorporated several factors that influence detectability and introduce potential bias in abundance estimation and inference.
In this context, the detection probability and other sources of variation that must be estimated have at least two components: one related to the searching for sign and another related to counts of animals using video cameras at the focal observations. Sources of potential bias in abundance estimations using video cameras could be related to restricted viewpoints \[[@pone.0128133.ref017]\], changes in behavior of target individuals caused by the presence of the device \[[@pone.0128133.ref018]\] and temporal variation in detectability due to animal movement \[[@pone.0128133.ref013]\]. However, these biases could be minimized due to resource influence in the case of animal counting around shelters or other resources that concentrate individuals. The camera may record most of the individuals if: it covers the main resource influence area; behavior is normalized when individuals tend to accept a new static object near shelters or food sources \[[@pone.0128133.ref019]\]; and the sampling schedule is designed to include patterns of resource use, which would reduce temporal variation \[[@pone.0128133.ref013], [@pone.0128133.ref020]\]. In addition, although using continuously recording video cameras avoids the false negative bias related with triggers, the influence of the length of a sampling period over detection or counting persists. Unfortunately, perfect detectability should not be assumed even if all these sources of bias could be controlled or minimized. Detectability may be estimated using the proposed approach by repeated observations \[[@pone.0128133.ref001]\].
The mara, an endemic mammal of the Argentine semi-desert, was classified as 'Near Threatened´ according to the 2008 IUCN Red List assessment due to population decline \[[@pone.0128133.ref021]\]. The reported trend was based on expert knowledge according to field observations given the absence of systematic data aimed to describe population abundance and direct abundance estimates. This species is a good model to apply the proposed two-step protocol and to assess sources of variation because of its communal breeding behavior and the distribution pattern associated with this behavior. The same behavior also makes it impossible or impracticable to use well-established methods. Maras form long-term monogamous couples that spend most of the year dispersed over wide areas with a radius of up to 2000 m around warrens, avoiding other couples \[[@pone.0128133.ref008], [@pone.0128133.ref009]\], and this dispersion makes them difficult to detect. Thus, encounter rates are too low (Section A in [S1 File](#pone.0128133.s004){ref-type="supplementary-material"}) to make accurate estimations using distance sampling, and the assumption of independent detection events could be violated because of poorly defined clusters around warrens \[[@pone.0128133.ref004]\]. Mark recapture models \[[@pone.0128133.ref002]\] are not applicable because live-trapping of maras results in a high mortality rates \[[@pone.0128133.ref022]\] and it is not possible to recognizing individuals visually. On the other hand, couples stay close to communal warrens during the breeding season, from August to December where they can be more easily counted.
Warrens are tunnels in the ground where pups of several couples spend the night and most of the day, using it as a shelter to avoid predators. In contrast, adults do not actively use the warrens for shelter but spend part of the day next to the warren nursing their pups or on alert while pups play and feed on vegetation \[[@pone.0128133.ref014]\]. Warrens are easier to detect than maras and could persist, even when not actually in use by maras, causing false-positive detection \[[@pone.0128133.ref016]\]. Moreover, the number of maras associated with a warren is variable among warrens and through time. Warrens accommodate pups from one to more than ten couples \[[@pone.0128133.ref008], [@pone.0128133.ref009], [@pone.0128133.ref015]\]. In addition, a breeding pair visits the warren at least once a day to attend to their pups \[[@pone.0128133.ref014]\], but because it is not possible to recognize individuals, repeated visits cannot be differentiated from new couples approaching. Thus, the number of adults near warrens could vary through time during the day depending on how many couples are visiting their pups together and also because some environmental circumstances are perceived as threats that may make adults leave the warren.
Previous studies on the ecology and behavior of maras have utilized direct counts made by an observer to quantify the number of adults associated with studied warrens \[[@pone.0128133.ref009], [@pone.0128133.ref014], [@pone.0128133.ref015]\]. In recent studies, the observers have been replaced by surveillance video cameras that register the activity around warrens and allow researchers to count the maximum number of adults visiting the warren together and the number of resident pups of different age classes \[[@pone.0128133.ref008]\]. The use of video cameras provides significant logistical and methodological benefits because it makes it possible to observe several warrens at the same time with no need for multiple observers \[[@pone.0128133.ref023]\], reducing fieldwork costs, avoiding bias due to differences among observers and diminishing disturbances in the study area \[[@pone.0128133.ref024]--[@pone.0128133.ref026]\]. As mentioned above, this technique might be biased due to i) the detection capacity of the camera, ii) the perception of the camera as a threat, and iii) variation in the timing and frequency of adult visits.
The aim of this study was to evaluate possible bias and influence of several factors over detectability when applying the two-step protocol using the mara as a case study. Specifically, our objectives were to quantify detectability of warrens performing line-transect samplings within defined plots. We also estimated the bias related to false positive detection due to misidentification or the persistence in the field of abandoned warrens that have no associated couples (non-active warrens). We evaluated the detection capacity of our camera against an observer in the field. This way we were able to evaluate changes in visiting behavior due to the presence of the camera. Finally we evaluated and quantified temporal variation in the number of adults. Temporal variation in the detection probability could be related to: i) the length of sampling period, ii) the diurnal trend in visiting dynamics and iii) the inclusion of inactive warrens in estimations.
Materials and Methods {#sec002}
=====================
Study site {#sec003}
----------
The study was carried out in Península Valdés (Argentinean Patagonia), a 4,000 km^2^ provincial protected area declared a UN World Natural Heritage Site ([Fig 1](#pone.0128133.g001){ref-type="fig"}). Península Valdés currently has the IUCN category of 'managed ecosystem' and consists of private properties where sheep ranching is the main productive activity. Within Península Valdés, vegetation structure varies but can be described by three main landscape configurations: shrubland, shrub-grass mosaic and grassland.
{#pone.0128133.g001}
Warren detection {#sec004}
----------------
We used volunteer observers to assess the detectability of warrens by applying a line transect sampling design within a given area (searching plots). We searched for warrens while walking along 20 parallel transects that were 5 km long and 200 m apart in four 2000-ha areas in Península Valdés (42.48°S 62.12°W, 42.55°S 63.76°W, 42.61°S 63.64°W and 42.67°S 63.69°W). Volunteer observers were briefly trained in the field before walks by showing them warrens, feces and footprints of mara and by comparing warrens with peludo (*Chaetophactus villosus*) burrows because these structures can be easily confused. We selected the size of our sampling area in proportion with the home range size (193 ha \[[@pone.0128133.ref009]\]) so it was large enough to possibly contain 10 warrens or more. Study sites represented the two main contrasting landscapes within the protected area with suitable habitat characteristics for mara \[[@pone.0128133.ref008]\]: shrub-grass mosaic and grassland ([Fig 1](#pone.0128133.g001){ref-type="fig"}). We placed study sites where we knew that the species was present, based on knowledge from local people and previous visits, because i) we were interested in sources of variation in the counting of warrens rather than species occurrence, and ii) random points within Península Valdés produced few warren encounters with high cost (Section B in [S1 File](#pone.0128133.s004){ref-type="supplementary-material"}). After the line transect survey, but in the same period between reproductive seasons to avoid the digging of new warrens (closed population), the author (VAR) surveyed the areas intensively, searching for undetected warrens, walking in several directions through located warrens, walking in several directions within areas where warrens were not found and visiting with local people in sectors where they had usually seen maras while working. In addition, we visited each point marked by volunteer observers to check if it was a mara warren or a case of misidentification. We calculated the detectability of warrens based on the mark-recapture concept and double sampling approach \[[@pone.0128133.ref001]\] applying the following equation: $$\beta = \text{m}_{2}/\text{n}_{2}$$ where β is detectability, m~2~ is the number of warrens "marked" during the line transect survey and n~2~ is the total number of warrens known after the intensive second survey (i.e. double sample). We also calculated the error associated with each observer because of misidentification as the proportion of wrongly marked warrens over the total number of marked points.
We surveyed identified warrens through the reproductive season to obtain the proportion of active warrens in order to evaluate bias related to detection of false positives of abandoned warrens. We set continuously recording video cameras for one day every 15 days in 27 warrens and repeatedly visited the remaining warrens searching for changes in signs of activity, such as recently removed soil at the warren entrance, new footprints or feces \[[@pone.0128133.ref015]\]. Camera features and settings were the same used to assess counting (see next section for details).
Counting adults {#sec005}
---------------
### Surveillance setting {#sec006}
Twelve mara warrens were surveyed during the middle of the mara breeding season (between 22 October and 10 November in 2011) to assess possible bias and the effects of several factors on the detectability of adult mara using continuously recording video cameras. These warrens were outside the searched areas described above, were previously known, were active in previous reproductive seasons and were easily accessed. All warrens were located within a private sheep ranch in the southwest of Península Valdés (Patagonia, Argentina; [Fig 1](#pone.0128133.g001){ref-type="fig"}, 42.62°S 63.71°W), where the predominant vegetation is a shrub-grass mosaic of *Chuquiraga avellanedae* and *Stipa tenuis* \[[@pone.0128133.ref027]\].
Each warren was monitored using a surveillance camera with 2 MPixels (Vivotec, IP7160) placed approximately 15 m from the entrance ([Fig 2](#pone.0128133.g002){ref-type="fig"}) to register the activity for 13 hours per day (7--20 h). The cameras were powered by 12 V batteries with a photocell that cut off power at night, saving energy and memory storage capacity for daylight hours when maras are active \[[@pone.0128133.ref028]\]. Surveys were stored in one minute long mp4 format videos.
{#pone.0128133.g002}
### Detection capacity of camera vs. direct counting {#sec007}
We compared the maximum number of adults recorded simultaneously by direct observation and by the camera within the same period of time (7:30--11:30) to evaluate the accuracy of counting from video with respect to direct counting. Direct observations were conducted from a platform located more than 50 m from the warren that was simultaneously monitored by the camera. The comparison was performed by means of a paired Wilcoxon's signed-rank test.
### The effect of a camera on mara behavior {#sec008}
We recorded mara behavior when cameras were present and when they were absent. First, to evaluate if the camera was inhibiting the approach of maras to the warren, we used direct observation to count the number of maras distant from and close to the warren using a circle centered at the warren with radius equal to the distance between the warren and the camera (approximately 15 m) as a reference. Second, to evaluate if the camera was perceived as a threat by maras, we counted the number of alert maras (head up and looking around) and relaxed maras (resting, feeding or interacting with pups). We built contingency tables to compare the number of distant/close and alert/relaxed individuals with and without the camera present by means of Χ^2^ tests, discarding data from warrens with null observations and applying the Yates' continuity correction when the number of observations was less than 200 \[[@pone.0128133.ref029]\].
### Temporal variation {#sec009}
The temporal variation in the number of adults detected by the camera was evaluated by surveying each warren for three days (not always consecutive). Given that the videos were stored in one-minute files, the raw data consisted of the maximum number of adults recorded simultaneously in each minute. We used the videos to differentiate active warrens, such as the ones in which resident pups were registered, from inactive warrens that have no couples or pups associated with them but could be visited occasionally by non-resident individuals.
To evaluate the effect of the length of the sampling period on detection probability, we examined the data set taking the maximum number of adults observed in random time-frames of increased duration by 10 minutes, from 10 to 770 minutes. We took different starting points at random within the data set from each sample unit (a given warren in a given day) for the 77 possible time-frames, performing one thousand iterations of that process. We expressed the maximum obtained in each time-frame and iteration as a proportion of the maximum registered in the corresponding warren during the three days of surveillance. The mean of this proportion was plotted against the length of the sample period to identify the time-frame where the calculated proportions reached within 0.05 of the asymptotic value (i.e., shorter sampling period with highest accuracy). All the calculations were performed using R statistical software \[[@pone.0128133.ref030]\].
To evaluate if there is a diurnal trend in the visiting dynamics, we compared the maximum number of adults per minute detected by the camera during the morning (7--11 h), the middle of the day (11--15 h) and the afternoon (15--19 h). We fitted a linear mixed model, including time of day as a predictor and the warren and the date in which the observation was made as hierarchical random factors to account for the variation due to heterogeneity among experimental units. We used a temporally structured variance term to address autocorrelation. We considered only the active warrens because registration of non-resident individuals could mask visiting patterns of resident individuals.
Finally, we quantify the variation in estimates of the number of adults associated with a warren due to the estimations taking place on different days. We calculate the maximum number of adults registered in each day of observation in each warren and fit a linear mixed model, including the warren identity as a random factor. To evaluate the possibility that the variation was overestimated due to the registration of non-resident individuals, the model was fitted twice: once considering all warrens and again considering only the seven warrens where resident pups were registered (active warrens).
We used R statistical software \[[@pone.0128133.ref030]\] and the R-package nlme \[[@pone.0128133.ref031]\] to fit the models. In both cases, the natural logarithm of the maximum number of adults plus one was the dependent variable. Normal distribution of the error was chosen because a plot of the mean versus the variance of the response variable \[[@pone.0128133.ref032]\] showed that the Poisson distribution was an inappropriate choice. Given that the mixed models decompose the sources of variation due to the random effect and residual variation, we obtained estimates of the variation due to differences in warren characteristics (random factor), to temporal factors and to the detection capability of the method (residual variation). To quantify variation in biological units applicable to other estimations, we expressed it as coefficients of variation (CV).
Results {#sec010}
=======
During the line transect search with volunteer observers we found 12 and 16 warrens within grassland sites, areas A and B, respectively, and 12 and 16 warrens within shrub-grass sites, areas C and D, respectively, giving an average encounter rate of 0.14 km^-1^. The posterior intensive survey showed 6 undetected warrens in area B and 3 in each other area. Therefore, detectability was 0.8 in area A, 0.73 in B, 0.8 in C and 0.84 in D. The error associated with each observer because of misidentification was 0.47 on average, varying between less well trained observers (0.62) and highly trained observers (0.22). The error due to the persistence of abandoned warrens was 0.92 on average, given that only one of the located warrens was active in area A and B, none in C and four in D.
The maximum number of adults registered by direct observation and by the camera were not significantly different (Wilcoxon's signed-rank test: *T* = 9, *P* = 0.21). We did not find evidence indicating that the presence of the camera affected the behavior of mara approaching the warren or that the camera was perceived as a threat by maras. The proportion of distant individuals did not show significant differences with and without the camera present in three warrens ([Table 1](#pone.0128133.t001){ref-type="table"}), and it was lower with the camera present in two warrens (0.13 and 0.46 with camera, 0.73 and 0.77 without camera, respectively). Only in one warren were more distant individuals registered with the camera present (0.43 with camera and 0.13 without camera). Regarding the proportion of alert individuals, there were no significant differences with and without camera present in four warrens ([Table 1](#pone.0128133.t001){ref-type="table"}), and the proportion was lower with camera present in one warren (0.37 with camera and 0.54 without camera). Only in one warren were more alert individuals registered with the camera present (0.63 with camera and 0.36 without camera).
10.1371/journal.pone.0128133.t001
###### Camera effect over mara behavior.
{#pone.0128133.t001g}
Proximity Alertness
------ ----------- ----------- -------- ----------
1 0.049 0.824 3.444 0.064
2 c 5.223 0.022 0.003 0.955
3 c 0.009 0.925 2.012 0.156
4 0.424 0.515 12.672 3.00E-04
7 c 36.796 1.31E-09 0.036 0.849
12 c 8.564 0.003 7.984 0.005
Results of Χ^2^ tests based on contingency tables comparing the number of distant/close and alert/relaxed individuals with and without the camera present. The informed X^2^ statistic (*X* ^*2*^) and the associated probability (*P*) are given, and subscript c in the warren's number indicates cases where Yates's correction was applied.
The number of adults registered by minute was highly variable, with the maximum recorded only during short periods of time each day ([Fig 3](#pone.0128133.g003){ref-type="fig"}). Any sampling period much shorter than an entire day likely underestimates the number of adults associated with each warren. The shortest sampling period capable of registering 95% of the maximum number of adults from entire-day surveys was 650 minutes when considering all the warrens and 600 minutes when only warrens with pups were considered ([Fig 4A and 4B](#pone.0128133.g004){ref-type="fig"}). We did not find evidence of a diurnal trend in the visiting dynamics, because the fitted model did not show significant differences among the three periods of day tested ([Table 2](#pone.0128133.t002){ref-type="table"}). The temporal variation in estimates of the number of adults associated with a warren caused by the estimation on different days was CV = 0.12, while the variation due to warren characteristics (random component) was CV = 0.45. Both variation components were increased when inactive warrens were considered for model fitting (temporal variation CV = 0.42; random component CV = 1.12).
{#pone.0128133.g003}
{#pone.0128133.g004}
10.1371/journal.pone.0128133.t002
###### Diurnal trend in visiting dynamics.
{#pone.0128133.t002g}
b~i~ S.E. *t* ~*16135*~ *P*
--------------------- -------- ------- --------------- -------
Morning (Intercept) 1.242 0.160 7.745 0.000
Noon 0.057 0.038 1.489 0.136
Afternoon -0.004 0.052 -0.080 0.936
Estimated parameters for predictors of adult abundance comparing three times of day according to the fitted model. The informed estimated parameter (b~i~), standard error (S.E.), t statistic (*t* ~*d*.*f*.~) and associated probability (*P*) are given.
Discussion {#sec011}
==========
The two-step protocol was successfully applied to *D*. *patagonum* within the Península Valdés protected area, which was logistically suitable and allowed to warrens to be found and the associated adults to be counted, estimating detection probability. Both components of the two-step protocol revealed important sources of variation that could affect detectability. In this section, we discuss the results related to the different sources of variation and the alternative monitoring designs to cope with them.
Warren detectability was approximately 80%, with little variation, and the encounter rate was 0.14 km^-1^, much higher than the individual encounter rate of 0.05 km^-1^ (Section C in [S1 File](#pone.0128133.s004){ref-type="supplementary-material"}). These results confirm that it is more convenient to count warrens than individuals in the open field. The estimated detection probabilities showed that line transect sampling was effective for warren detection within sampled areas (searching plots), despite the fact that distance among consecutive transects (200 m) may seem too large in a shrubby landscape. In addition, given the little variation in detectability among landscape configurations, it could be possible to pool strata together, in which case the encounter rate could be high enough to apply distance sampling \[[@pone.0128133.ref004]\] to obtain the warren density and detection probability. Nevertheless, we acknowledge that our sample size (four sites) is small to make general statements and is conditioned to the species presence in the area. Thus, further investigation across the species range is needed to better understand the effect of landscape configurations over warren detectability and the limitations that the encounter rate could impose on sampling design and methodology.
False positives (misidentification and abandoned warrens) were much more important to accurate estimation than imperfect detection. Misidentification error could be reduced by more extensively training volunteer observers or by doing all searching with a smaller group of highly trained observers, which also would reduce the variation among observers. In both cases we still recommend re-checking (double sampling) warrens detected in a proportion of the searching areas to estimate misidentification error and include it in abundance estimations.
The largest source of error was abandoned warrens, which is functionally equivalent to the more common case of a suitable patch of habitat not being occupied by the target species. This source of variation could be included in the abundance estimation as the probability that a located warren has associated adults, randomly selecting warrens to observe with cameras and getting the proportion of active warren. However, the proportion of abandoned warrens is so high that counts in random sampled warrens would result in too many zero counts. This causes problems in abundance modelling, even assuming zero inflated distributions. The necessary effort to survey, select, and confirm that warrens are active is cost effective in terms of reducing the variability of counts, as discussed later.
Misidentification is a typical bias when the signs used are breeding or resting sites (e.g., mara warrens could be confused with peludo burrows or squirrel dreys with bird nests) and must be included in the protocol to avoid overestimations \[[@pone.0128133.ref016]\]. On the other hand, it is also important to not include abandoned sites and to understand the dynamic of their use \[[@pone.0128133.ref033]\]. Some rare species use the same breeding areas repeatedly and, therefore, are relatively easy to census; however, because some new nesting sites could be used each year, a sampling design should allow for its detection and census \[[@pone.0128133.ref033]\].
Our results show that surveillance cameras could replace an *in situ* observer to estimate the number of adults associated with a warren since we saw no differences in their detection capability. Similar results were observed in other species. For example, visual census and high definition video transects were compared for monitoring coral reef fish assemblages in marine ecosystems (e.g., \[[@pone.0128133.ref034]\]). Our results indicate that maras do not perceive the camera as a threat, and its presence would not affect the visiting behavior of adults to the warren. We saw two possible instances of reaction to the cameras (more distant individuals in warren 2 and more alert individuals in warren 12). These differences were not seen in the majority of warrens. In these two cases, the results may have been affected by other factors during observations, such as vehicular traffic in a road close to both warrens.
Other studies have also considered cameras a non-intrusive tool \[[@pone.0128133.ref019], [@pone.0128133.ref035], [@pone.0128133.ref036]\]. However, studies performing specific analyses to evaluate possible effects of the camera over individuals' behavior are scarce. Although scarce, results seems to consistently show no negative effects, similar to our own findings; for example, beavers (*Castor canadensis*) did not show significant interactions with cameras and did not leave monitored warrens \[[@pone.0128133.ref035]\] and red foxes (*Vulpes vulpes*) showed a fast adaptation to the presence of cameras after initial negative effects in their behavior \[[@pone.0128133.ref037]\].
We found that samples shorter than a day (\< 10 daylight hours) will underestimate the number of adults. This variation occurs because the maximum activity around warrens is short in duration, happens only a few times or just once during each considered day and is unpredictable because diurnal trends in visiting behavior are not evident. As a result, we do not recommend the use of movement- or time-triggered cameras for this species given that the probability of missing the maximum number of animals is high.
Repeated counts of the maximum number of adults registered simultaneously during a single-day observation period showed little variation, with a detection probability of approximately 90%. This modest amount of temporal variation, which would be associated with the continuous movement of adults around the monitored area, is part of the residual variation of the method. A similar result was reported in reef fish, given that the most relevant variation in fish counts was observed over a very short time period \[[@pone.0128133.ref013], [@pone.0128133.ref038]\]. This is clearly related to the high mobility of studied animals and with differences in the detection capacity of observers \[[@pone.0128133.ref038]\]. With the sampling design used here, it is not possible to differentiate short-term variation due to continuous movement from the methodological component of the variance related with the detection capacity of the camera.
Another source of temporal variation when counts are performed over temporary aggregations, such as around breeding sites, is related with the asynchrony of the individual breeding season. A single count on any day of the season underestimates the entire population, given that there is never a day when all individuals are present \[[@pone.0128133.ref039]\]. Although this source of variation was not assessed in our study, it has been studied in other cases, and models have been developed based on a few counts throughout the breeding period as well as independent evidence on the length of time individuals remain in an aggregation \[[@pone.0128133.ref039]\]. This information can be obtained from the scheme of the two-step protocol as proposed for repeating mara observations using cameras in the reproductive season. It can be possible to measure the permanence of couples using a warren by following the number of pups of different age classes \[[@pone.0128133.ref008]\] from a crèche born until they leave the warren (approximately six weeks old \[[@pone.0128133.ref014]\]).
The main variation in counts of adults was among warrens, once the temporal variation was reduced by taking observations over the entire day. The spatial variability was also mentioned as a key factor in birds counts that has to be included to correctly assess population trends \[[@pone.0128133.ref040]\]. This variation in the case of mara is not due to imperfect detection but to ecological processes. Previous studies have shown that the differences in the number of adults among warrens is related to the effects of environmental factors, such as vegetation and distance to anthropogenic structures, effects of socio-spatial interactions among warrens, the long term colonization dynamic, and site history in relation with sheepherding or hunting \[[@pone.0128133.ref008], [@pone.0128133.ref009], [@pone.0128133.ref015]\]. However, incorporating spatial variation using a stratified sampling design \[[@pone.0128133.ref033]\] is not yet possible because the importance and strength of those factors is not adequately quantified. Thus, if to estimate abundance it is necessary to take a sample of located warrens instead of performing counts in all of them, monitoring designs should include the largest number of warrens possible to improve the accuracy of the estimated mean number of adults that will be associated with all found warrens.
Temporal and among warren variation was increased when inactive warrens were included in estimations. Inactive warrens could be confused with active warrens due to the use of signs of mara presence (recently removed soil and fresh footprints or feces close by) that produce an incorrect assessment of warren status. Signs would be useful to recognize areas with maras \[[@pone.0128133.ref008]\], but not to identify reproductively active warrens, because many warrens are visited, but not all are used every year \[[@pone.0128133.ref014]\]. Some abandoned warrens can be discarded after several visits to search for new signs that should appear if it is active. If new signs appear, camera records are useful to confirm mara presence around the warren and also that they are breeding in it. It is important to confirm that pups are using the warren because only then will the adults consistently visit the warren. In addition, the occasionally sighted individual should not be included in estimations because they could be breeding in another warren, in which case they would be double counted, or they could be not breeding, and thus they would not be part of the target population.
The proposed method to estimate abundance is directed to the reproductively active portion of the population, which probably stays closer to the warren where they can be counted. Alternatively, abundance could be estimated without this bias using randomly arranged camera traps in the study area and modelling the process of contact between animals and the camera according to random encounter models \[[@pone.0128133.ref003]\]. However, given that the daily range of maras averages 1.7 km \[[@pone.0128133.ref009]\] and that estimated densities within our study area were lower than 1 km^-1^ \[[@pone.0128133.ref008]\], more than 1000 camera-days would be needed to obtain the minimum number of encounters required by the method \[[@pone.0128133.ref003]\]. In contrast, only 23 camera-days were needed to check and count breeding individuals within the study area.
Even though our two-step protocol resulted in estimates of detection and insight into potential pitfalls, it would not be wise to apply these estimates to other areas and times. The process of estimating detection probability and other source of variability could be incorporated in any monitoring protocol by double searching a proportion of the surveyed areas and double counting a proportion of the founded warrens \[[@pone.0128133.ref001]\]. Based on our results and following Pollock et al. \[[@pone.0128133.ref001]\] to calculate the allocation of sampling effort between collecting data on the count index and collecting the more detailed data to do the detectability estimation, we find that 22% of the surveyed areas should be double searched and 18% of warrens should be observed twice for the counting of adults. We believe that the application of this monitoring protocol over the mara range could provide reliable and systematic information about population abundance and spatial dynamics. Through the means of warren searches and surveillance with cameras within a given area, it is possible to obtain a minimum local population size and detect if there are new warrens, changes in the number of active warrens, or changes in the number of couples in each warren over a period of years. This information is critical for obtaining information about variation in the reproductively active population and long term trends in population abundance beyond this variation.
Supporting Information {#sec012}
======================
###### Number of adults by minute recorded by the camera in each warren and sampling day.
(XLS)
######
Click here for additional data file.
###### Maximum number of adults recorded by the camera and the observer in each warren.
(XLS)
######
Click here for additional data file.
###### Number of alert adults, relaxed adults, adults distant from and close to the warren when cameras were present and when they were absent in each warren.
(XLS)
######
Click here for additional data file.
###### Alternative methods and designs applied to mara within Península Valdés.
Section A: Distance sampling. Section B: Random design to find signs of mara presence. Section C: Encounter rate of adult mara individuals in walking transects.
(DOC)
######
Click here for additional data file.
We are grateful to the owners of the ranches La Anita, Laguna Grande, Valdes Creek, San Jorge and La Corona for allowing us to conduct sampling on their properties, and especially to Luis Porcel for helping with accommodations and researchers' "survival issues". To P. Torres, P. Contreras, S. Días, Robin, Paula, Celeste, Macarena, M. Zamero, N. Velazques, C. Pascheta, A. Formoso, L. Feugeas, E. Galván, P. Hackerman, L. Beltramino, and R. D'Agostino for assistance in the field. To A. Parma, A. Marino, A. Irigoyen, P. Hackerman and D. Deutschman for comments on early versions of the manuscript that helped to improve it. Centro Nacional Patagónico and Fundación Patagonia Natural provided logistical support. The study was authorized by the Dirección General de Conservación de Areas Protegidas del Chubut and the Dirección General de Fauna del Chubut.
[^1]: **Competing Interests:**This study was funded in part by ALUAR ([www.aluar.com.ar](http://www.aluar.com.ar)). There are no patents, products in development or marketed products to declare. This does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.
[^2]: Conceived and designed the experiments: VAR LB DEG. Performed the experiments: VAR LB. Analyzed the data: VAR LB DEG. Contributed reagents/materials/analysis tools: VAR. Wrote the paper: VAR LB DEG.
| |
The start of the year is always the most popular time for people to reflect back on their habits and craft some resolutions to help them reach new (or stubborn) goals. It’s a fresh and blank slate – 365 days for us to become better versions of ourselves, and the same goes for our kids. Even though this is a prime time for goal setting, there are studies showing that only a mere 25% of people actually follow-through with them.
What most people don’t realize is that getting enough sleep is the key to maintaining and actually successfully completing our goals for the new year, and it’s even more key to our children’s development. As a year is a long time for growth when it comes to our little ones, the first goal on your and their resolution list for 2020 should be HEALTHY SLEEP.
Why is sleep so important as a foundation to all of our goals?
It’s simple really. Getting the right amount of sleep, in a stable bedtime routine, promotes wellness, and improves our moods, behaviors, nutrition, and overall development.
Sleep deprivation leads us to feeling more tired during the day, enhancing the desire to consume sweet and sugary treats for the immediate, but not long-lasting, energy boost. The lack of sleep has also been proven by the American Journal of Clinical Nutrition to activate a hunger hormone which increases food cravings. They found these food cravings to fall in line with energy-dense, high-carb foods which in turn leads to unhealthy lifestyles. These cravings also apply to our little ones who are already more likely to ask for sweet snacks.
Sleep helps us and our kids recharge our brains, giving us the energy we need to overcome those cravings. Every day after we receive the optimal amount of sleep helps creates the perfect environment for a lift in our health and development, both young and old.
To find out more about why sleep is so important, check out our blog Why We All Need Sleep.
Here are some fundamental ways to ensure your little ones and you start getting enough and better sleep in the new year:
Maintain a regular sleep and wake schedule.
Having a bedtime routine is essential to our brain’s reset process. It helps our mind and body know it’s time for rest, thus allowing us to drift off to dreamland quicker and have a more restful sleep. If your little one struggles to fall asleep after you read a story and turn out the lights, try adding audio-bedtime stories or soothing sounds to finish off their bedtime routine. Giving their minds something to focus on other than needing to sleep helps them relax quicker, and in return allows you to head to bed at a more sensible time.
Our app, Moshi Sleep, has a range of sleepy sounds and melodic stories designed specifically for children and busy minds.
Get moving for at least 30 minutes a day.
Exercise is a great way to burn off excess energy, and exposure to natural light and fresh air helps keep our bodies healthy. Sustaining a healthy lifestyle promotes a more restful sleep, which gives us the energy needed to get moving the next day as well. Be sure to wind down any physical activities you and your children are doing at least two hours before bedtime to allow our bodies to shift back into a restful state.
Be on the lookout for signs of insufficient sleep.
Because some children are less likely to show signs of sleepiness, they may instead struggle with attentiveness or have issues with focus during school. Other kids may become overly tired and actually end up falling asleep while at school or taking excessive naps on the weekends. It’s important to keep an eye out for these signs as a lack of sleep can also affect our moods and mental wellbeing. Regardless of age, everyone can feel a bit down sometimes and insufficient sleep can amplify those emotions.
The start of a new year is an opportunity for us to get on track and make strides towards becoming healthier and happier people. It’s important to lay a foundation of better sleep for ourselves and our children to ensure the goals we set this year are actually achieved, not put on the back burner.
We hope to see you reach all of your goals in 2020! | https://www.moshisleep.com/blog/why-sleep-should-be-your-1-new-years-resolution/ |
Abstract: The main goal of the paper is to present a ‘revisit’ to the integrated marketing communications (IMC) concept, based on the key changes in the communication environment with the emergence of online media (incl. social media and online trade platforms). The author reviews the opportunity of IMC to serve not only as a ‘side’ element in the sales process but to guide it, simultaneously acting as a means of communication, a sales channel and a data-gathering tool for the effectiveness of the synergic system (communications and trade). The highlight of the paper is the detailed study of the potential to put the brand into the center of IMC as a complex and universally functioning marketing and cultural construct. Brand owners can use IMC as an engagement tool. Engagement, in terms of psychology, is closely related to emotional decision-making. Therefore, the brand concept aspects reviewed are closely related to the IMC – the brand represents an integrated system, in which consumers can decide on following different journeys to a brands of their choice.
Keywords: integrated marketing communications (IMC), brand, market mix, advertising, social media. | https://journal.rhetoric.bg/?p=2252 |
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Hamid, K., & Babak, T. (2014). How significant are users’ opinions in social media? International Journal of Accounting & Information Management, 22(4), 254-272. doi:10.1108/IJAIM-02-2014-0010
Irena Pletikosa Cvijikj, F. M. (2011). A Case Study of the Effects of Moderator Posts within a Facebook Brand Page. Paper presented at the Social Informatics. In: Datta A., Shulman S., Zheng B., Lin SD., Sun A., Lim EP. (eds) Social Informatics. SocInfo 2011. Lecture Notes in Computer Science, vol 6984. Springer, Berlin, Heidelberg
Jayasingh, S., & Venkatesh, R. (2015). Customer engagement factors in facebook brand pages. Asian Social Science, 11(26), 19, 11. doi:10.5539/ass.v11n26p19.
Juhee, K., Liang, T., & Marie, F. A. (2015). Restaurant brand pages on Facebook: Do active member participation and monetary sales promotions matter? International Journal of Contemporary Hospitality Management, 27(7), 1662-1684. doi:10.1108/IJCHM-02-2014-0075
Kao, L., Huang, Y., & Sandnes, F.E. (2016). Mining time-dependent influential users in Facebook fans group. 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC), 000718-000723.
Kim, Y., & Chandler, J. D. (2018). How social community and social publishing influence new product launch: The case of Twitter during the Playstation 4 and Xbox ONE launches. Journal of Marketing Theory and Practice. doi:10.1080/10696679.2017.1389238
Kwok, L., & Yu, B. (2012). Spreading Social Media Messages on Facebook: An Analysis of Restaurant Business-to-Consumer Communications. Cornell Hospitality Quarterly, 54(1), 84-94. doi:10.1177/1938965512458360
Lipsman, A., Mudd, G., Rich, M., & Bruich, S. (2012). The Power of “Like”. Journal Of Advertising Research, 52(1), 40. doi:https://doi.org/10.2501/JAR-52-1-040-0520710
Liu, J.-W., Wang, Y.-H., Tsai, J. C. A., & Chang, J. Y. T. (2017). Ambidextrous Innovation and Game Market Fit Performance: Feedback from Game Testers. Journal of Computer Information Systems, 1-10. doi:10.1080/08874417.2017.1330127
M. Gainsbury, S., Hing, N., Delfabbro, P. H., & King, D. L. (2014). A taxonomy of gambling and casino games via social media and online technologies. International Gambling Studies, 14(2), 196-213. doi:10.1080/14459795.2014.890634
Machado, A. T. (2017). Consumer Engagement on Facebook Brand Page: the multiplier effect of comments. Paper presented at the Congresso IBERCOM, XV.
Na, S., Dennis, R., & Bixuan, S. (2015). How to make your Facebook posts attractive: A case study of a leading budget hotel brand fan page. International Journal of Contemporary Hospitality Management, 27(8), 1772-1790. doi:10.1108/IJCHM-06-2014-0302
Nguyen, N. P. D. (2015). Integration of social networking sites (with focus on Facebook) into CRM. First Monday. doi:https://doi.org/10.5210/fm.v4i9.690
Permana, S. D. H. (2017). E-marketing strategy in game industry with social media using business model. IMC 2016 Proceedings, 1(1), 258-263.
Phan Trong, N., & Myungsik, Y. (2014, 10-12 Feb. 2014). The lexicon-based sentiment analysis for fan page ranking in Facebook. Paper presented at the The International Conference on Information Networking 2014 (ICOIN2014) , Phuket, 2014, pp. 444-448.
doi: 10.1109/ICOIN.2014.6799721
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Rangan, K., Huang, Y.-K., Zhang, F., Yu, B., Maharabhushanam, P., & Kwok, L. (2015). Documenting business-to-consumer (B2C) communications on Facebook: What have changed among restaurants and consumers? Worldwide Hospitality and Tourism Themes, 7(3), 283-294. doi:10.1108/WHATT-03-2015-0018
S., V. B., M., E., & Veris. (2011). Social media around the world 2011. InSites Consulting. Retrieved. Retrieved from http://www.slideshare.net/stevenvanbelleghem/social-media-around-the-world2011?lead=394fd930572c9b62fb082021af5a6d0922046ec4.
Sabate, F., Berbegal-Mirabent, J., Canabate, A., & Lebherz, P. R. (2014). Factors influencing popularity of branded content in Facebook fan pages. European Management Journal, 32(6), 1001-1011. doi:https://doi.org/10.1016/j.emj.2014.05.001
?velch, J., & Krobova, T. (2017). Who Is the Note-Worthy Fan? Featuring Players in the Official Facebook Communication of Mainstream Video Games. Replay. The Polish Journal of Game Studies, 3, 81-100. | http://ir.lib.ncu.edu.tw:88/thesis/view_etd.asp?URN=105421025&fileName=GC105421025.pdf |
Many living in the state of Florida understand why wetlands are treasures of the state. But according to the U.S Fish and Wildlife Service, nearly half of 20.3 million wetland acres have been lost since settlers began draining and diking wetlands in the 19th century. And of course the growing population threatens what remains. So USDA’s Natural Resources Conservation service (NRCS) in Florida wants landowners to know there are programs to help preserve the state treasure.
The Agricultural Conservation Easement Program (ACEP) helps preserve and restore this precious resource. A landowner can sell a conservation easement to limit land uses, restore wetlands, protect wildlife habitat and prevent property development. Agricultural producers also conserve and protect water quality, reduce soil erosion and create wildlife habitat with financial and technical assistance from the Environmental Quality Incentive Program (EQIP).
NRCS notes those wetlands work hard for us, cleaning pollutants out of the water, storing it and controlling flooding. Coastal tidal salt marshes, mangrove swamps, inland southern swamps, freshwater marshes and riparian wetlands provide habitat for a vast array of rare plants and animals found nowhere else in the world.
Learn more about ACEP at your local NRCS office, or go to their website. | https://southeastagnet.com/2019/06/06/saving-florida-treasured-wetlands/?shared=email&msg=fail |
The "Epigenetic- Lab” performs the molecular analysis of various human diseases associated with epigenetic and / or genetic alterations.
In particular, the diagnostic activity of the “Epigenetic-Lab” includes 2 macro-areas.
BRAIN TUMORS
Molecular characterization of adult brain cancer.
Using an innovative approach, based on the analysis of the epigenomic profile (METHYLOME), it is possible to characterize a brain tumor.
This investigation, validated using reference technologies (proficency test), allows us to answer 3 fundamental diagnostic questions for the neuro-oncologist who must define the "management" of the patient with glioma /glioblastoma
These are:
MGMT methylation status (MSP + Methyloma)
IDH1-IDH2 mutation (direct sequencing + Methyloma)
1p-19q co-deletion (MLPA + Methyloma)
Furthermore, the genome wide methylome analysis allows us to obtain a series of additional information that help one to define and characterize the tumor sample (gene amplifications and loss of genes, such as EGFR amplification, CDKN2A/ B deletion, presence of H3K27 mutation, H3FA ...)
We also provide methylome profiles and mutation status of hTERT gene in Meningiomas, that together represent an unique tool for early recognition of aggressive tumor features and high recurrence risk, thus allowing a prompt and appropriate therapy.
GENETIC DISEASES
With MS-MLPA technique we provide genetic and epigenetic data for loci associated with various imprinting disorders (IDs).
In particular:
• Detection of epigenetic and genetic alteration in the region 15q11-q13 (diagnosis of Prader Willi Syndrome /Angelman's Syndrome)
• Detection of epigenetic and genetic alteration in the 11p15.5 (Beckwith Wideman's /Silver Russel Syndromes)
• Detection of uniparental disomy at chromosome 7 (UPD7) associated with rare forms of Silver Russell Syndrome.
• Targeted methylation analysis, for genetic diseases linked to DNA methylation defects
Identification of genetic mutations associated with Fabry disease, an X-linked lysosomal disorder, associated with defects in the alpha galactosidase A enzyme.
The investigation is carried out by direct sequencing of all exons (7) of the GLA gene and validation by Next-Generation sequencing.
The enzymatic assay that evaluates the activity of GLA is usually performed to screen potential affected males before genetic analysis. | https://www.ceinge.unina.it/en/epigenetic-lab |
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