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Arnav Tripathy
|
https://ncatlab.org/nlab/source/Arnav+Tripathy
|
* [webpage](https://www.math.harvard.edu/people/tripathy-arnav/)
## Selected writings
Systematic construction of [[Ricci tensor|Ricci flat]] [[Riemannian metrics]] on [[K3]] [[orbifolds]]:
* [[Shamit Kachru]], [[Arnav Tripathy]], [[Max Zimet]], _K3 metrics_ ([arXiv:2006.02435](https://arxiv.org/abs/2006.02435))
* [[Arnav Tripathy]], [[Max Zimet]], _A plethora of K3 metrics_ ([arXiv:2010.12581](https://arxiv.org/abs/2010.12581))
category: people
|
Arne Ostvaer
|
https://ncatlab.org/nlab/source/Arne+Ostvaer
|
__Paul Arne Østvær__ is a Norwegian mathematician specialized in abstract [[homotopy theory]]. Paul Arne used Quillen [[model categories]] to study questions in [[algebraic topology]], [[operator algebras]] and [[algebraic geometry]]. In algebraic geometry this includes $A^1$-homotopy theory, [[motive]]s, and even [[noncommutative motive]]s. In operator algebras he introduced a Quillen model structure on certain category of cubical presheaves on a category of operator algebras.
* [web](http://folk.uio.no/paularne)
* _Homotopy theory of $C^*$-algebras_, Frontiers in Mathematics,
Springer Basel, 2010, ([arxiv/0812.0154](http://arxiv.org/abs/0812.0154), [pdf](http://folk.uio.no/paularne/file0.pdf))
(on the [[model structure on operator algebras]])
* Markus Spitzweck, Paul Arne Østvær, _Motivic twisted K-theory_, [arxiv/1008.4915](http://arxiv.org/abs/1008.4915)
[[!redirects Paul Arne Østvær]]
[[!redirects Paul Arne Ostvaer]]
[[!redirects Arne Østvær]]
[[!redirects Paul Østvær]]
|
Arne Strøm
|
https://ncatlab.org/nlab/source/Arne+Str%C3%B8m
|
* [webpage](https://www.sv.uio.no/econ/english/people/aca/arnest/)
## Selected writings
Early discussion and results on [[Hurewicz cofibrations]] and [[NDR pairs]]:
* [[Arne Strøm]], _Note on cofibrations_, Math. Scand. **19** (1966) 11-14 ([jstor:24490229](https://www.jstor.org/stable/24490229), [dml:165952](https://eudml.org/doc/165952), MR0211403)
* [[Arne Strøm]], _Note on cofibrations II_, Math. Scand. **22** (1968) 130--142 ([jstor:24489730](https://www.jstor.org/stable/24489730), [dml:166037](https://eudml.org/doc/166037), MR0243525)
Introducing the [[Strøm model structure]] [[model structure on topological spaces|on topological spaces]] and discussion of the [[classical homotopy category]]:
* [[Arne Strøm]], _The homotopy category is a homotopy category_, Archiv der Mathematik 23 (1972) ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/strom-the-homotopy-category-is-a-homotopy-category-1972.pdf), [[Strom_HomotopyCategory.pdf:file]])
Introducing the
* {#Strom72} [[Arne Strøm]], _The homotopy category is a homotopy category_, Archiv der Mathematik 23 (1972) ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/strom-the-homotopy-category-is-a-homotopy-category-1972.pdf), [[Strom_HomotopyCategory.pdf:file]])
## Related $n$Lab entries
* [[Strøm model structure]]
* [[Strøm's theorem]]
[[!redirects Arne Strom]]
|
Arnold conjecture
|
https://ncatlab.org/nlab/source/Arnold+conjecture
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
(...)
## Related concepts
* [[symplectic geometry]]
## Literature
Review and introduction:
* Roman Golovko, *On variants of the Arnold conjecture*, Archivum Mathematicum, Tomus 56 (2020), 277–28 ([pdf](https://www.emis.de/journals/AM/20-5/Golovko_2020.pdf), [doi:10.5817/AM2020-5-277](http://dx.doi.org/10.5817/AM2020-5-277))
Via [[algebraic topology]]/[[stable homotopy theory]] and specifically in relation to [[Morava K-theory]]:
* [[Mohammed Abouzaid]], [[Andrew J. Blumberg]], _Arnold Conjecture and Morava K-theory_, ([arXiv:2103.01507](https://arxiv.org/abs/2103.01507))
|
Arnold Neumaier
|
https://ncatlab.org/nlab/source/Arnold+Neumaier
|
[Arnold Neumaier](
http://arnold-neumaier.at)
I am interested in the foundations of mathematics
(building an automatic mathematical research system),
the foundations of theoretical physics (interpretation
of quantum mechanics and statistical physics),
and in applied mathematics of all sorts (from protein folding
to cattle breeding to uncertainty modeling in space system design).
I wrote books on topics in finite geometry, numerical analysis, and Lie algebras (different ones for each of
these subjects).
## Selected writings
On the foundations and [[interpretation of quantum mechanics|interpretation]] of [[quantum physics]]:
* [[Arnold Neumaier]], *Introduction to [[coherent product|coherent spaces]]*, ([arXiv:1804.01402](https://arxiv.org/abs/1804.01402))
* [[Arnold Neumaier]], *Coherent Quantum Physics: A Reinterpretation of the Tradition*, 1st edition, De Gruyter (2019) [ISBN:9783110667295, [doi:10.1515/9783110667387](https://doi.org/10.1515/9783110667387)]
On [[generalized quantum measurement]] via [[POVMs]] (and [[Born's rule]]):
* [[Arnold Neumaier]], *Born's rule and measurement* [[arXiv:1912.09906](https://arxiv.org/abs/1912.09906)]
category: people
|
Arnold Nordsieck
|
https://ncatlab.org/nlab/source/Arnold+Nordsieck
|
* [Wikipedia entry](https://en.wikipedia.org/wiki/Arnold_Nordsieck)
## related $n$Lab entries
* [[infrared divergence]], [[quantum electrodynamics]]
category: people
|
Arnold Shapiro
|
https://ncatlab.org/nlab/source/Arnold+Shapiro
|
* [Wikipedia entry](https://en.wikipedia.org/wiki/Arnold_S._Shapiro)
## Selected writings
Introducing the [[Atiyah-Bott-Shapiro orientation]] [[MSpin]]$\to$[[KO]] and [[MSpin^c|MSpin<sup><i>c</i></sup>]]$\to$[[KU]]:
* {#AtiyahBottShapiro64} [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]], _Clifford modules_, Topology Volume 3, Supplement 1, July 1964, Pages 3-38 (<a href="https://doi.org/10.1016/0040-9383(64)90003-5">doi:10.1016/0040-9383(64)90003-5</a>, [pdf](http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf))
category: people
|
Arnold Sommerfeld
|
https://ncatlab.org/nlab/source/Arnold+Sommerfeld
|
One of the founding fathers of [[quantum mechanics]].
* [Wikipedia entry](http://en.wikipedia.org/wiki/Arnold_Sommerfeld)
## Related entries
* [[Bohr-Sommerfeld quantization]]
category: people
|
Arnold-Kuiper-Massey theorem
|
https://ncatlab.org/nlab/source/Arnold-Kuiper-Massey+theorem
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Statements
+-- {: .num_prop}
###### Proposition
**(Arnold-Kuiper-Massey theorem)**
The [[4-sphere]] is the [[quotient space]] of the [[complex projective plane]] by the [[finite group of order 2|O(1)]]-[[action]] by [[complex conjugation]] (on homogeneous coordinates):
$$
\mathbb{C}P^2 / \mathrm{O}(1)
\simeq
S^4
$$
=--
([Arnold 71](#Arnold71), [Massey 73](#Massey73), [Kuiper 74](#Kuiper74), [Arnold 88](#Arnold88))
In fact, this is is the beginning of a small pattern indexed by the [[real normed division algebras]]:
+-- {: .num_prop}
###### Proposition
The [[7-sphere]] is the [[quotient space]] of the (right-)[[quaternionic projective plane]] by the left multiplication [[action]] by [[U(1)]] $\subset$ [[Sp(1)]]:
$$
\mathbb{H}P^2 / \mathrm{U}(1)
\simeq
S^7
$$
=--
([Arnold 99](#Arnold99), [Atiyah-Witten 01, Sec. 5.5](#AtiyahWitten01))
+-- {: .num_prop}
###### Proposition
The [[13-sphere]] is the [[quotient space]] of the (right-)[[octonionic projective plane]] by the left multiplication [[action]] by [[Sp(1)]]:
$$
\mathbb{O}P^2 / \mathrm{Sp}(1)
\simeq
S^{13}
$$
=--
([Atiyah-Berndt 02](#AtiyahBerndt02))
## References
### AKM-theorem for the complex projective plane
The original proof that the [[4-sphere]] is a quotient of the [[complex projective plane]] by an action of [[cyclic group of order 2|Z/2]]:
* {#Arnold71} [[Vladimir Arnold]], _On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms_, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.
* {#Massey73} [[William Massey]], _The quotient space of the complex projective space under conjugation is a 4-sphere_, Geometriae Didactica 1973 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/massey5.pdf))
* {#Kuiper74} [[Nicolaas Kuiper]], _The quotient space of $\mathbb{C}P(2)$ by complex conjugation is the 4-sphere_, Mathematische Annalen, 1974 ([doi:10.1007/BF01432386](https://doi.org/10.1007/BF01432386))
* {#Arnold88} [[Vladimir Arnold]], _Ramified covering $\mathbb{C}P^2 \to S^4$, hyperbolicity and projective topology_, Siberian Math. Journal 1988, V. 29, N 5, P.36-47
See also
* José Seade, Section V.5 in: _On the Topology of Isolated Singularities in Analytic Spaces_, Progress in Mathematics, Birkhauser 2006 ([ISBN:978-3-7643-7395-5](https://www.springer.com/gp/book/9783764373221))
* J. A. Hillman, _An explicit formula for a branched covering from $\mathbb{C}P^2$ to $S^4$_ ([arXiv:1705.05038](https://arxiv.org/abs/1705.05038))
The [[SO(3)]]-[[equivariant function|equivariant]] enhancement:
* Le, [[José Seade]], [[Alberto Verjovsky]], _Quadrics, orthogonal actions and involutions in complex projective space_, L'Enseignement Mathématique, t. 49 (2003) ([e-periodica:001:2003:49::488](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#488))
### Generalization to the quaternionic projective plane
The generalization to the [[7-sphere]] being a [[U(1)]]-quotient of the [[quaternionic projective plane]] is due to
* {#Arnold99} [[Vladimir Arnold]], _Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation_, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67; English translation: Proceedings of the Steklov Institute of Mathematics, 1999, 224, 46–56 ([mathnet:tm691](http://mi.mathnet.ru/eng/tm691))
and independently due to
* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]], Section 5.5 of: _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177), [doi:10.4310/ATMP.2002.v6.n1.a1]( https://dx.doi.org/10.4310/ATMP.2002.v6.n1.a1))
> (in the context of [[M-theory on G2-manifolds]])
### Generalization to the octonionic projective plane
Another proof of these cases and further generalization to the [[13-sphere]] being an [[Sp(1)]]-quotient of the [[octonionic projective plane]]:
* {#AtiyahBerndt02} [[Michael Atiyah]], [[Jürgen Berndt]], *Projective planes, Severi varieties and spheres*, in: Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 ([arXiv:math/0206135](https://arxiv.org/abs/math/0206135), [doi:10.4310/SDG.2003.v8.n1.a1](https://dx.doi.org/10.4310/SDG.2003.v8.n1.a1))
[[!redirects AKM theorem]]
[[!redirects AKM-theorem]]
|
Aron C. Wall
|
https://ncatlab.org/nlab/source/Aron+C.+Wall
|
* [personal page](http://www.wall.org/~aron/)
## Selected writings
Argument that the image of *[[wormhole]] traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:
* [[Ping Gao]], [[Daniel Louis Jafferis]], [[Aron C. Wall]], *Traversable Wormholes via a Double Trace Deformation*, Journal of High Energy Physics **2017** 151 (2017) [[arXiv:1608.05687](https://arxiv.org/abs/1608.05687), <a href="https://doi.org/10.1007/JHEP12(2017)151">doi:10.1007/JHEP12(2017)151</a>]
On [[holographic tensor networks]] and [[quantum error correcting codes]] in full [[AdS-CFT]]:
* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_, JHEP 2019:69 ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))
* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Holographic Tensor Networks in Full AdS/CFT_ ([arXiv:1902.10157](https://arxiv.org/abs/1902.10157))
category: people
[[!redirects Aron Wall]]
|
Aron J. Beekman
|
https://ncatlab.org/nlab/source/Aron+J.+Beekman
|
* [personal page](https://abeekman.nl/)
## Selected writings
On [[Abrikosov vortices]] in [[superconductors]] as [[strings]]:
* {#BeekmanZaanen11} [[Aron J. Beekman]], [[Jan Zaanen]], *Electrodynamics of Abrikosov vortices: the field theoretical formulation*, Front. Phys. **6** (2011) 357–369 [[doi:10.1007/s11467-011-0205-0](https://doi.org/10.1007/s11467-011-0205-0)]
category: people
[[!redirects Aron Beekman]]
|
Arpon Raksit
|
https://ncatlab.org/nlab/source/Arpon+Raksit
|
* [personal page](https://www.mit.edu/~arpon/)
## Selected writings
On the [[J-homomorphism]]:
* [[Arpon Raksit]], _Vector fields and the J-homomorphism_, 2014 ([pdf](http://stanford.edu/~arpon/math/files/vfields.pdf), [[Raksit_JHomomorphism.pdf:file]])
On [[transchromatic characters]] in [[global equivariant homotopy theory]]:
* {#Raksit15} [[Arpon Raksit]], _Characters in global equivariant homotopy theory_, 2015 ([pdf](https://www.math.harvard.edu/media/raksit.pdf), [[Raksit_Characters.pdf:file]])
category: people
|
arrangement of hyperplanes
|
https://ncatlab.org/nlab/source/arrangement+of+hyperplanes
|
#Contents#
* table of contents
{:toc}
## Idea
An arrangement of [[hyperplanes]] is a finite set of hyperplanes in a (finite-dimensional) linear, affine or projective space. Usually (such as for [[configuration spaces of points]]) one studies in fact the __[[complement]] of the union of the hyperplanes__, and its topological and other properties. This space is the basis of many interesting [[fiber bundles]] appearing in [[conformal field theory]], study of [[hypergeometric functions]] (Aomoto, Gelfand, Varchenko), [[quantum groups]] etc., for more on which see at
*[[Knizhnik-Zamolodchikov equation]]*.
## References
### General
* Wikipedia [arrangement of hyperplanes](http://en.wikipedia.org/wiki/Arrangement_of_hyperplanes)
* [[eom]]: [arrangement of hyperplanes](http://eom.springer.de/A/a110700.htm)
* Peter Orlik, Hiroaki Terao, _Arrangements of Hyperplanes_, Grundlehren der Mathematischen Wissenschaften __300__, Springer 1992, MR1217488
* [[I. M. Gelfand]], M. M. [[Kapranov]], [[A. Zelevinsky]], _Discriminants, resultants and multidimensional determinants_, Birkhäuser 1994, 523 pp.
* Corrado De Concini, Claudio Procesi, _Topics in hyperplane arrangements, polytopes and box-splines_, Universitext 223, Springer 2010.
Stanley's survey focuses instead more on combinatorics of the intersection poset of an arrangement (as well as arrangements in the case of vector space over a finite field):
* Richard P. Stanley, _An introduction to hyperplane arrangements_, in: Geometric combinatorics, 389–496, IAS/Park City Math. Ser., 13, Amer. Math. Soc. 2007, [pdf](http://www.math.umn.edu/~ezra/PCMI2004/stanley.jcp.pdf), [errata](http://www-math.mit.edu/~rstan/arrangements/errata.pdf)
* Richard Randell, _Morse theory, Milnor fibers and minimality of hyperplane arrangements_, [math.AG/0011101](http://arxiv.org/abs/math/0011101)
* [[Daniel C. Cohen]], Michael Falk, Richard Randell, _Discriminantal bundles, arrangement groups, and subdirect products of free groups_, [arxiv/1008.0417](http://arxiv.org/abs/1008.0417)
### Relating to configuration spaces of points
See the references on *[Braid representatioons via twisted de Rham cohomology of configuration spaces](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces)*
[[!redirects arrangements of hyperplanes]]
[[!redirects hyperplane arrangement]]
|
arrow
|
https://ncatlab.org/nlab/source/arrow
|
The term _arrow_ is sometimes used as a synonym for _[[morphism]]_, _[[map]]_, and also for directed edge (in a [[directed graph]] or [[quiver]]).
In [[computer science]] it may also refer to a concept generalizing [[monad (in computer science)|monads]], see at _[[arrow (in computer science)]]_.
Abbreviations for the class of all arrows of a category $\mathsf{C}$ used in the literature include $\mathrm{Arr}(\mathsf{C})$, $\mathrm{Ar}(\mathsf{C})$, and $\mathrm{Mor}(\mathsf{C})$. Note that this is the class of [[objects]] of the [[arrow category]] of $\mathsf{C}$, and the same notations are sometimes used for that whole category.
category: disambiguation
[[!redirects arrows]]
|
arrow (in computer science)
|
https://ncatlab.org/nlab/source/arrow+%28in+computer+science%29
|
#Contents#
* table of contents
{:toc}
## Idea
In [[computer science]], the notion of *arrows* [[Hughes 2000](#Hughes00)] generalizes that of _[[monads in computer science]]_.
More concretely, the notion of *arrow* abstracts that of the [[hom-functor|hom-]][[profunctor]] of the [[Kleisli category]] $\mathrm{Kl}(T)$ for a [[strong monad]] $T \colon \mathbf C \to \mathbf C$ as an endoprofunctor on $\mathbf C$.
Therefore, an arrow $A \colon \mathbf C^{op} \times \mathbf C \to \mathbf{Set}$ can be thought as a putative replacement of $\mathrm{Hom}_{\mathbf C} \colon \mathbf C^{op} \times \mathbf C \to \mathbf{Set}$. In fact an arrow comes equipped with a transformation $\mathrm{arr} : \mathrm{Hom}_{\mathbf C} \Rightarrow A$ that lift morphisms of $\mathbf C$ to the 'augmented' morphisms given by $A$, and with a transformation $\ggg : A \circ A \Rightarrow A$ that behaves like a composition operation. Then one requires that $\ggg$ is associative and that $\mathrm{arr}(1_a)$ is the unit of this operation.
These data makes $A$ a [[promonad]] on $\mathbf C$, i.e. a monad on $\mathbf C$ in the bicategory of [[profunctors]]. Since every monad on $\mathbf{Set}$ (or on any reasonable enriching base where programmers work) is strong, arrows traditionally generalize the hom-profunctor of [[strong monads]]. The additional requirement that $A$ is a [[strong profunctor]] (and the laws it is required to satisfy) imply that an arrow is a [[strong monad|strong monad]] in the bicategory of [[profunctors]] ([Asada10](#Asada10)).
## Definition
\begin{definition}
An arrow $A$ on a [[monoidal category]] $\mathbf C$ is a monoid in the category of [[strong profunctor|strong endoprofunctors]] on $\mathbf C$, i.e. it's a functor $\mathbf C^{op} \times \mathbf C \to \mathbf{Set}$ equipped with the following structure:
1. A natural family of morphisms $\mathrm{arr} : \mathbf C(a,b) \to A(a,b)$ (the unit of the monoid),
2. A natural family of morphisms called *composition* $\ggg : A(a,b) \times A(b,c) \to A(a,c)$ (the multiplication of the monoid), which satisfy an associative law and for which $\mathrm{arr}_{a,a}(1_a)$ is the unit,
3. A family of morphisms called *strength* $s_{a,b,m} : A(a,b) \to A(m \otimes a, m \otimes b)$, which is [[dinatural transformation|dinatural]] in $m$ and natural in $a, b$, and satisfies coherence laws that make $(A, s)$ a [[strong profunctor]].
\end{definition}
## References
Arrows originate in Section 2 of
* {#Hughes00} [[John Hughes]], *Generalising monads to arrows*, Sci. Comput. Program. **37** 1-3 (2000) 67-111 [[doi:10.1016/S0167-6423(99)00023-4](https://doi.org/10.1016/S0167-6423%2899%2900023-4), [pdf](http://www.cse.chalmers.se/~rjmh/Papers/arrows.pdf)]
The 'justification' of arrows as [[strong monads]] in $\mathbf{Prof}$ is given in
* {#Asada10} Kazuyuki Asada, _Arrows are strong monads_, 2010 ([pdf](http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf))
Comparisons of monads with [[applicative functors]] (also known as idioms) and with [[arrows (in computer science)]] are in
* [[Sam Lindley]], [[Philip Wadler]], [[Jeremy Yallop]], _Idioms are Oblivious, Arrows are Meticulous, Monads are Promiscuous_, Electron. Notes Theor. Comput. Sci. 229(5), pp. 97--117, 2011 ([doi:10.1016/j.entcs.2011.02.018](https://doi.org/10.1016/j.entcs.2011.02.018))
* [[Exequiel Rivas]], _Relating Idioms, Arrows and Monads from Monoidal Adjunctions_, MSFP@FSCD 2018, pp. 18--33 ([arXiv:1807.04084](https://arxiv.org/abs/1807.04084), [doi:10.4204/EPTCS.275.3](https://doi.org/10.4204/EPTCS.275.3))
[[!redirects arrows (in computer science)]]
[[!redirects arrows in computer science]]
|
arrow (∞,1)-category
|
https://ncatlab.org/nlab/source/arrow+%28%E2%88%9E%2C1%29-category
|
#Contents#
* table of contents
{:toc}
## Defintion
For $\mathcal{C}$ an [[(∞,1)-category]], its **arrow $(\infty,1)$-category** is the [[(∞,1)-category of (∞,1)-functors]]
$$
\mathcal{C}^{\Delta^1}
\coloneqq
Func_\infty(\Delta^1, \mathcal{C})
\,.
$$
## Related concepts
* [[slice (∞,1)-category]]
* [[arrow category]]
* [[arrow (∞,1)-topos]]
|
arrow (∞,1)-topos
|
https://ncatlab.org/nlab/source/arrow+%28%E2%88%9E%2C1%29-topos
|
#Contents#
* table of contents
{:toc}
## Definition
For $\mathbf{H}$ an [[(∞,1)-topos]], its _arrow $(\infty,1)$-topos_ is its [[arrow (∞,1)-category]].
## Properties
For $(X_{def} \stackrel{\iota_X}{\to} X_{bulk})$ and $(A_{def} \stackrel{\mathbf{F}}{\to} A_{bulk})$ in $\mathbf{H}^{\Delta^1}$, there is an [[(∞,1)-pullback]] diagram
$$
\array{
\mathbf{H}^{\Delta^1}(\iota_X, \mathbf{F})
&\stackrel{}{\to}&
\mathbf{H}(X_{def}, A_{def})
\\
\downarrow &\swArrow& \downarrow
\\
\mathbf{H}(X_{bulk}, A_{bulk})
&\to&
\mathbf{H}(X_{def}, A_{bulk})
}
\,.
$$
Note: 'def' and 'bulk' indicate terms of fields in a [[QFT with defects]].
## Related concepts
* [[slice (∞,1)-topos]]
* [[relative cohomology]]
* [[Sierpinski topos]]
|
arrow category
|
https://ncatlab.org/nlab/source/arrow+category
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
Every [[category]] $C$ gives rise to an _arrow category_ $Arr(C)$ such that the [[objects]] of $Arr(C)$ are the [[morphisms]] (or _arrows_, hence the name) of $C$.
## Definition
For $C$ any [[category]], its **arrow category** $Arr(C)$ is the category such that:
* an [[object]] $a$ of $Arr(C)$ is a [[morphism]] $a\colon a_0 \to a_1$ of $C$;
* a [[morphism]] $f\colon a \to b$ of $Arr(C)$ is a [[commutative square]]
$$ \array {
a_0 & \overset{f_0}\to & b_0 \\
\llap{a}\downarrow & & \rlap{b}\downarrow \\
a_1 & \underset{f_1}\to & b_1
} $$
in $C$;
* [[composition]] in $Arr(C)$ is given simply by placing commutative squares side by side to get a commutative oblong.
This is isomorphic to the [[functor category]]
$$
Arr(C) := Funct(I,C) = [I,C] = C^I
$$
for $I$ the [[interval category]] $\{0 \to 1\}$. $Arr(C)$ is also written $[\mathbf{2},C]$, $C^{\mathbf{2}}$, $[\Delta[1],C]$, or $C^{\Delta[1]}$, since $\mathbf{2}$ and $\Delta[1]$ (for the $1$-[[simplex]]) are common notations for the interval category.
## Properties
\begin{proposition}
The arrow category $Arr(C)$ is equivalently the [[comma category]] $(id/id)$ for the case that $id\colon C \to C$ is the [[identity functor]].
\end{proposition}
\begin{remark}
$Arr(C)$ plays the role of a [[directed homotopy theory|directed]] [[homotopy|path object]] for categories in that functors
$$
X \to Arr(Y)
$$
are the same as [[natural transformations]] between functors between $X$ and $Y$.
\end{remark}
\begin{example}\label{SliceCategoriesAndArrowCategory}
**(arrow category is [[Grothendieck construction]] on [[slice categories]])**
\linebreak
For $\mathcal{S}$ any category, let
$$
\mathcal{S}_{(-)}
\,\colon\,
\mathcal{S}
\longrightarrow
Cat
$$
be the [[pseudofunctor]] which sends
* an [[object]] $B \,\in\, \mathcal{S}$ to the [[slice category]] $\mathcal{S}_{/B}$,
* a [[morphism]] $f \colon B \to B'$ to the left [[base change]] [[functor]] $f_! \,\colon\, \mathcal{C}_{B} \to \mathcal{C}_{/B'}$ given by post-[[composition]] in $\mathcal{C}$.
The [[Grothendieck construction]] on this functor is the [[arrow category]] $Arr(\mathcal{S})$ of $\mathcal{S}$:
$$
Arr(\mathcal{S})
\;\;\;
\simeq
\;\;\;
\int_{B \in \mathcal{S}}
\mathcal{S}_{/B}
\mathrlap{\,.}
$$
This follows readily by unwinding the definitions.
In the refinement to the [[Grothendieck construction for model categories]] (here: [[slice model categories]] and [[model structures on functors]]) this equivalence is also considered for instance in [Harpaz & Prasma (2015), above Cor. 6.1.2](Grothendieck+construction+for+model+categories#HarpazPrasma15).
The correponding [[Grothendieck fibration]] is also known as the *[[codomain fibration]]*.
\end{example}
## Related concepts
* [[path space object]]
* [[slice category]], [[undercategory]]
* [[arrow (∞,1)-category]]
* [[arrow (∞,1)-topos]]
* [[twisted arrow category]]
[[!redirects arrow category]]
[[!redirects arrow categories]]
[[!redirects category of morphisms]]
[[!redirects categories of morphisms]]
|
arrow structure
|
https://ncatlab.org/nlab/source/arrow+structure
|
# Arrow frames, arrow languages, arrow models and arrow logics
* automatic table of contents goes here
{:toc}
## Idea
These structures are intended to allow one to talk about 'all such objects
as may be represented in a picture by arrows', (Venema).
## Arrow frames
+-- {: .un_defn}
###### Definition
An **arrow frame** is a [[frame (modal logic)|frame]], $\mathfrak{F} = (W, C, R, I )$, such that $C$ is a ternary relation, (so $C\subseteq
W \times W \times W$), $R$ is a binary relation, and $I$ a unary one, so $I\subseteq W$.
=--
## Gloss
The interpretation intended for these relations is similar to the structure of a [[groupoid]], but from a relational point of view.
* For $C(a,b,c)$ think $a = b\star c$, 'the' _composite_ of $b$ and $c$, but, of course, $a$ need not be uniquely defined by this;
* For $R(a,b)$, think $b$ is a 'reverse' arrow for $a$;
* For $I(a)$ i.e. $a\in I$, think $a$ is an 'identity arrow'.
This defines a frame by a relational signature. It does not specify any formulae to be satisfied, nor is there a [[geometric models for modal logics|valuation]] around to give some 'meaning' to the structure.
## Arrow languages
The arrow language follows the general form of the [[modal logic|modal languages]], but note that there are several 'arities' of relation in the structure, so we need several different types of modal operators.
+-- {: .un_defn}
###### Definition
The **arrow language** is defined by the rule
$$\phi ::= p \mid \bot \mid \neg \phi \mid \phi_1 \vee \phi_2 \mid \phi\circ \psi \mid \otimes\phi\mid 1',$$
where the $p$ are the propositional variables, as usual, and in addition
* $1'$ is 'identity' and is a nullary modality or modal constant;
* $\otimes$ (an unfortunate notation, but which seems fairly standard in the sources used for this) is the 'converse' operator and is a 'diamond', i.e. a unary operator, and
* $\circ$ is the 'composition operator' and is a dyadic operator.
=--
The possible interpretations or readings of these include
$1'$ is SKIP;
$\otimes \phi$ is $phi$ conversely; and
$\phi\circ \psi$ is 'first $\phi$ then $\psi$'.
## Arrow models
The usual rules for modal semantics apply. An **arrow model**, $\mathfrak{M}$, consists of an arrow frame, $\mathfrak{F} = (W, C, R, I )$, (and a valuation $V$, which plays a somewhat behind the scenes role in this general situation). In what follows, $a\in W$,
* $\mathfrak{M},a \models 1'$ if and only if $Ia$;
* $\mathfrak{M},a \models \otimes\phi$ if and only if $\mathfrak{M},b \models\phi$ for some $b$ with $R a b$;
* $\mathfrak{M},a \models\phi\circ \psi$ if and only if $\mathfrak{M},b \models\phi$ and $\mathfrak{M},c \models\psi$ for some $b,c$ with $C a b c$.
## References
A general treatment of these ideas can be found in
* P. [[Blackburn]], M. de Rijke and Y. [[Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001,
whilst a short introduction is
* Y. [[Venema]], [_A crash course in Arrow Logic_](http://staff.science.uva.nl/~yde/papers/arrow.pdf), in: M Marx, L Pólos and M Masuch (editors), Arrow Logic and Multi-Modal Logic,
Studies in Logic, Language and Information, CSLI Publications, Stanford (1996) 3--34.
[[!redirects arrow structure]]
[[!redirects arrow structures]]
[[!redirects arrow logic]]
[[!redirects arrow logics]]
|
arsmath
|
https://ncatlab.org/nlab/source/arsmath
|
I'm Walt. I blog at [Ars Mathematica](http://arsmathematica.net).
category:people
|
Arthur Bartels
|
https://ncatlab.org/nlab/source/Arthur+Bartels
|
* [website](http://wwwmath.uni-muenster.de/u/bartelsa/)
category: people
|
Arthur Besse
|
https://ncatlab.org/nlab/source/Arthur+Besse
|
Arthur L. Besse is a collective of mathematicians who worked together closely in round-table meetings at various institutions in France, focused mainly on topics in [[differential geometry]]. His middle name is Lancelot, but this is never written. The collective has consisted of: Marcel Berger, Hermann Karcher, Jean-Pierre Bourguignon, Geneviève Averous, Nigel Hitchin, Jerry Kazdan, Pierre Pansu, Paul Gauduchon, Dennis DeTurck, Lionel Bérard-Bergery, Andrei Derdzinski, Josette Houillot, Norihito Koiso, Albert Polombo, John A. Thorpe, Jacques Lafontaine, Liane Valère. From the preface of (besse) we find the acknowledgement:
"Enfin, qu'il me soit permis de saluer ici mon prédécesseur et homonyme Jean Besse, de Zürich, qui s'est illustré dans la théorie des fonctions d'une variable complexe..." (Finally, here let me salute my predecessor and namesake Jean Besse of Zürich, who has illustrated himself in the theory of the functions of a complex variable...)
at which point the author(s) refer to the article:
Besse, Jean. _Sur le domaine d'existence d'une fonction analytique_, Commentarii Mathematici Helvetici 10 (1938).
## selected publications
* Arthur L. Besse {#besse}, _Einstein Manifolds_, Springer-Verlag 1987.
## related $n$Lab entries
* [[holonomy]], [[special holonomy]]
* [[curvature]]
* [[spectral theory]] of the [[Laplacian]]
* [[Alexandrov space]]
* [[Riemannian geometry]]
* [[subriemannian geometry]]
* [[Einstein manifolds]]
* [[quaternionic manifold]], [[hyperKähler manifold]], [[quaternionic-Kähler manifold]]
category: people
|
Arthur Cayley
|
https://ncatlab.org/nlab/source/Arthur+Cayley
|
_Arthur Cayley_ (16 August 1821 – 26 January 1895) is one of the principal and most prolific mathematicians of 19-th century, especially in [[algebra]]. He also worked on questions in [[algebraic geometry]], [[graph theory]], ...
* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Cayley)
## Selected writings
Introducing the [[octonions]]:
* {#Cayley1845} [[Arthur Cayley]], _On certain results relating to quaternions_, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 3 Volume 26, 1845 - Issue 171 ([doi:10.1080/14786444508562684](https://doi.org/10.1080/14786444508562684))
Cayley's 1848 paper on elimination theory contains probably the first serious application of [[homological algebra]], including [[Koszul complexes]]. It is reproduced and commented in the book
* [[I. M. Gelfand]], [[Mikhail Kapranov|M. M. Kapranov]], A. Zelevinsky, _Discriminants, [[resultants]], and multidimensional determinants_, Birkhäuser 1994, 523 pp.
## Related $n$Lab entries
* [[resultant]]
* [[hyperdeterminant]]
* [[Cayley form]]
* [[weak Cayley table]]
* [[Cayley-Hamilton theorem]]
[[!redirects Cayley]]
category: people
|
Arthur Eddington
|
https://ncatlab.org/nlab/source/Arthur+Eddington
|
[[!redirects Eddington]]
* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Eddington)
category: people
|
Arthur Greenspoon
|
https://ncatlab.org/nlab/source/Arthur+Greenspoon
|
* [[Eric Sharpe]], [[Arthur Greenspoon]], _[Advances in String Theory: The First Sowers Workshop in Theoretical Physics](http://www.ams.org/bookstore-getitem/item=AMSIP-44)_, AMS 2008
|
Arthur Gretton
|
https://ncatlab.org/nlab/source/Arthur+Gretton
|
* [webpage](https://www.is.mpg.de/~bs)
## Selected writings
On [[Bochner's theorem]] generalized to [[non-abelian groups]] in the context of [[kernel methods]]:
* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))
category: people
|
Arthur H. Stone
|
https://ncatlab.org/nlab/source/Arthur+H.+Stone
|
> Not to be confused with [[Arthur Lewis Stone]].
Arthur Harold Stone was a [[general topology|general topologist]].
He got his PhD degree in 1941 from Princeton University,
advised by [[Solomon Lefschetz]].
## Selected writings
On [[numerable covers]]:
* {#Stone} [[Arthur H. Stone]], _Paracompactness and product spaces_, Bulletin of the American Mathematical Society 54:10 (1948), 977–983 ([doi:10.1090/S0002-9904-1948-09118-2](https://doi.org/10.1090/S0002-9904-1948-09118-2))
## Related entries
* [[numerable cover]]
[[!redirects Arthur Harold Stone]]
|
Arthur Jaffe
|
https://ncatlab.org/nlab/source/Arthur+Jaffe
|
* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Jaffe)
## Selected writings
On [[non-perturbative quantum field theory]]:
* [[Gerard ’t Hooft]], [[Arthur Jaffe]], [[Gerhard Mack]], [[P. K. Mitter]], [[Raymond Stora]] (eds.) *Nonperturbative Quantum Field Theory*, NATO Science Series B **185** (1988) [[doi:10.1007/978-1-4613-0729-7](https://doi.org/10.1007/978-1-4613-0729-7)]
On the relation between the practice of [[mathematics]] and [[physics]] (for historical context see [Hitchin 20, Sec. 8](Michael+Atiyah#Hitchin20)):
* {#JaffeQuinn93} [[Arthur Jaffe]], [[Frank Quinn]], _"Theoretical Mathematics": Towards a cultural synthesis of mathematics and theoretical physics_, Bulletin of the AMS, Volume 29,Number 1, July 1993 ([arXiv:math/9307227](http://arxiv.org/abs/math/9307227))
> we claim that the role of rigorous [[proof]] in mathematics is functionally [[analogy|analogous]] to the role of [[experiment]] in the natural sciences ([Jaffe-Quinn 93, p. 2](JaffeQuinn93))
On the [[mass gap]]-problem in [[Yang-Mills theory]]:
* {#JaffeWitten00} [[Arthur Jaffe]], [[Edward Witten]], _Quantum Yang-Mills theory_ (2000) [[[JaffeWitten-QuantumYangMills.pdf:file]]]
## Related entries
* [[constructive quantum field theory]]
category: people
|
Arthur L. Stone
|
https://ncatlab.org/nlab/source/Arthur+L.+Stone
|
> Not to be confused with [[Arthur Harold Stone]], a general topologist.
* [Mathematics Genealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=12504)
## Selected writings
On [[adjunctions]], their induced ([[comonad|co]]-)[[monads]] and their decomposition, such as of [[idempotent adjunctions]]:
* {#MacDonaldStone82} [[John L. MacDonald]], [[Arthur L. Stone]], *The tower and regular decomposition*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 23 (1982) no. 2, pp. 197-213 ([numdam:CTGDC_1982__23_2_197_0](http://www.numdam.org/item/?id=CTGDC_1982__23_2_197_0))
category: people
[[!redirects Arthur Stone]]
[[!redirects Arthur Lewis Stone]]
|
Arthur Ogus
|
https://ncatlab.org/nlab/source/Arthur+Ogus
|
* [webpage](http://math.berkeley.edu/~ogus/)
## related $n$Lab entries
* [[logarithmic geometry]]
category: people
[[!redirects Ogus]]
|
Arthur Parzygnat
|
https://ncatlab.org/nlab/source/Arthur+Parzygnat
|
* [webpage](https://arthurparzygnat.com/)
## Selected writings
On [[higher parallel transport]] for [[connections on a 2-bundle]]:
* {#Parzygnat18} [[Arthur Parzygnat]], _Two-dimensional algebra in lattice gauge theory_, Journal of Mathematical Physics 60, 043506 (2019) ([arXiv:1802.01139](https://arxiv.org/abs/1802.01139), [doi:10.1063/1.5078532](https://doi.org/10.1063/1.5078532))
## Related $n$Lab entries
* [[Gelfand-Naimark-Segal construction]]
* [[higher parallel transport]], [[connection on a 2-bundle]]
* [[tensor network]]
category: people
[[!redirects Arthur J. Parzygnat]]
|
Arthur Prior
|
https://ncatlab.org/nlab/source/Arthur+Prior
|
A logician and philosopher, noted for his treatment of tense logic, a form of [[temporal logic]].
* [SEP entry](http://plato.stanford.edu/entries/prior/)
|
Arthur Sale
|
https://ncatlab.org/nlab/source/Arthur+Sale
|
[[!redirects Arhtur Sale]]
* [webpage](http://www.utas.edu.au/information-and-communication-technology/people/individual-details/_nocache?user=ahjs)
## related $n$Lab entries
* [[computer science]]
* [[type theory]]
category: people
|
Arthur Sard
|
https://ncatlab.org/nlab/source/Arthur+Sard
|
Arthur Sard was a mathematician at Queens College.
He got his PhD in 1936 from Harvard University, advised by [[Marston Morse]],
in which he proved the [[Morse–Sard theorem]], named after him and [[Anthony P. Morse]] (no relation to Marston Morse).
* [Wikipedia entry](https://en.wikipedia.org/wiki/Arthur_Sard)
## Selected writings
On the [[Morse–Sard theorem]]:
* [[Arthur Sard]], _The measure of the critical values of differentiable maps_, Bulletin of the American Mathematical Society, 48:12 (1942), 883–890, [doi](https://doi.org/10.1090/S0002-9904-1942-07811-6).
category: people
|
Arthur Schopenhauer
|
https://ncatlab.org/nlab/source/Arthur+Schopenhauer
|
* [Wikipedia entry](https://en.wikipedia.org/wiki/Arthur_Schopenhauer)
## related $n$Lab entries
* [[philosophy]]
* [[first law of thought]]
category: people
|
Arthur Wasserman
|
https://ncatlab.org/nlab/source/Arthur+Wasserman
|
* [webpage](https://lsa.umich.edu/math/people/emeritus-faculty/awass.html)
## Selected writings
On [[equivariant differential topology]]:
* {#Wasserman69} [[Arthur Wasserman]], section 3 of: _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 (<a href="https://doi.org/10.1016/0040-9383(69)90005-6">doi:10.1016/0040-9383(69)90005-6</a>[pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))
## Related $n$Lab entries
* [[differential topology]], [[equivariant homotopy theory]]
* [[equivariant bordism]]
* [[equivariant cohomotopy]]
category: people
|
Arthur Wightman
|
https://ncatlab.org/nlab/source/Arthur+Wightman
|
* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Wightman)
* [a biography at princeton.edu](https://www.princeton.edu/physics/arthur-wightman)
## Selected writings
On [[non-perturbative quantum field theory|non-perturbative]]/[[AQFT|algebraic]] [[quantum field theory]] and its key [[theorems]] ([[spin-statistics theorem]], [[PCT theorem]], ...):
* [[Raymond F. Streater]], [[Arthur S. Wightman]], *PCT, Spin and Statistics, and All That*, Princeton University Press (1989, 2000) [[ISBN:9780691070629](https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that), [jstor:j.ctt1cx3vcq](https://www.jstor.org/stable/j.ctt1cx3vcq)]
## Related entries
* [[Wightman axioms]], [[AQFT]]
category: people
[[!redirects Arthur S. Wightman]]
|
Arthur-Selberg trace formula
|
https://ncatlab.org/nlab/source/Arthur-Selberg+trace+formula
|
A generalization of the [[Selberg trace formula]], developed by [[James Arthur]].
## References
* Steve Gelbart, _Lectures on the Arthur--Selberg Trace Formula_, ([pdf](http://arxiv.org/abs/math/9505206))
* Wikipedia, _[Arthur-Selberg trace formula](http://en.wikipedia.org/wiki/Arthur%E2%80%93Selberg_trace_formula)_
|
Artin gluing
|
https://ncatlab.org/nlab/source/Artin+gluing
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
**Artin gluing** is a fundamental construction in [[locale]] theory and [[topos]] theory. The original example is the way in which a [[topological space]] or [[locale]] $X$ may be glued together from an [[open subspace]] $i \colon U \hookrightarrow X$ and its [[closed subspace|closed]] [[complement]] $j \colon K \hookrightarrow X$. The analogous construction for toposes gives the [[sheaf topos]] $Sh(X)$ via a gluing together of $Sh(U)$ and $Sh(K)$, and applies more generally to give a sense of how to put two toposes together so that one becomes an [[open geometric morphism|open]] [[subtopos]] and the other a [[closed subtopos]] of the gluing.
## The topological case
{#TheTopologicalCase}
Let us consider first the case of topological spaces. Let
* $X$ be a [[topological space]],
* $i \colon U \hookrightarrow X$ an [[open subspace]], and
* $j \colon K \hookrightarrow X$ the [[complement|complementary]] [[closed subspace]].
Let $O(X)$ denote the topology of $X$. There is an [[injective]] [[map]]
$$
\array{
\langle i^\ast, j^\ast \rangle
&\colon\;&
O(X) &\longrightarrow& O(U) \times O(K)
\\
&&
V
&\mapsto&
(U \cap V, K \cap V)
}
$$
that is a [[homomorphism]] of [[frame|frames]]. The general problem is to characterize the [[image]] of this map: in terms of structure pertaining to $O(U)$ and $O(K)$, which pairs $(W, W')$ of relatively open sets in $U$ and $K$ "glue together" to form an open set $W \cup W'$ in $X$?
Let $int_X \colon P(X) \to P(X)$ denote the [[interior]] operation, assigning to a [[subset]] of $X$ its interior. This is a [[left exact functor|left exact]] [[comonad]] on $P(X)$. (Indeed, topologies on the set $X$ are in [[natural bijection]] with [[left exact functor|left exact]] [[comonads]] on $P(X)$.) Our problem is to understand when the inclusion
$$
W \cup W'
\;\xhookrightarrow{\phantom{--}}\;
int_X(W \cup W')
$$
obtains. Since $W \in O(U)$ is already open when considered as a subset of $X$, this condition boils down to the condition that
\[
\label{ConditionOne}
W'
\;\xhookrightarrow{\phantom{--}}\;
int_X(W \cup W')
\]
+-- {: .un_thm}
######Proposition
A necessary and sufficient condition for (eq:ConditionOne) is that the inclusion $W' \hookrightarrow int_X(W \cup K)$ obtains.
=--
+-- {: .proof}
######Proof
The necessity is clear since $W' \subseteq K$. The sufficiency is equivalent to having an inclusion
$$
W' \cap int_X(W \cup K)
\;\xhookrightarrow{\phantom{--}}\;
int_X(W \cup W')
\,.
$$
Since $W'$ is relatively open in the subspace $K$, we may write $W' = K \cap V$ for some $V \in O(X)$, and so we must check that there is an inclusion
$$
(K \cap V) \cap int_X(W \cup K)
\;\xhookrightarrow{\phantom{--}}\;
int_X(W \cup (K \cap V))
$$
or in other words -- using distributivity and the fact that $int_X$ preserves intersections -- an inclusion
$$
K \cap V \cap int_X(W \cup K)
\;\xhookrightarrow{\phantom{--}}\;
int_X(W \cup K) \cap int_X(W \cup V)
\,.
$$
But this is clear, since we have
$$
K \cap V \cap int_X(W \cup K)
\;\xhookrightarrow{\phantom{--}}\;
int_X(W \cup K)
$$
and
$$
K \cap V \cap int_X(W \cup K)
\;\xhookrightarrow{\phantom{--}}\;
V
\;\xhookrightarrow{\phantom{--}}\;
W \cup V
\;=\;
int_X(W \cup V)
\,,
$$
where to derive the last equation, we use the fact that $W \in O(U)$ and $V$ are open in $X$.
=--
+-- {: .un_thm}
######Proposition
The operation
$$
O(U) \ni W
\;\mapsto\;
int_X(W \cup K)
\;=\;
int_X(W \cup \neg U) \in O(X)
$$
is the [[right adjoint]] $i_\ast$ to $i^\ast \colon O(X) \to O(U)$.
=--
This is well-known.
+-- {: .proof}
######Proof
For $V \in O(X)$ we have
$$
\frac
{V \subseteq int_X(W \cup \neg U) \qquad \text{in} \: O(X)}
{V \subseteq W \cup \neg U \qquad \text{in} \: P(X)}
\,,
$$
but the last condition is equivalent to having $U \cap V \subseteq W$ in $P(X)$, or to $i^\ast(V) = U \cap V \subseteq W$ in $O(X)$.
=--
Summarizing, the gluing condition (1) above (for $W' \in O(K)$, $W \in O(U)$) translates into saying that there is an inclusion
$$
W'
\;\xhookrightarrow{\phantom{--}}\;
j^\ast i_\ast W
\,.
$$
where $i^\ast, j^\ast$ are [[restriction]] maps and $i^\ast \dashv i_\ast$.
For future reference, observe that the operator $j^\ast i_\ast \colon O(U) \to O(K)$ is [[left exact functor|left exact]].
We can turn all this around. Suppose $U$ and $K$ are topological spaces, and suppose $f \colon O(U) \to O(K)$ is left exact. Then we can manufacture a topological space $X$ which contains $U$ as an open subspace and $K$ as its closed complement, and (letting $i$, $j$ being the inclusions as above) such that $f = j^\ast i_\ast$. The open sets of $X$ may be identified with pairs $(W, W') \in O(U) \times O(K)$ such that $W' \subseteq f(W)$; here we are thinking of $(W, W')$ as a stand-in for $W \cup W'$. In particular, open sets $W$ of $U$ give open sets $(W, \varnothing)$ of $X$, while open sets $W'$ of $K$ also give open sets $U \cup W'$ of $X$.
## The localic case
The development given above generalizes readily to the context of [[locales]].
Thus, let $X$ be a locale, with corresponding [[frame]] denoted by $O(X)$. Each element $U \in O(X)$ gives rise to two distinct frames:
* The frame whose elements are [[algebra for a monad|algebras]] ([[fixed point|fixed points]]) of the left exact [[idempotent monad]] $U \vee - \colon O(X) \to O(X)$. The corresponding locale is the [[closed sublocale]] $\neg U$ (more exactly, the frame surjection $O(X) \to Alg(U \vee -)$ is identified with a sublocale $\neg U \to X$).
* The frame whose elements are algebras of the left exact idempotent monad $U \Rightarrow - \colon O(X) \to O(X)$. (NB: for topological spaces, this is $U \Rightarrow V = int_X(V \cup \neg U)$. This is isomorphic as a frame (but not as a subset of $O(X)$) to the [[principal ideal]] of $O(X)$ generated by $U$, which is more obviously the topology of $U$.) The sublocale corresponding to the frame surjection $O(X) \to Alg(U \Rightarrow -)$ is the [[open sublocale]] corresponding to $U$.
Put $K = \neg U$, and let $i^\ast: O(X) \to O(U)$, $j^\ast: O(X) \to O(K)$ be the frame maps corresponding to the open and closed sublocales attached to $U$, with [[right adjoints]] $i_\ast$, $j_\ast$. Again we have a left exact functor
$$f = j^\ast i_\ast \colon O(U) \to O(K).$$
Observe that this gives rise to a left exact comonad
$$O(U) \times O(K) \to O(U) \times O(K): (W, W') \mapsto (W, W' \wedge f(W)) \qquad (2)$$
whose coalgebras are pairs $(W, W')$ such that $W' \leq f(W)$. The coalgebra category forms a frame.
+-- {: .un_thm}
######Theorem
The frame map $\langle i^\ast, j^\ast \rangle \colon O(X) \to O(U) \times O(K)$ is identified with the comonadic functor attached to the comonad (2). In particular, $O(X)$ can be recovered from $O(U)$, $O(K)$, and the comonad (2).
=--
Since $O(U + K) \cong O(U) \times O(K)$, we can think of the frame map $\langle i^\ast, j^\ast \rangle$ as giving a localic surjection $U + K \to X$.
Again, we can turn all this around and say that given any two locales $U$, $K$ and a left exact functor
$$f \colon O(U) \to O(K),$$
we can manufacture a locale $X$ whose frame $O(X)$ is the category of coalgebras for the comonad
$$1_{O(U)} \times \wedge \circ (f \times 1_{O(K)}) \colon O(U) \times O(K) \to O(U) \times O(K) \qquad (3)$$
so that $U$ is naturally identified with an open sublocale of $X$, $K$ with the corresponding closed sublocale, and with a localic surjection $U + K \to X$. This is the (Artin) **gluing construction** for $f$.
## The toposic case
Now suppose given [[topos|toposes]] $E$, $E'$ and a [[left exact functor]] $\Phi \colon E \to E'$. There is an induced left exact [[comonad]]
$$E \times E' \stackrel{\delta \times 1}{\to} E \times E \times E' \stackrel{1 \times \Phi \times 1}{\to} E \times E' \times E' \stackrel{1 \times product}{\to} E \times E' \qquad (3)$$
whose category of coalgebras is again (by a basic theorem of topos theory; see for instance [here](http://ncatlab.org/toddtrimble/published/Three+topos+theorems+in+one)) a topos, called the **Artin gluing** construction for $\Phi$, denoted $\mathbf{Gl}(\Phi)$.
Objects of $\mathbf{Gl}(\Phi)$ are triples $(e, e', f \colon e' \to \Phi(e))$. A morphism from $(e_0, e_0^', f_0)$ to $(e_1, e_1^', f_1)$ consists of a pair of maps $g \colon e_0 \to e_1$, $g'\colon e_0^' \to e_1^'$ which respects the maps $f_0, f_1$ :
$$
\array{
e_0^' &\overset{f_0}{\to} & \Phi (e_0)
\\
g'\downarrow &&\downarrow \Phi(g)
\\
e_1^' &\underset{f_1}{\to} & \Phi(e_1)
}
$$
In other words, the Artin gluing is just the [[comma category]] $E' \downarrow \Phi$. (In fact, this comma category is a topos whenever $\Phi$ preserves pullbacks.)
On the other hand, if $E$ is a topos and $U\in E$ is a [[subterminal object]], then it generates two [[subtoposes]] that are [[complement|complements]] in the [[lattice of subtoposes]], namely, an [[open subtopos]] whose [[reflector]] is $(-)^U$, and a [[closed subtopos]] whose reflector is the [[pushout]] $A\mapsto A +_{A\times U} U$. If $E=Sh(X)$ is the topos of sheaves on a locale, then $U$ corresponds to an element of $O(X)$, hence an open sublocale with complement $K$ (say), and the open subtopos can be identified with $Sh(U)$ and the closed one with $Sh(K)$.
Returning to the general case, let us denote the [[geometric embedding]] of the open subtopos by $i\colon E_U \hookrightarrow E$ and that of the closed subtopos by $j\colon E_{\neg U}\hookrightarrow E$. Then we have a composite functor, sometimes called the **fringe functor**,
$$ E_U \xrightarrow{i_*} E \xrightarrow{j^*} E_{\neg U} $$
which is left exact.
+-- {: .un_thm}
######Theorem
Let $U$ be a subterminal object of a topos $E$, as above. Then the left exact left adjoint
$$\langle i^\ast, j^\ast \rangle \colon E \to E_U \times E_{\neg U}$$
is canonically identified with the comonadic gluing construction $\mathbf{Gl}(j^\ast i_\ast) \to E_U \times E_{\neg U}$. In particular, $E$ can be recovered from $E_U$, $E_{\neg U}$, and the functor $j^* i_*$.
=--
For a proof, see A4.5.6 in the [[Elephant]].
Once again, the import of this theorem may be turned around. If $f \colon E \to F$ is any left exact functor, then the projection
$$\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} E$$
is easily identified with a [[logical functor]] $\mathbf{Gl}(f) \to \mathbf{Gl}(f)/X$ where $X$ is the [[subterminal object]] $(1, 0, 0 \to f(1))$. This realizes $E$ as an [[open subtopos]] of $\mathbf{Gl}(f)$.
On the other hand, for the same subterminal object $X \hookrightarrow 1$, the corresponding classifying map
$$[X] \colon 1 \to \Omega$$
induces a [[Lawvere-Tierney topology]] $j$ given by
$$\Omega \cong 1 \times \Omega \stackrel{[X] \times 1}{\to} \Omega \times \Omega \stackrel{\wedge}{\to} \Omega.$$
Then, the category of sheaves $Sh(j)$, or more exactly the left exact left adjoint $\mathbf{Gl}(f) \to Sh(j)$ to the category of sheaves, is naturally identified with the projection
$$\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} F,$$
thus realizing $F$ as equivalent to the [[closed subtopos]] ([[Elephant]], A.4.5, pp. 205-206) attached to the subterminal object $X$.
### Some details and further adjunctions
The sheaves in $\mathbf{Gl}(f)$ corresponding to the open resp. closed subtoposes can be described explicitly. Recall that the objects of $\mathbf{Gl}(f)$ have the form $(X, Y, u:Y\to f(X))$: then the open copy of $E$ corresponds to the subcategory on those objects $(X, Y, u:Y\to f(X))$ with $u$ an isomorphism in $F$ and the closed copy of $F$ to the subcategory with objects $(X, Y, u:Y\to f(X))$ such that $X\simeq 1$ in $E$.
The [[open subtopos]] corresponding to $E$ is [[dense subtopos|dense]] in $\mathbf{Gl}(f)$ precisely if $f:E\to F$ preserves the initial object since $(0,0,0\to f(0))$ is the initial object in $\mathbf{Gl}(f)$ and $0\to f(0)$ is an isomorphism precisely if $f$ preserves $0$.
To summarize: given a left exact $f\colon E\to F$ we get an open inclusion of $E$ with a further left adjoint:
$$ i_\ast \colon E\to \mathbf{Gl}(f) \qquad X\mapsto (X,f(X),id_{f(X)})$$
$$ i^\ast\colon \mathbf{Gl}(f)\to E \qquad (X,Y,u)\mapsto X$$
$$ i_{!} \colon E\to \mathbf{Gl}(f) \qquad X\mapsto (X,0,0\to f(X)) \quad ,$$
and a closed inclusion of $F$ into $\mathbf{Gl}(f)$ with
$$ j_\ast \colon F\to \mathbf{Gl}(f) \qquad X\mapsto (1,X,X\to 1)$$
$$ j^\ast\colon\mathbf{Gl}(f)\to F \qquad (X,Y,u)\mapsto Y$$
that will lack the left adjoint $j_!$ in general. The situation when $j_!$ exists is characterized by the following observation:
+-- {: .un_thm}
######Proposition
The closed inclusion $j$ is essential i.e. $j^\ast$ has a left adjoint $j_!$ precisely if the fringe functor $f$ has a left adjoint $l$.
=--
+-- {: .proof}
######Proof
Suppose $j_!$ exists. The fringe functor $f$ is up to natural isomorphism just $j^\ast i_\ast$ and $i^\ast j_!\dashv j^\ast i_\ast$ since adjoints compose.
Conversely, suppose that $l\dashv f$ with $\eta\colon id\to f{l}$ the corresponding [[unit of an adjunction|unit]]. Define
$$ j_{!} \colon F\to \mathbf{Gl}(f) \qquad Y\mapsto (l(Y),Y,\eta_Y\colon Y\to f{l}(Y)) \quad .$$
Now given morphisms $\alpha\colon Y_1\to Y$ and $u\colon Y\to f(X)$ in $F$ by general properties of a unit there is precisely one morphism $\overline{u\circ\alpha}\colon l(Y_1)\to X$ corresponding to $u\circ\alpha$ under the adjunction such that the following diagram commutes:
$$
\array{
Y_1 &\overset{\eta_{Y_1}}{\to} & f{l}(Y_1)
\\
\alpha\downarrow &&\downarrow f(\overline{u\circ\alpha})
\\
Y &\underset{u}{\to} & f(X)
}
$$
This establishes a bijective correspondence between $Y_1\to j^\ast(X,Y,u)$ and $j_!(Y_1)\to (X,Y,u)$ which is natural since $\eta$ is.
=--
In particular, the left adjoint $j_!$ exists if the fringe functor $f$ is the [[direct image]] of a geometric morphism, or the [[inverse image]] of an [[essential geometric morphism]].
$$j_!\dashv j^\ast\dashv j_\ast \colon \mathbf{Gl}(f)\to F\quad .$$
The existence of a _right_ adjoint for the fringe functor: $f\dashv r\colon F\to E$, on the other hand, corresponds to the existence of an 'amazing' right adjoint for the open subtopos inclusion: $i_\ast\dashv i^!:\mathbf{Gl}(f)\to E$.
One direction follows again from the composition of adjoints: $j^\ast i_\ast\dashv i^! j_\ast$ , whereas for the other direction we define:
$$
i^!\colon \mathbf{Gl}(f)\to E \qquad (X,Y,u)\mapsto r(Y) \times_{rf(X)} X\quad .
$$
When $f$ is fully faithful, the unit $X \to rf(X)$ is an iso, and so we can instead use
$$
i^!\colon \mathbf{Gl}(f)\to E \qquad (X,Y,u)\mapsto r(Y)\quad .
$$
Note that in this case $E$ is dense and we get a 'co-cohesive' adjoint string
$$i_!\dashv i^\ast\dashv i_\ast \dashv i^!\colon \mathbf{Gl}(f)\to E$$
where $i_!$ and $i_\ast$ are fully faithful.
### Examples
Since $i:\ast\hookrightarrow E$ is left exact where $\ast$ is the degenerate topos with one identity morphism, _every_ topos $E$ is trivially a result of Artin gluing: $E\simeq E\downarrow i$.
Of course, more interesting examples of the gluing construction abound as well. Here are a few:
* Let $E$ be an (elementary, not necessarily Grothendieck) topos, and let $\hom(1, -): E \to Set$ represent the terminal object $1$ -- this of course is left exact. The gluing construction $\mathbf{Gl}(\hom(1, -))$ is called the **scone** (Sierpinski cone), or the [[Freyd cover]], of $E$.
* If $E$ is a Grothendieck topos and $\Delta \colon Set \to E$ is the (essentially unique) left exact left adjoint, then we have a gluing construction $E \downarrow \Delta$. This gluing may be regarded as the result of _attaching a generic open point_ to $E$.
* A concrete instance of the constructions in both the preceding examples is the [[Sierpinski topos]] $Set^{\to}$ corresponding e.g. to $Set\downarrow id_{Set}$: its objects are functions $X\to Y$ between sets $X,Y$ and the closed copy of $Set$ sits on the objects of the form $X\to 1$ and the open copy on the objects $X\overset{\simeq}{\to}Y$.
* Since a topos $\mathcal{E}$ is finitely bicomplete, the product functor $\sqcap:\mathcal{E}\times\mathcal{E}\to\mathcal{E}$ with $(X,Y)\mapsto X\times Y$ is part of an adjoint string $\sqcup\dashv\triangle\dashv\sqcap$ involving the [[diagonal functor]] and the coproduct functor. Since $\sqcap$ is left exact, Artin gluing applies. In the case $\mathcal{E}=Set$ , $\mathbb{Gl}(\sqcap)$ yields the [[hypergraph|topos of hypergraphs]]; this example is discussed in detail at [[hypergraph]]. These cases are somewhat unusual in that the fringe functor here has a left adjoint which itself has a further left adjoint.
### Remarks
* Artin gluing for toposes carries over in some slight extra generality, replacing left exact functors $f$ by _pullback-preserving functors_.
* Artin gluing applies also to other [[doctrine|doctrines]]: [[regular category|regular categories]], [[pretopos|pretoposes]], [[quasitopos|quasitoposes]], etc. See ([Carboni-Johnstone](#CJ)) and ([Johnstone-Lack-Sobocinski](#JLS07)).
## Related entries
* [[Freyd cover]]
* [[comma category]]
* [[recollement]]
## References
* [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (exposé IV, sect. 9; in particular: sect. 9.5) {#SGA4}
* [[Gavin Wraith]], _Artin Glueing_ , JPAA **4** (1974) pp.345-358 [link](http://www.sciencedirect.com/science/article/pii/0022404974900140).
* [[Aurelio Carboni]], [[Peter Johnstone]], _Connected limits, familial representability and Artin glueing_ , Mathematical Structures in Computer Science **5** (1995) pp.441-459. ([pdf](http://journals.cambridge.org/article_S0960129500001183))
{#CJ}
* [[Aurelio Carboni]], [[Peter Johnstone]], _Corrigenda to 'Connected limits...'_ , Mathematical Structures in Computer Science **14** (2004) pp.185-187.
* Peter F. Faul, Graham R. Manuell, _Artin Glueings of Frames as Semidirect Products_ , arXiv:1907.05104 (2019). ([abstract](https://arxiv.org/abs/1907.05104))
* Peter F. Faul, Graham Manuell, José Siqueira, _Artin glueings of toposes as adjoint split extensions_ , arXiv:2012.04963 (2020). ([abstract](https://arxiv.org/abs/2012.04963))
* [[Matthias Hutzler]], _Syntactic presentations for glued toposes and for crystalline toposes_, Phd. diss. Universität Augsburg 2021. ([arXiv:2206.11244](https://arxiv.org/abs/2206.11244))
* [[Mamuka Jibladze]], _Lower Bagdomain as a Glueing_ , Proc. A. Razmadze Math. Inst. **118** (1998) pp.33-41. ([pdf](http://www.rmi.ge/~jib/pubs/baglue.pdf))
* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014). (section 4.2, pp.107-112)
* [[Peter Johnstone]], _[[Sketches of an Elephant]]_ , Oxford UP 2002. (sec. A2.1.12, pp.82-84; A4.5.6, p.208)
{#Johnstone}
* {#JLS07}[[Peter Johnstone]], [[Steve Lack]], [[Pawel Sobocinski]], _Quasitoposes, Quasiadhesive Categories and Artin Glueing_ , pp.312-326 in LNCS **4624** Springer Heidelberg 2007. ([preprint](http://users.ecs.soton.ac.uk/ps/papers/calco07.pdf))
* [[Anders Kock|A. Kock]], T. Plewe, _Glueing analysis for complemented subtoposes_ , TAC **2** (1996) pp.100-112. ([pdf](http://www.tac.mta.ca/tac/volumes/1996/n9/n9.pdf))
* J. C. Mitchell, A. Scedrov, _Notes on sconing and relators_ , Springer LNCS **702** (1993) pp.352-378. ([ps-draft](ftp://ftp.cis.upenn.edu/pub/papers/scedrov/rel.ps.Z)) [PDF](https://pdfs.semanticscholar.org/ba4e/b38b05470481dc3c912fb37f0598bbda94fe.pdf)
* [[Susan Niefield]], _The glueing construction and double categories_ , JPAA **216** no.8/9 (2012) pp.1827-1836.
[[!redirects Artin-Wraith gluing]]
[[!redirects Artin-Wraith glueing]]
[[!redirects Artin glueing]]
|
Artin L-function
|
https://ncatlab.org/nlab/source/Artin+L-function
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _Artin L-function_ $L_\sigma$ ([Artin 23](#Artin23)) is an [[L-function]] associated with a [[number field]] $K$ and induced from the choice of an $n$-dimensional [[Galois representation]], hence a [[linear representation]]
$$
\sigma\;\colon\; Gal(L/K) \longrightarrow GL_n(\mathbb{C})
$$
of the [[Galois group]] for some finite [[Galois extension]] $L$ of $K$: it is the product ("[[Euler product]]") over all [[prime ideals]] $\mathfrak{p}$ in the [[ring of integers]] of $K$, of, essentially, the [[characteristic polynomials]] of the [[Frobenius homomorphism]] $Frob_p$ regarded (see [here](Frobenius+morphism#AsElementsOfGaloisGroup)) as elements of [[Galois group]]
$$
L_{K,\sigma} \colon s
\mapsto
\underset{\mathfrak{p}}{\prod}
det
\left(
id - (N (\mathfrak{p}))^{-s}
\sigma(Frob_{\mathfrak{p}})
\right)^{-1}
\,
$$
([e.g. Gelbhart 84, II.C.2](#Gelbhart84), [Snyder 02, def. 2.1.3](#Snyder02)).
> discussion of ramified primes needs to be added
For $\sigma = 1$ the [[trivial representation]] then the Artin L-function reduces to the [[Dedekind zeta function]] (see [below](#RelationToDedekindZeta)). So conversely one may think of Artin L-functions as being Dedekind zeta functions which are "twisted" by a [[Galois representation]]. (Notice that Galois representations are the analog in [[arithmetic geometry]] of [[flat connections]]/[[local systems of coefficients]]).
For $\sigma$ any 1-dimensional [[Galois representation]] (hence the case $n = 1$) then there is a _[[Dirichlet character]]_ $\chi$ such that the Artin L-function $L_\sigma$ is equal to the [[Dirichlet L-function]] $L_\chi$ -- this relation is part of _[[Artin reciprocity]]_.
For $\sigma$ any $n$-dimensional representation for $n \geq 1$ then the [[conjecture]] of _[[Langlands correspondence]]_ is that for each $n$-dimensional [[Galois representation]] $\sigma$ there is an [[automorphic representation]] $\pi$ such that the Artin L-function $L_\sigma$ equals the [[automorphic L-function]] $L_\pi$ (e.g [Gelbhart 84, pages 5-6](#Gelbhart84)).
## Properties
### For irreducible representations -- Artin's conjecture
_Artin's conjecture_ is the statement that for a nontrivial _[[irreducible representation]]_ $\sigma$ the Artin L-function $L_{K,\sigma}$ is not just a [[meromorphic function]] on the complex plane, but in fact an [[entire holomorphic function]].
e.g. ([Ram Murty 94, p. 3](#RamMurty94))
> or rather with at most a pole at $s = 1$ [Murty-Murty 12, page 29 in chapter 2](#MurtyMurty12)
### For induced representations
{#ForInducedRepresentations}
Let $H \hookrightarrow Gal(L/K)$ be [[subgroup]] of the Galois group $G \coloneqq Gal(L/K)$ and write $L^H \hookrightarrow L$ for the subfield of elements fixed by $H$.
Let $\sigma$ be a representation of $H = Gal(L/L^H)$ and write
$Ind_H^G\sigma$ for the [[induced representation]] of $G$. Then the corresponding Artin L-functions are equal:
$$
L_{K,{Ind_H^{G}\sigma}} = L_{L^H, \sigma}
\,.
$$
(e.g. ([Murty-Murty 12, equation (2) in chapter 2](#MurtyMurty12))).
### Relation to the Dedekind zeta function
{#RelationToDedekindZeta}
For $\sigma = 1$ the [[trivial representation]] then $\sigma(Frob_{\mathfrak{p}}) = id$ identically, and hence in this case the definition of the Artin L-function becomes verbatim that of the [[Dedekind zeta function]] $\zeta_K$:
$$
L_{L,1} = \zeta_L
\,.
$$
If $L/K$ is a [[Galois extension]], the by the behaviour of Artin L-functions for induced representation as [above](#ForInducedRepresentations) this is also the Artin L-function of $K$ itself for the [[regular representation]] of $Gal(L/K)$
$$
\zeta_L
=
L_{L,1}
=
L_{K,{Ind_1^{Gal(L/K)}1}}
$$
(e.g. ([Murty-Murty 12, below (2) in chapter 2](#MurtyMurty12)))
### Analogy with Selberg/Ruelle zeta-functions
{#AnalogyWithSelbergZeta}
The [[Frobenius morphism]] $Frob_p$ giving an element in the [[Galois group]] means that one may think of it as an element of the [[fundamental group]] of the given [[arithmetic curve]] (see at _[[algebraic fundamental group]]_). There is a direct analogy between Frobenius elements at prime numbers in arithmetic geometry and parallel transport along [[prime geodesics]] in hyperbolic geometry ([Brown 09, p. 6](#Brown09)).
Under this interpretation, a [[Galois connection]] corresponds to a [[flat connection]] ([[local system of coefficients]]) on an arithmetic curve, and its Artin L-function is a product of [[characteristic polynomials]] of the [[monodromies]]/[[holonomies]] of that flat connection.
Now, in [[differential geometry]], given a suitable odd-dimensional [[hyperbolic manifold]] equipped with an actual [[flat bundle]] over it, then associated with it is the _[[Selberg zeta function]]_ and _[[Ruelle zeta function]]_. Both are (by definition in the latter case and by theorems in the former) [[Euler products]] of [[characteristic polynomials]] of [[monodromies]]/[[holonomies]]. See at _[Selberg zeta function -- Analogy with Artin L-function](Selberg+zeta+function#AnalogyWithArtinLFunction)_ and at _[Ruelle zeta function -- Analogy with Artin L-function](Ruelle+zeta+function#AnalogyToTheArtinLFunction)_ for more on this.
See also ([Brown 09, page 6](#Brown09), [Morishita 12, remark 12.7](#Morishita12)).
(The definition also has some similarity to that of the [[Alexander polynomial]], see at _[[arithmetic topology]]_.)
## Related concepts
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]
## References
The original article is
* {#Artin23} [[Emil Artin]], _Über eine neue Art von L Reihen_. Hamb. Math. Abh. 3. (1923) Reprinted in his collected works, ISBN 0-387-90686-X. English translation in ([Snyder 02, section A](#Snyder02))
Reviews include
* Wikipedia, _[Artin L-function](http://en.wikipedia.org/wiki/Artin_L-function)_
* {#MurtyMurty12} M. Ram Murty, V. Kumar Murty, _Non-vanishing of L-functions and applications_, Modern Birkhäuser classics 2012 ([chapter 2 pdf](http://www.beck-shop.de/fachbuch/leseprobe/9783034802734_Excerpt_001.pdf))
* {#Snyder02} [[Noah Snyder]], _Artin L-Functions: A Historical Approach_, 2002 ([pdf](http://www.math.columbia.edu/~nsnyder/thesismain.pdf))
and in the context of the [[Langlands program]]
* {#Gelbhart84} [[Stephen Gelbart]], _An elementary introduction to the Langlands program_, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))
Further development includes
* {#RamMurty94} M. Ram Murty, _Selberg's conjectures and Artin -functions_, Bull. Amer. Math. Soc. 31 (1994), 1-14 ([web](http://www.ams.org/journals/bull/1994-31-01/S0273-0979-1994-00479-3/home.html))
The analogy with the [[Selberg zeta function]] is discussed in
* {#Brown09} Darin Brown, _Lifting properties of prime geodesics_, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 ([euclid](http://projecteuclid.org/euclid.rmjm/1239113439))
* {#Morishita12} [[Masanori Morishita]], section 12.1 of _Knots and Primes: An Introduction to Arithmetic Topology_, 2012 ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))
The analogies between Alexander polynomial and L-functions and touched upon in
* Ken-ichi Sugiyama, _The properties of an L-function from a geometric point of view_, 2007 [pdf](http://geoquant2007.mi.ras.ru/sugiyama.pdf); _A topological $\mathrm{L}$ -function for a threefold_, 2004 [pdf](http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1376-12.pdf)
[[!redirects Artin L-functions]]
|
Artin reciprocity law
|
https://ncatlab.org/nlab/source/Artin+reciprocity+law
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[Emil Artin]]'s reciprocity law is a [[reciprocity law]] in [[class field theory]] for [[global fields]]. It states, in consequence, that for each 1-dimensional [[Galois representation]] $\sigma$ there exists a [[Dirichlet character]] $\chi$ such that the [[Artin L-function]] $L_\sigma$ equals the [[Dirichlet L-function]] $L_\chi$. (The generalization of this reciprocal correspondence to higher dimensional Galois representations is the content of the conjectured _[[Langlands correspondence]]_).
For $K$ a [[global field]] there is a canonical map
$$
\mathbb{I}_K \longrightarrow Gal(K^{ab}/K)
$$
from its [[group of ideles]] to the [[abelianization]] of its [[Galois group]], given by
$$
(\cdots, a_v, \cdots) \mapsto \prod_v Frob_v^{ord_v(a_v)}
\,.
$$
For $K$ a [[number field]], _Artin's reciprocity law_ says that this map is surjective, that it factors through the [[idele class group]] $K^\times \backslash \mathbb{I}_K$ and moreover that further quotienting this by the connected component $\mathcal{O}$ of 1 in the idele class group yields an [[isomorphism]]
$$
K^\times \backslash \mathbb{I}_K / \mathcal{O}
\stackrel{\simeq}{\longrightarrow}
Gal(K^{ab}/K)
\,.
$$
For $K = \mathbb{Q}$ this is also the statement of the [[Kronecker-Weber theorem]], and together this is a starting point of the [[Langlands correspondence]] [[conjecture]], see there for more.
For $K$ a [[function field]] the map is no longer surjective, but yields on the quotient by the [[restricted product]] $\prod_v \mathcal{O}_v^\times$ an injection with dense image
$$
K^\times \backslash \mathbb{I}_K / \prod_v \mathcal{O}_v^\times
\hookrightarrow
Gal(K^{ab}/K)
\,.
$$
([e.g. Toth 11, p. 3](#Toth11))
Notice that the double quotients appearing here are by the [[Weil uniformization theorem]] analogous to [[moduli stacks of bundles]] on the [[arithmetic curve]] on which $K$ is the field of [[rational functions]]. Under this [[function field analogy]] the analog of Artin's reciprocity law plays a central role in the [[geometric Langlands correspondence]].
## Properties
### Function field analogy
[[!include function field analogy -- table]]
## Related concepts
* [[Artin L-function]]
## References
* [[Serge Lang]], _Algebraic number theory_, GTM 110, Springer 1970, 1986, 1994, 2000
* {#Snyder02} [[Noah Snyder]], section 2.3 of _Artin L-Functions: A Historical Approach_, 2002 ([pdf](https://web.archive.org/web/20120204005924/http://www.math.columbia.edu/~nsnyder/thesismain.pdf))
* Wikipedia: en:[Artin reciprocity law](http://en.wikipedia.org/wiki/Artin_reciprocity_law), de:[Artinsches_Reziprozitaetsgesetz](http://de.wikipedia.org/wiki/Artinsches_Reziprozit%C3%A4tsgesetz)
A discussion with an eye towards [[geometric class field theory]] is in
* {#Toth11} Peter Toth, _Geometric abelian class field theory_, 2011 ([web](http://dspace.library.uu.nl/handle/1874/206061))
[[!redirects Artin's reciprocity law]]
[[!redirects Artin reciprocity]]
|
Artin representability theorem
|
https://ncatlab.org/nlab/source/Artin+representability+theorem
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A list of sufficient conditions for a [[presheaf]] on the [[opposite category]] of [[commutative rings]] to be [[representable functor|represented]] by a suitable [[algebraic space]]/[[scheme]]. Recalled e.g. as [Lurie Rep, theorem 1 ](#LurieRep).
## Related concepts
* [[Grothendieck existence]]
* [[Artin-Lurie representability theorem]]
## References
* {#Artin69} [[Michael Artin]], _Algebraization of formal moduli. I_, Global Analysis (Papers in Honor of K. Kodaira), Princeton Univ. Press, Princeton, N.J., 1969, pp. 21-71. [MR0260746](http://www.ams.org/mathscinet-getitem?mr=0260746) (41:5369)
* {#LurieRep} [[Jacob Lurie]], _[[Representability theorems]]_
Generalization to [[supergeometry]]:
* Nadia Ott, *Artin's theorems in supergeometry* ([arXiv:2110.12816](https://arxiv.org/abs/2110.12816))
[[!redirects Artin's representability theorem]]
|
Artin stack
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https://ncatlab.org/nlab/source/Artin+stack
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#### Higher geometry
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[[!include higher geometry - contents]]
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#Contents#
* table of contents
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## Idea
An _Artin stack_ is a [[geometric stack]] over the [[étale site]]: the [[stack]] quotient of a [[scheme]] whose [[automorphism group]]s are [[algebraic group]]s.
This is the general notion of [[algebraic stack]]. But often the latter term is used in a more restrictive sense to mean [[Deligne-Mumford stack]].
## Properties
* [[Artin representability theorem]]
## References
The original article:
* [[Michael Artin]], *Versal deformations and algebraic stacks*, Invent Math **27** (1974) 165–189 [[doi:10.1007/BF01390174](https://doi.org/10.1007/BF01390174), [eudml:142310](https://eudml.org/doc/142310), [pdf](http://math.uchicago.edu/~drinfeld/Artin_on_stacks.pdf)]
Review in the general context of [[algebraic stacks]]:
* {#LaumontMoret-Bailly} [[Gérard Laumon]], [[Laurent Moret-Bailly]], _Champs algébriques_, Ergebn. der Mathematik und ihrer Grenzgebiete **39**, Springer (2000) [[doi:10.1007/978-3-540-24899-6](https://doi.org/10.1007/978-3-540-24899-6)]
* [[Bertrand Toen]], _[[Master course on algebraic stacks]]_
and in the broader context of [[derived algebraic geometry]]:
* [[Dennis Gaitsgory]], [[Nick Rozenblyum]], Section I.2.4 in: *[[A study in derived algebraic geometry]]*
[[!redirects Artin stacks]]
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Artin-Lam induction exponent
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https://ncatlab.org/nlab/source/Artin-Lam+induction+exponent
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### Context
#### Group Theory
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[[!include group theory - contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
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## Definition
\begin{definition}
For $G$ a [[finite group]], its *Artin-Lam induction exponent* is the [[minimum]] $e(G)$ among [[positive numbers|positive]] [[natural numbers]] such that the $e(G)$-dimensional [[trivial representation]] is in the [[ideal]] of the [[rational number|rational]] [[representation ring]] of $G$ which is generated by $G$-representations that are [[induced representation|induced]] by [[cyclic group|cyclic]] [[subgroups]] of $G$.
\end{definition}
> This definition according to the abstract of [Madsen, Thomas & Wall 1983](#MadsenThomasWall83). Other authors take the smallest $e(G)$ such that $e(G) \cdot \chi$ is in this ideal for *all* rational representations $\chi$ (e.g. [Yamauchi 70](#Yamauchi70)), in which case $e(G)$ is the [[exponent of a group|exponent]] of the additive group underlying the quotient by the ideal, whence the terminology.
## Related concepts
* [[free group actions on n-spheres]]
## References
The original article:
* {#Lam68} [[Tsit-Yuen Lam]], *Artin exponent of finite groups*, Journal of Algebra **9** (1968) 94-119 (<a href="https://doi.org/10.1016/0021-8693(68)90007-0">doi:10.1016/0021-8693(68)90007-0</a>, [pdf](https://core.ac.uk/download/pdf/82635891.pdf))
Further discussion:
* {#Yamauchi70} Kenichi Yamauchi, *On the 2-part of Artin Exponent of Finite Groups*, Science Reports of the Tokyo Kyoiku Daigaku, Section A
Vol. 10, No. 263/274 (1970), 234-240 ([jstor:43698746](https://www.jstor.org/stable/43698746))
* K. K. Nwabueze, *Some definition of the Artin exponent of finite groups* ([arXiv:math/9611212](https://arxiv.org/abs/math/9611212))
* S. Jafari and H. Sharifi, *On the Artin exponent of some rational groups*, Communications in Algebra, 46:4, 1519-152 ([doi:10.1080/00927872.2017.1347665](https://doi.org/10.1080/00927872.2017.1347665))
Textbook account:
* Charles Curtis, Irving Reiner, *Methods of representation theory -- With applications to finite groups and orders -- Vol. I*, Wiley 1981 ([pdf](https://webusers.imj-prg.fr/~olivier.dudas/src/parisa1.pdf))
Application to discussion of [[free group actions on n-spheres]]:
* {#MadsenThomasWall83} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *Topological spherical space form problem III: Dimensional bounds and smoothing*, Pacific J. Math. 106(1): 135-143 (1983) ([pjm:1102721110](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-106/issue-1/Topological-spherical-space-form-problem-III-Dimensional-bounds-and-smoothing/pjm/1102721110.full))
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Artin-Lurie representability theorem
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https://ncatlab.org/nlab/source/Artin-Lurie+representability+theorem
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#### Higher geometry
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#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
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## Idea
The generalization of the [[Artin representability theorem]] from [[algebraic geometry]] to [[spectral algebraic geometry]].
## Statement
Write $CAlg^{cn}$ for the [[(∞,1)-category]] of [[connective]] [[E-∞ rings]], and [[∞Grpd]] for that of [[∞-groupoids]].
+-- {: .num_theorem}
###### Theorem
Necessary and sufficient conditions for an [[(∞,1)-presheaf]]
$$
\mathcal{F} \;\colon\; CAlg^{cn}\longrightarrow \infty Grpd
$$
over some $Spec R$, on the [[opposite (∞,1)-category]] of [[connective]] [[E-∞ rings]] to be [[representable functor|represented]] by a [[spectral Deligne-Mumford stack|spectral Deligne-Mumford n-stack]] locally of almost finite presentation over $R$:
1. For every discrete commutative ring, $\mathcal{F}(A)$ is [[n-truncated object in an (infinity,1)-topos|n-truncated]].
1. $\mathcal{F}$ is an [[∞-stack]] for the [[étale (∞,1)-site]].
1. $\mathcal{F}$ is nilcomplete, integrable, and an [[infinitesimally cohesive (∞,1)-presheaf on E-∞ rings]].
1. $\mathcal{F}$ admits a [[connective]] [[cotangent complex]].
1. the natural transformation to $Spec R$ is locally almost of finite presentation.
=--
([Lurie Rep, theorem 2](#LurieRep))
+-- {: .num_remark}
###### Remark
The condition that $\mathcal{F}$ be [[infinitesimally cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive]] implies that the [[Lie differentiation]] around any point, given by restriction to [[local Artin rings]] ([[formal duals]] of [[infinitesimally thickened points]]), is a [[formal moduli problem]], hence equivalently an [[L-∞ algebra]].
=--
## Applications
The motivating example of the Artin-Lurie representability theorem is the re-proof of the _[[Goerss-Hopkins-Miller theorem]]_. See there for more.
## Related concepts
* [[Grothendieck existence]]
* [[Artin representability theorem]]
## References
* {#LurieRep} [[Jacob Lurie]], _[[Representability theorems]]_
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Artin-Mazur codiagonal
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https://ncatlab.org/nlab/source/Artin-Mazur+codiagonal
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[[!redirects Artin-Mazur codiagonal ]]
The Artin-Mazur codiagonal of a bisimplicial set is another name often given to the _[[total simplicial set]]_ of a bisimplicial set, and more discussion of it is found at [[bisimplicial set]].
(We will tend not to use this term because of possible confusion with other uses of [[codiagonal]].)
##References
The construction seems first to be used in
* [[M. Artin]], [[B. Mazur]], _On the Van Kampen theorem_ , Topology 5 (1966) 179–189.
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Artin-Mazur formal group
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https://ncatlab.org/nlab/source/Artin-Mazur+formal+group
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###Context###
#### Formal geometry
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[[!include formal geometry -- contents]]
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#### Group Theory
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[[!include group theory - contents]]
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# Contents
* table of contents
{: toc}
## Idea
Every [[variety]] in [[positive characteristic]] has a [[formal group]] attached to it, called the _Artin-Mazur formal group_. This group is often related to arithmetic properties of the variety such as being ordinary or supersingular.
The Artin-Mazur formal group in dimension $n$ is a [[formal group]] version of the [[Picard infinity-group|Picard n-group]] of flat/holomorphic [[circle n-bundles]] on the given variety. Therefore for $n = 1$ one also speaks of the _[[formal Picard group]]_ and for $n = 2$ of the _[[formal Brauer group]]_.
## Definition
{#Definition}
### Deformations of higher line bundles (of $H^n(-,\mathbb{G}_m)$-cohomology)
Let $X$ be a [[smooth scheme|smooth]] [[proper scheme|proper]] $n$ [[dimension|dimensional]] [[variety]] over an [[algebraically closed field]] $k$ of [[positive number|positive]] [[characteristic]] $p$.
Writing $\mathbb{G}_m$ for the [[multiplicative group]] and $H_{et}^\bullet(-,-)$ for [[etale cohomology]], then $H_{et}^n(X,\mathbb{G}_m)$ classifies $\mathbb{G}_m$-[[principal infinity-bundle|principal n-bundles]] ([[line n-bundles]], [[bundle 2-gerbe|bundle (n-1)-gerbes]]) on $X$. Notice that, by the discussion at _[Brauer group -- relation to étale cohomology](Brauer%20group#RelationToEtaleCohomology)_, for $n = 1$ this is the [[Picard group]] while for $n = 2$ this contains (as a [[torsion subgroup]]) the [[Brauer group]] of $X$.
Accordingly, for each [[Artin algebra]] regarded as an [[infinitesimally thickened point]] $S \in ArtAlg_k^{op}$ the [[cohomology group]] $H_{et}^n(X\times_{Spec(k)} S,\mathbb{G}_m)$ is that of [[equivalence classes]] of $\mathbb{G}_n$-[[principal infinity-bundle|principal n-bundles]] on a formal thickening of $X$.
The defining inclusion $\ast \to S$ of the unique global point induces a restriction map $H^n_{et}(X\times_{Spec(k)} S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))$ which restricts an $n$-bundle on the formal thickening to just $X$ itself. The [[kernel]] of this map hence may be thought of as the group of $S$-parameterized infinitesimal [[deformations]] of the trivial $\mathbb{G}_m$-$n$-bundle on $X$.
(For $n = 1$ this is an [[infinitesimal neighbourhood]] of the neutral element in the [[Picard scheme]] $Pic_X$, for higher $n$ one will need to genuinely speak about [[Picard stacks]] and higher stacks.)
As $S$ varies, these groups of deformations naturally form a [[presheaf]] on "[[infinitesimally thickened points]]" ([[Isbell duality|formal duals]] to [[Artin algebras]]).
+-- {: .num_defn}
###### Definition
For $X$ an [[algebraic variety]] as above, write
$$
\Phi_X^n \;\colon\; ArtAlg_k \to Grp
$$
$$
\Phi_X^n(S) \coloneqq \mathrm{ker}(H^n_{et}(X\times_{Spec(k)} S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))
\,.
$$
=--
([Artin-Mazur 77, II.1 "Main examples"](#ArtinMazur77))
The fundamental result of ([Artin-Mazur 77, II](#ArtinMazur77)) is that under the above hypotheses this presheaf is [[prorepresentable functor|pro-representable]] by a [[formal group]], which we may hence also denote by $\Phi_X^n$. This is called the **Artin-Mazur formal group** of $X$ in degree $n$.
More in detail:
+-- {: .num_prop #SufficientConditionsForRepresentability}
###### Proposition
Let $X$ be an [[algebraic variety]] [[proper morphism|proper]] over an [[algebraically closed field]] $k$ of [[positive number|positive]] [[characteristic]].
A sufficient condition for $\Phi_X^k$ to be [[prorepresentable functor|pro-representable]] by a [[formal group]] is that $\Phi_X^{k-1}$ is [[formally smooth morphism|formally smooth]].
In particular if $dim H^{k-1}(X,\mathcal{O}_X) = 0$ then $\Phi^{k-1}(X)$ vanishes, hence is trivially formally smooth, hence $\Phi^k(X)$ is representable
=--
The first statement appears as ([Artin-Mazur 77, corollary (2.12)](#ArtinMazur77)). The second as ([Artin-Mazur 77, corollary (4.2)](#ArtinMazur77)).
+-- {: .num_remark #Dimension}
###### Remark
The [[dimension]] of $\Phi^k_X$ is
$$
dim(\Phi^k_X) = dim H^k(X,\mathcal{O}_X)
\,.
$$
=--
([Artin-Mazur 77, II.4](#ArtinMazur77)).
### Deformations of higher line bundles with connection (of Deligne cohomology)
{#DeformationsOfDeligneCohomology}
In ([Artin-Mazur 77, section III](#ArtinMazur77)) is also discussed the formal [[deformation theory]] of [[line n-bundles with connection]] (classified by [[ordinary differential cohomology]], being [[hypercohomology]] with [[coefficients]] in the [[Deligne complex]]). Under suitable conditions this yields a [[formal group]], too.
Notice that by the discussion at _[intermediate Jacobian -- Characterization as Hodge-trivial Deligne cohomology](http://ncatlab.org/nlab/show/intermediate%20Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology)_ the formal deformation theory of Deligne cohomology yields the formal completion of [[intermediate Jacobians]] (all in suitable degree).
## Examples
### General
+-- {: .num_remark}
###### Remark
* For a [[curve]] $X$ (i.e. $dim(X)= 1$), the Artin-Mazur group is often called the **[[formal Picard group]]** $\widehat{\mathrm{Pic}}$.
* For a [[surface]] $X$ (i.e. $dim(X) =2$), the Artin-Mazur group is called the **[[formal Brauer group]]** $\widehat{Br}$.
=--
### Of Calabi-Yau varieties
{#OfCalabiYauVarieties}
+-- {: .num_example #AMfgOfStrictCalabiYau}
###### Example
Let $X$ be a strict [[Calabi-Yau variety]] in [[positive characteristic]] of [[dimension]] $n$ (strict meaning that the [[Hodge numbers]] $h^{0,r} = 0$ vanish for $0 \lt r \lt n$, i.e. over the [[complex numbers]] that the [[holonomy group]] exhausts $SU(n)$, this is for instance the case of relevance for [[supersymmetry]], see at _[[supersymmetry and Calabi-Yau manifolds]]_).
By prop. \ref{SufficientConditionsForRepresentability} this means that
the Artin-Mazur formal group $\Phi^n_X$ exists.
Since moreover $h^{0,n} = 1$ it follows by remark \ref{Dimension} that it is of [[dimension]] 1
=--
For discussion of $\Phi_X^n$ for [[Calabi-Yau varieties]] $X$ of [[dimension]] $n$ and in [[positive number|positive]] [[characteristic]] see ([Geer-Katsura 03](#GeerKatsura03)).
## Related concepts
* [[formal group]]
* [[height of a variety]]
[[!include moduli of higher lines -- table]]
## References
The original article is
* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 ([numdam:ASENS_1977_4_10_1_87_0](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663))
Further developments are in
* {#Stienstra87} [[Jan Stienstra]], _Formal group laws arising from algebraic varieties_, American Journal of Mathematics, Vol. 109, No.5 (1987), 907-925 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/stienstra1.pdf))
Lecture notes touching on the cases $n = 1$ and $n = 2$ include
* {#Liedtke14} Christian Liedtke, example 6.13 in _Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem_ ([arXiv.1403.2538](http://arxiv.org/abs/1403.2538))
Discussion of Artin-Mazur formal groups for all $n$ and of [[Calabi-Yau varieties]] of [[positive number|positive]] [[characteristic]] in [[dimension]] $n$ is in
* {#GeerKatsura03} [[Gerard van der Geer]], T. Katsura, _On the height of Calabi-Yau varieties in positive characteristic_, Documenta Math. 8. 97-113 (2003) ([arXiv:math/0302023](http://arxiv.org/abs/math/0302023))
[[!redirects Artin-Mazur formal group]]
[[!redirects Artin-Mazur formal groups]]
[[!redirects Artin–Mazur formal group]]
[[!redirects Artin–Mazur formal groups]]
[[!redirects Artin--Mazur formal group]]
[[!redirects Artin--Mazur formal groups]]
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Artin-Schreier sequence
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https://ncatlab.org/nlab/source/Artin-Schreier+sequence
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#Contents#
* table of contents
{:toc}
## Definition
Let $X$ be a [[reduced scheme|reduced]] [[scheme]] of [[characteristic]] the [[prime number]] $p$, hence such that for all points $x \in X$
$$
p \cdot \mathcal{O}_{X,x} = 0
\,.
$$
Write
$$
F - id \coloneqq (-)^p - (-) \;\colon\; (\mathbb{G}_a)_X \longrightarrow (\mathbb{G}_a)_X
$$
for the [[endomorphism]] of the [[additive group]] over the [[étale site]] $X_{et}$ of $X$ (the [[structure sheaf]] regarded as just a [[sheaf of abelian groups]]) which is the [[Frobenius endomorphism]] $F(-) \coloneqq (-)^p$ minus the identity.
+-- {: .num_prop}
###### Proposition
There is a [[short exact sequence]] of [[abelian sheaves]] over the [[étale site]]
$$
0 \to (\mathbb{Z}/p\mathbb{Z})_X \to (\mathbb{G}_a)_X \stackrel{F-id}{\to} (\mathbb{G}_a)_X \to 0
\,.
$$
=--
This is called the _Artin-Schreier sequence_ (e.g. [Tamme, section II 4.2](#Tamme), [Milne, example 7.9](#Milne)).
+-- {: .proof}
###### Proof
By the discussion at [category of sheaves -- Epi-/Mono-morphisms](category+of+sheaves#EpiMonoIsomorphisms) we need to show that the left morphism is an injection over any [[étale morphism]] $U_Y \to X$, and that for every element $s \in \mathcal{O}_X$ there exists an [[étale site]] [[covering]] $\{U_i \to X\}$ such that $(-)^p- (-)$ restricts on this to a morphism which hits the restriction of that element.
The first statement is clear, since $s = s^p$ says that $s$ is a constant section, hence in the image of the [[constant sheaf]]
$\mathbb{Z}/p\mathbb{Z}$ and hence for each connected $U_Y \to X$ the left morphism is the inclusion
$$
\mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathcal{O}_{X'}
$$
induced by including the unit [[section]] $e_{X'}$ and its multiples $r e_{X'}$ for $0 \leq r \lt p$. (This uses the "[[freshman's dream]]"-fact that in [[characteristic]] $p$ we have $(a + b)^p = a^p + b^p$).
This is injective by assumption that $X$ is of characteristic $p$.
To show that $(-)^p - (-)$ is an epimorphism of sheaves, it is sufficient to find for each element $s \in \mathcal{O}_X = A$ an [[étale site|étale cover]] $Spec(B) \to Spec(A)$ such that its restriction along this cover is in the image of $(-)^p - (-) \colon B \to B$. The choice
$$
B \coloneqq A[t]/(t- t^p - s)
$$
by construction has the desired property concerning $s$, the preimage of $s$ is the equivalence class of $t$.
To see that with this choice $Spec(B) \to Spec(A)$ is indeed an [[étale morphism of schemes]] it is sufficient to observe that it is a [[morphism of finite presentation]] and a [[formally étale morphism]]. The first is true by construction. For the second observe that for a ring homomorphism $B \to T$ the generator $t$ cannot go to a nilpotent element since otherwise $s$ would have to be nilpotent. This implies [[formally étale morphism|formal étaleness]] analogous to the discussion at [étale morphism of schemes -- Open immersion is Etale](etale+morphism+of+schemes#OpenImmersionIsEtale).
=--
## Related concepts
* [[étale cohomology]]
* [[Kummer sequence]]
* [[Kummer-Artin-Schreier-Witt exact sequence]]
* [[exponential exact sequence]]
## References
* [[Günter Tamme]], section II 4.2 _[[Introduction to Étale Cohomology]]_
* [[James Milne]], example 7.9 of _[[Lectures on Étale Cohomology]]_
[[!redirects Artin-Schreier sequences]]
[[!redirects Artin-Schreier exact sequence]]
[[!redirects Artin-Schreier exact sequences]]
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Artin-Tate lemma
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https://ncatlab.org/nlab/source/Artin-Tate+lemma
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## Statement
\begin{proposition}\label{TheLemma}
**([Artin & Tate 1951](#ArtinTate51))**
\linebreak
Given a [[Noetherian ring|Noetherian]] [[commutative ring]] $A$, a [[commutative algebra|commutative]] [[associative algebra|algebra]] $B$ over $A$, and a commutative algebra $C$ over $B$ (and thus a commutative algebra over $A$), if $C$ is a [[finitely generated algebra]] over $A$ and $C$ is a [[finitely generated module]] over $B$, then $B$ is a finitely generated algebra over $A$. (If $A$ is a commutative subring of commutative ring $B$, then $B$ is a commutative algebra over $A$.)
\end{proposition}
## See also
* [[Noetherian ring]]
* [[commutative algebra]]
## References
The original article:
* {#ArtinTate51} [[Emil Artin]], [[John T. Tate]], *A note on finite ring extensions*, J. Math. Soc Japan, Volume **3** (1951) 74–77 $[$[doi:10.2969/jmsj/00310074](https://doi.org/10.2969/jmsj/00310074), [pdf](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-3/issue-1/A-Note-on-Finite-Ring-Extensions/10.2969/jmsj/00310074.pdf)$]$
Review:
* [[Michael Atiyah]], [[Ian G. Macdonald]], _Introduction to commutative algebra_, (1969, 1994) $[$[pdf](http://math.univ-lyon1.fr/~mathieu/CoursM2-2020/AMD-ComAlg.pdf), [ISBN:9780201407518](https://www.routledge.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518)$]$
* David Eisenbud, *Commutative Algebra with a View Toward Algebraic Geometry*, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN:0-387-94268-8
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Artinian bimodule
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https://ncatlab.org/nlab/source/Artinian+bimodule
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## Idea
An *Artinian bimodule* is a bimodule which satisfies the [[descending chain condition]] on its subbimodules.
## Definition
Given [[rings]] $R$ and $S$ and an $R$-$S$-[[bimodule]] $B$, let $\mathrm{Mono}(B)$ be the [[category]] whose [[objects]] are $R$-$S$-[[subbimodules]] of $B$ and whose [[morphisms]] are $R$-$S$-[[bimodule monomorphisms]]. A descending chain of $R$-$S$-subbimodules is an [[inverse sequence]] of $R$-$S$-subbimodules in $\mathrm{Mono}(B)$, a [[sequence]] of $R$-$S$-subbimodules $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following [[dependent type|dependent sequence]] of $R$-$S$-bimodule monomorphisms: for [[natural number]] $n \in \mathbb{N}$, a dependent $R$-$S$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.
An $R$-$S$-[[bimodule]] $B$ is **Artinian** if it satisfies the [[descending chain condition]] on its subbimodules: for every descending chain of $R$-$S$-subbimodules $(A, i_n)$ of $B$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the $R$-$S$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an $R$-$S$-[[bimodule isomorphism]].
## Examples
A [[ring]] $R$ is [[Artinian ring|Artinian]] if it is Artinian as a $R$-$R$-[[bimodule]] with respect to its canonical bimodule structure, with its [[left action]] $\alpha_L:R \times R \to R$ and [[right action]] $\alpha_R:R \times R \to R$ defined as its multiplicative binary operation and its [[biaction]] $\alpha:R \times R \times R \to R$ defined as its ternary product:
$$\alpha_L(a, b) \coloneqq a \cdot b$$
$$\alpha_R(a, b) \coloneqq a \cdot b$$
$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$
## See also
* [[bimodule]]
* [[Noetherian bimodule]]
* [[Artinian ring]]
[[!redirects Artinian bimodules]]
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Artinian local ring
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https://ncatlab.org/nlab/source/Artinian+local+ring
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[[!redirects Artin ring]]
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#### Algebra
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#### Formal geometry
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[[!include formal geometry -- contents]]
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## Definition
A **Artinian local ring** or **local Artinian ring** or **local Artin ring** or **Artin local ring** or **Weil ring** is a [[local ring]] which satisfies one of the equivalent conditions
* The local ring is an [[Artinian ring]]; it satisfies the [[descending chain condition]]
* The [[Jacobson radical]] of the local ring is a [[nilradical]]
These are also called **Artinian local algebras**, **local Artinian algebras**, **Artin local algebras**, or **local Artin algebras** or **Weil algebras**, since Artinian local rings are [[commutative algebras]] over the residue field $\mathbb{K}$.
In [[synthetic differential geometry]], the term Weil algebra is used for real Artinian local algebras, see _[[infinitesimally thickened point]]_ for more details.
## Properties
Every Artinian local $\mathbb{K}$-algebra $A$ has a [[maximal ideal]] $\mathfrak{m}_A$, whose [[residue field]] $A / \mathfrak{m}_A$ is $\mathbb{K}$ itself. As a $\mathbb{K}$ [[vector space]] one has a splitting $A=\mathbb{K}\oplus \mathfrak{m}_A$. Moreover, the descending chain condition implies that $(\mathfrak{m}_A)^n=0$ for some $n\gg 0$, a consequence of [[Nakayama lemma]]. This implies that the maximal ideal is a [[nilradical]].
Given a field $K$ and a Artinian local $K$-algebra $A$, let $I$ be the [[nilradical]] of $A$. There is a function $v:I \to \mathbb{N}$ which takes a [[nilpotent element]] $r \in I$ to the least natural number $n$ such that $r^{v(r) + 1} = 0$, such that given [[nilpotent elements]] $r \in I$ and $s \in I$ and non-zero scalars $a \in K$ and $b \in K$,
* $v(a r + b s) = v(r) + v(s)$
* $v(r s) = \min(v(r), v(s))$
### Spectra
Passing from commutative rings to their spectra (in the sense of algebraic geometry), Artinian local algebras correspond to infinitesimal pointed spaces. As such, they appear as bases of deformations in [[infinitesimal object|infinitesimal]] [[deformation theory]]. For instance $Spec(\mathbb{K}[\epsilon]/(\epsilon^2))$ is the base space for 1-dimensional first order deformations. Similarly, $Spec(\mathbb{K}[\epsilon]/(\epsilon^{n+1}))$ is the base space for 1-dimensional $n$-th order deformations.
An Artinian local algebra has a unique [[prime ideal]], which means that its [[spectrum]] consists of a single point, i.e., $Spec(A)$ is trivial as a topological space. It is however non-trivial as a ringed space, since its ring of functions is $A$. By this reason spectra of Artinian local algebras are occasionally called _[[infinitesimally thickened point|fat points]]_ in the literature.
## Examples
* A classical example is the ring of [[dual numbers]] $\mathbb{K}[\epsilon]/(\epsilon^2)$ over a field $\mathbb{K}$.
* Every [[prime power local ring]] is a local Artinian ring.
## See also
| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |
## References
* [[The Stacks Project]]
* [10.100. Flatness criteria over Artinian rings](http://stacks.math.columbia.edu/tag/051E)
Local Artinian $\infty$-algebras are discussed in
* [[Jacob Lurie]], _Formal moduli problems_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf))
[[!redirects Weil ring]]
[[!redirects Weil rings]]
[[!redirects local Artinian ring]]
[[!redirects local Artinian rings]]
[[!redirects Artinian local ring]]
[[!redirects Artinian local rings]]
[[!redirects local Artinian algebra]]
[[!redirects local Artinian algebras]]
[[!redirects Artinian local algebra]]
[[!redirects Artinian local algebras]]
[[!redirects local Artin ring]]
[[!redirects local Artin rings]]
[[!redirects Artin local ring]]
[[!redirects Artin local rings]]
[[!redirects local Artin algebra]]
[[!redirects local Artin algebras]]
[[!redirects Artin local algebra]]
[[!redirects Artin local algebras]]
|
artinian ring
|
https://ncatlab.org/nlab/source/artinian+ring
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### Artinian rings
Every [[ring]] $R$ has a canonical $R$-$R$-[[bimodule]] structure, with [[left action]] $\alpha_L:R \times R \to R$ and [[right action]] $\alpha_R:R \times R \to R$ defined as the multiplicative binary operation on $R$ and [[biaction]] $\alpha:R \times R \times R \to R$ defined as the ternary product on $R$:
$$\alpha_L(a, b) \coloneqq a \cdot b$$
$$\alpha_R(a, b) \coloneqq a \cdot b$$
$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$
Let $\mathrm{TwoSidedIdeals}(R)$ be the [[category of two-sided ideals]] in $R$, whose objects are [[two-sided ideals]] $I$ in $R$, [[subbimodules|sub-$R$-$R$-bimodules]] of $R$ with respect to the canonical bimodule structure on $R$, and whose [[morphisms]] are $R$-$R$-[[bimodule monomorphisms]].
A descending chain of two-sided ideals in $R$ is an [[inverse sequence]] of two-sided ideals in $R$, a [[sequence]] of two-sided ideals $A:\mathbb{N} \to \mathrm{TwoSidedIdeals}(R)$ with the following [[dependent sequence]] of $R$-$R$-bimodule monomorphisms: for natural number $n \in \mathbb{N}$, a dependent $R$-$R$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.
A [[ring]] $R$ is **Artinian** if it satisfies the [[descending chain condition]] on its two-sided ideals: for every descending chain of two-sided ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the $R$-$R$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an $R$-$R$-[[bimodule isomorphism]].
### Left Artinian rings
Let $\mathrm{LeftIdeals}(R)$ be the category of [[left ideals]] in $R$, whose objects are [[left ideals]] $I$ in $R$, sub-left-$R$-modules of $R$ with respect to the canonical [[left module]] structure $(-)\cdot(-):R \times R \to R$ on $R$, and whose [[morphisms]] are left $R$-module monomorphisms.
A descending chain of left ideals in $R$ is an [[inverse sequence]] of left ideals in $R$, a [[sequence]] of left ideals $A:\mathbb{N} \to \mathrm{LeftIdeals}(R)$ with the following [[dependent sequence]] of left $R$-module monomorphisms: for natural number $n \in \mathbb{N}$, a dependent left $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.
A [[ring]] $R$ is **left Artinian** if it satisfies the [[descending chain condition]] on its left ideals: for every descending chain of left ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the left $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an left $R$-module isomorphism.
### Right Artinian rings
Let $\mathrm{RightIdeals}(R)$ be the category of [[right ideals]] in $R$, whose objects are [[right ideals]] $I$ in $R$, sub-right-$R$-modules of $R$ with respect to the canonical [[right module]] structure $(-)\cdot(-):R \times R \to R$ on $R$, and whose [[morphisms]] are right $R$-module monomorphisms.
A descending chain of right ideals in $R$ is an [[inverse sequence]] of right ideals in $R$, a [[sequence]] of right ideals $A:\mathbb{N} \to \mathrm{RightIdeals}(R)$ with the following [[dependent sequence]] of right $R$-module monomorphisms: for natural number $n \in \mathbb{N}$, a dependent right $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.
A [[ring]] $R$ is **right Artinian** if it satisfies the [[descending chain condition]] on its right ideals: for every descending chain of right ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the right $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an right $R$-module isomorphism.
## Properties
In an artinian ring $R$ the [[Jacobson radical]] $J(R)$ is [[nilpotent ideal|nilpotent]]. A left artinian ring is semiprimitive if and only if the zero ideal is the unique nilpotent ideal.
## Artinian and Noetherian rings
A dual condition is noetherian: a __[[noetherian ring]]__ is a ring satisfying the ascending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian. For a converse there is a strong condition: a left (unital) ring $R$ is left artinian iff $R/J(R)$ is semisimple in $_R Mod$ and the Jacobson radical $J(R)$ is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.
## See also
* [[Artinian bimodule]]
* [[artinian object]]
| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |
## References
* [[The Stacks Project]]
* [10.100. Flatness criteria over Artinian rings](http://stacks.math.columbia.edu/tag/051E)
[[!redirects artinian ring]]
[[!redirects artinian rings]]
[[!redirects Artinian ring]]
[[!redirects Artinian rings]]
[[!redirects Artin ring]]
[[!redirects Artin rings]]
[[!redirects Artin algebra]]
[[!redirects Artin algebras]]
|
Artur Ekert
|
https://ncatlab.org/nlab/source/Artur+Ekert
|
* [personal page](https://www.arturekert.org/)
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Artur_Ekert)
## Selected writings
Introducing the notion of [[quantum key distribution]]:
* [[Artur K. Ekert]], *Quantum cryptography based on Bell’s theorem*, Phys. Rev. Lett. **67** 661 (1991) [[doi:10.1103/PhysRevLett.67.661](https://doi.org/10.1103/PhysRevLett.67.661)]
On the [[quantum physics]] of [[quantum information]]:
* [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]
category: people
[[!redirects Artur K. Ekert]]
|
Arturo Prat-Waldron
|
https://ncatlab.org/nlab/source/Arturo+Prat-Waldron
|
* [webpage](http://math.berkeley.edu/~aprat/)
category: people
|
Arun Debray
|
https://ncatlab.org/nlab/source/Arun+Debray
|
* [webpage](https://web.ma.utexas.edu/users/a.debray/)
## Selected writings
On [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[differential cohomology]] via [[sheaves of spectra]]:
* {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250))
On [[differential cohomology]] in [[mathematical physics]]:
* [[Arun Debray]], *Differential Cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.14338](https://arxiv.org/abs/2312.14338)]
category: people
|
Arun Kumar Pati
|
https://ncatlab.org/nlab/source/Arun+Kumar+Pati
|
* [Wikipedia entry](https://en.wikipedia.org/wiki/Arun_K._Pati)
* [Research group page](https://www.hri.res.in/~akpati/)
## Selected writings
Introducing the statement of the "[[no-deleting theorem]]" in [[quantum physics]]/[[quantum information theory]]:
* [[Arun Kumar Pati]], [[Samuel L. Braunstein]], *Impossibility of deleting an unknown quantum state*, Nature **404** (2000) 164-165 [[doi:10.1038/404130b0](https://doi.org/10.1038/404130b0), [arXiv:quant-ph/9911090](https://arxiv.org/abs/quant-ph/9911090)]
category: people
[[!redirects Arun K. Pati]]
[[!redirects Arun Pati]]
|
Arun Ram
|
https://ncatlab.org/nlab/source/Arun+Ram
|
__Arun Ram__ is a mathematician at the University of Melbourne (formerly at the Univ. of Wisconsin-Madison), a specialist in [[combinatorial representation theory]] and Lie theory (Lie groups and Lie algebras, quantum groups, Iwahori-Hecke algebras etc.).
* [web](http://www.ms.unimelb.edu.au/~ram)
## Selected writings
On [[quantum cohomology]] of [[flag varieties]]:
* [[Dale H. Peterson]] (notes by [[Arun Ram]]), *Quantum Cohomology of $G/H$*, MIT (1997) [[web](http://math.soimeme.org/~arunram/Resources/QuantumCohomologyOfGPL1-5.html), [pdf](http://math.soimeme.org/~arunram/Resources/PetersonGmodBcourse1997.pdf), [[Peterson-QuantumCohomology.pdf:file]]]
category: people
|
Arunas Liulevicius
|
https://ncatlab.org/nlab/source/Arunas+Liulevicius
|
* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6494)
## Selected writings
On the [[symplectic cobordism ring]]:
* [[Arunas Liulevicius]], _Notes on Homotopy of Thom Spectra_, American Journal of Mathematics
Vol. 86, No. 1 (1964), pp. 1-16 ([jstor:2373032](https://www.jstor.org/stable/2373032))
category: people
|
Arvind Rajaraman
|
https://ncatlab.org/nlab/source/Arvind+Rajaraman
|
* [webpage](http://www.physics.uci.edu/~arajaram/)
## Selected writings
On [[D-brane polarization]] in a context of [[AdS/QCD]]:
* [[Ofer Aharony]], [[Arvind Rajaraman]], _String Theory Duals for Mass-deformed $SO(N)$ and $USp(2N)$ $\mathcal{N}=4$ SYM Theories_, Phys. Rev. D62:106002, 2000 ([arXiv:hep-th/0004151](https://arxiv.org/abs/hep-th/0004151))
On [[black brane|black]] [[M2-M5 brane bound states]]:
* [[Arvind Rajaraman]], *Supergravity Solutions for Localised Brane Intersections*, JHEP 0109:018 (2001) [[arXiv:hep-th/0007241](https://arxiv.org/abs/hep-th/0007241), [doi:10.1088/1126-6708/2001/09/018](https://doi.org/10.1088/1126-6708/2001/09/018)]
## Related entries
* [[de Sitter spacetime]]
category: people
|
ascending chain condition
|
https://ncatlab.org/nlab/source/ascending+chain+condition
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Given a [[category]] $C$ and an object $B:C$, let $\mathrm{Mono}(B)$ be the [[category]] whose [[objects]] are [[subobjects]] of $B$ and whose [[morphisms]] are [[monomorphisms]]. A ascending chain of subobjects of $B$ is a [[direct sequence]] in $\mathrm{Mono}(B)$, a [[sequence]] of subobjects $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following [[dependent sequence]] of monomorphisms: for [[natural number]] $n \in \mathbb{N}$, a dependent monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$.
Given categories $C$ and $D$ with [[forgetful functor]] $F:C \to D$, an object $B$ is said to satisfy the **ascending chain condition** on subobjects of $F(B)$ if for every ascending chain of subobjects $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$ is an [[isomorphism]].
## Examples
There is a [[forgetful functor]] $F:Ring \to BMod$ from the category [[Ring]] of [[rings]] to the category $BMod$ of [[bimodules]], which forgets the multiplicative structure on the bimodule, that the canonical [[left action]] and [[right action]] of each ring $R$ have [[domain]] $R^2$ and are equal to each other and to the multiplicative binary operation of $R$, and that the canonical [[biaction]] of each ring $R$ has domain $R^3$.
Given a ring $R$, a two-sided ideal is a subobject of $F(R)$, a sub-$R$-$R$-bimodule. A [[ring]] $R$ is said to satisfy the ascending chain condition on two-sided ideals if for every ascending chain of two-sided ideals $(A, i_n)$ of $F(B)$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the monomorphism $i_n:A_{n} \hookrightarrow A_{n+1}$ is an [[isomorphism]].
Similarly, there are forgetful functors $G:Ring \to LeftMod$ from $Ring$ to the category $LeftMod$ of [[left modules]] and $H:Ring \to RightMod$ from $Ring$ to the category $RightMod$ of [[right modules]], and one could similarly define the ascending chain condition on [[left ideals]] and [[right ideals]] for rings.
## See also
* [[descending chain condition]]
* [[Noetherian bimodule]]
* [[Noetherian ring]]
* [[property sup]]
[[!redirects ascending chain conditions]]
|
Asgar Jamneshan
|
https://ncatlab.org/nlab/source/Asgar+Jamneshan
|
* [website](https://asgarjam.wixsite.com/home)
## Selected writings
On [[measure theory]] via (secretly) [[Boolean toposes]]:
* [[Asgar Jamneshan]], [[Terence Tao]], _Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration_ ([arXiv:2010.00681](arXiv:2010.00681))
On a generalization of the [[Furstenberg-Zimmer structure theorem]] via Boolean toposes:
* [[Asgar Jamneshan]], _An uncountable Furstenberg-Zimmer structure theory_ ([arXiv:2103.17167](https://arxiv.org/abs/2103.17167)).
category: people
|
Ash Asudeh
|
https://ncatlab.org/nlab/source/Ash+Asudeh
|
* [webpage](http://www.sas.rochester.edu/lin/sites/asudeh)
### Related entries
* [[monad (in linguistics)]]
category: people
|
Asher Peres
|
https://ncatlab.org/nlab/source/Asher+Peres
|
* [Wikipedia entry](http://en.wikipedia.org/wiki/Asher_Peres)
## Selected writings
Introducing the notion of [[quantum teleportation]]:
* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]
On the [[interpretation of quantum mechanics]]:
* {#Peres97} [[Asher Peres]], *Interpreting the Quantum World*, Stud. History Philos. Modern Physics **29** (1998) 611 [[arXiv:quant-ph/9711003](http://arxiv.org/abs/quant-ph/9711003)]
category: people
|
Ashkbiz Danehkar
|
https://ncatlab.org/nlab/source/Ashkbiz+Danehkar
|
Ashkbiz Danehkar is an astrophysicist studying black holes and galaxies.
* [webpage](https://www.danehkar.net/)
* [arXiv page](https://arxiv.org/a/danehkar_a_1.html)
* [INSPIRE-HEP](https://inspirehep.net/authors/1048604)
* [NASA ADS](https://ui.adsabs.harvard.edu/search/q=orcid%3A0000-0003-4552-5997)
* Scopus ID: [34972723700](https://www.scopus.com/authid/detail.uri?authorId=34972723700)
* ResearcherID: [C-2053-2009](https://www.webofscience.com/wos/author/rid/C-2053-2009)
* ORCID: [0000-0003-4552-5997](https://orcid.org/0000-0003-4552-5997)
## Selected writings
On [[Weyl tensor]]:
* [[Ashkbiz Danehkar]], _Covariant Evolution of Gravitoelectromagnetism_, Universe 8 (2022) 318 ([arXiv:2206.13946](https://arxiv.org/abs/2206.13946), [doi:10.3390/universe8060318](https://doi.org/10.3390/universe8060318))
* [[Ashkbiz Danehkar]], _On the Significance of the Weyl Curvature in a Relativistic Cosmological Model_, Mod.Phys.Lett.A 24 (2009) 3113-3127 ([arXiv:0707.2987](https://arxiv.org/abs/0707.2987), [doi:10.1142/S0217732309032046](https://doi.org/10.1142/S0217732309032046))
* [[Ashkbiz Danehkar]], _Gravitational Fields of the Magnetic-type_, Int.J.Mod.Phys.D 29 (2020) 2043001 ([arXiv:2006.13287](https://arxiv.org/abs/2006.13287), [doi:10.1142/S0218271820430014](https://doi.org/10.1142/S0218271820430014))
On [[electric-magnetic duality ]]:
* [[Ashkbiz Danehkar]], _Electric-magnetic duality in gravity and higher-spin fields_, Front. Phys. 6 (2019) 146 ([doi:10.3389/fphy.2018.00146](https://doi.org/10.3389/fphy.2018.00146))
* [[Ashkbiz Danehkar]], Hassan Alshal, [[Thomas Curtright]], _Dual fields of massive/massless gravitons in IR/UV completions_, Int.J.Mod.Phys.D 30 (2021) 2142021 ([arXiv:2109.05148](https://arxiv.org/abs/2109.05148), [doi:10.1142/S0218271821420219](https://doi.org/10.1142/S0218271821420219))
* Constantin Bizdadea, Eugen M. Cioroianu, [[Ashkbiz Danehkar]], Marius Iordache, Solange O. Saliu, Silviu C. Sararu, _Consistent interactions of dual linearized gravity in D=5: couplings with a topological BF model_, Eur.Phys.J.C 63 (2009) 491-519 ([arXiv:0908.2169](https://arxiv.org/abs/0908.2169), [doi:10.1140/epjc/s10052-009-1105-0](https://doi.org/10.1140/epjc/s10052-009-1105-0))
On [[black hole]] astrophysics:
* [[Ashkbiz Danehkar]], Michael A. Nowak, Julia C. Lee, Gerard A. Kriss, Andrew J. Young, Martin J. Hardcastle, Susmita Chakravorty, Taotao Fang, Joseph Neilsen, Farid Rahoui, Randall K. Smith, _The Ultra-fast Outflow of the Quasar PG 1211+143 as Viewed by Time-averaged Chandra Grating Spectroscopy_, Astrophys. J. 853 (2018) 165 ([arXiv:1712.07118](https://arxiv.org/abs/1712.07118), [doi:10.3847/1538-4357/aaa427](https://doi.org/10.3847/1538-4357/aaa427))
* Gerard A. Kriss, Julia C. Lee, [[Ashkbiz Danehkar]], Michael A. Nowak, Taotao Fang, Martin J. Hardcastle, Joseph Neilsen, Andrew Young, _Discovery of an Ultraviolet Counterpart to an Ultrafast X-Ray Outflow in the Quasar PG 1211+143_, Astrophys. J. 853 (2018) 166 ([arXiv:1712.08850](https://arxiv.org/abs/1712.08850), [doi:10.3847/1538-4357/aaa42b](https://doi.org/10.3847/1538-4357/aaa42b))
* Rozenn Boissay-Malaquin, [[Ashkbiz Danehkar]], Herman L. Marshall, Michael A. Nowak, _Relativistic Components of the Ultra-fast Outflow in the Quasar PDS 456 from Chandra/HETGS, NuSTAR, and XMM-Newton Observations_, Astrophys. J. 873 (2019) 29 ([arXiv:1901.06641](https://arxiv.org/abs/1901.06641), [doi:10.3847/1538-4357/ab0082](https://doi.org/10.3847/1538-4357/ab0082))
On [[galaxy]] [[astronomy]]:
* [[Ashkbiz Danehkar]], M. S. Oey, William J. Gray, _Catastrophic Cooling in Superwinds. II. Exploring the Parameter Space_, Astrophys. J. 921 (2021) 91 ([arXiv:2106.10854](https://arxiv.org/abs/2106.10854), [doi:10.3847/1538-4357/ac1a76](https://doi.org/10.3847/1538-4357/ac1a76))
* [[Ashkbiz Danehkar]], M. S. Oey, William J. Gray, _Catastrophic Cooling in Superwinds. III. Nonequilibrium Photoionization_, Astrophys. J. 937 (2022) 68 ([arXiv:2208.12030](https://arxiv.org/abs/2208.12030), [doi:10.3847/1538-4357/ac8cec](https://doi.org/10.3847/1538-4357/ac8cec))
* [[Ashkbiz Danehkar]], _Physical and Chemical Properties of Wolf-Rayet Planetary Nebulae_, Astrophys. J. Suppl. 257 (2021) 58 ([arXiv:2106.10762](https://arxiv.org/abs/2106.10762), [doi:10.3847/1538-4365/ac2310](https://doi.org/10.3847/1538-4365/ac2310))
* [[Ashkbiz Danehkar]], _Morpho-kinematic Properties of Wolf-Rayet Planetary Nebulae_, Astrophys. J. Suppl. 260 (2022) 14 ([arXiv:2107.03994](https://arxiv.org/abs/2107.03994), [doi:10.3847/1538-4365/ac5cca](https://doi.org/10.3847/1538-4365/ac5cca))
On [[plasma]] / [[computational physics]]:
* [[Ashkbiz Danehkar]], _Electron beam-plasma interaction and electron-acoustic solitary waves in a plasma with suprathermal electrons_, Plasma Phys. Control. Fusion 60 (2018) 065010 ([arXiv:1804.07299](https://arxiv.org/abs/1804.07299), [doi:10.1088/1361-6587/aabc40](https://doi.org/10.1088/1361-6587/aabc40))
* [[Ashkbiz Danehkar]], _Electrostatic solitary waves in an electron-positron pair plasma with suprathermal electrons_, Phys. Plasmas 24 (2017) 102905 ([arXiv:1711.01141](https://arxiv.org/abs/1711.01141), [doi:10.1063/1.5000873](https://doi.org/10.1063/1.5000873))
category: people
|
Ashok Das
|
https://ncatlab.org/nlab/source/Ashok+Das
|
* [Wikipedia entry](http://en.wikipedia.org/wiki/Ashok_Das)
## related $n$Lab entries
* [[gravity]]
* [[rubber-sheet analogy of gravity]]
category: people
|
Ashoke Sen
|
https://ncatlab.org/nlab/source/Ashoke+Sen
|
* [Wikipedia entry](http://en.wikipedia.org/wiki/Ashoke_Sen)
## Selected writings
On microscopic explanation of [[Bekenstein-Hawking entropy]] via [[geometric engineering of QFT|geometric engineering]] of [[black holes in string theory]] as [[bound states of D-branes]]:
* [[Ashoke Sen]], _Extremal black holes and elementary string states_, Mod. Phys. Lett. A10: 2081-2094, 1995 ([arXiv:hep-th/9504147](http://arxiv.org/abs/hep-th/9504147))
* {#Sen07} [[Ashoke Sen]], _Black Hole Entropy Function, Attractors and Precision Counting of Microstates_, Gen. Rel. Grav. 40: 2249-2431, 2008 ([arXiv:0708.1270](http://arxiv.org/abs/0708.1270))
Introducing [[Sen's conjecture]] on [[open string]] [[tachyon condensation]] and the decay of [[D-brane]]/[[anti-D-brane]] pairs in [[superstring theory]] via open [[superstring]] [[tachyon condensation]]:
* {#Sen98} [[Ashoke Sen]], _Tachyon Condensation on the Brane Antibrane System_, JHEP 9808:012,1998 ([arXiv:hep-th/9805170](https://arxiv.org/abs/hep-th/9805170))
* {#Sen99} [[Ashoke Sen]], _Universality of the Tachyon Potential_, JHEP 9912:027, 1999 ([arXiv:hep-th/9911116](http://arxiv.org/abs/hep-th/9911116))
with review in:
* [[Ashoke Sen]], _Tachyon Dynamics in Open String Theory_, Int. J. Mod. Phys. A20:5513-5656, 2005 ([arXiv:hep-th/0410103](https://arxiv.org/abs/hep-th/0410103))
Discussion of [[gravitational waves]] in relation to the [[soft graviton theorem]]:
* Arnab Priya Saha, Biswajit Sahoo, [[Ashoke Sen]], _Proof of the Classical Soft Graviton Theorem in $D=4$_ ([arXiv:1912.06413](https://arxiv.org/abs/1912.06413))
* [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], section 2.1 of _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))
* Sergei Alexandrov, Jan Manschot, [[Boris Pioline]], _D3-instantons, Mock Theta Series and Twistors_, JHEP 1304 (2013)
002 ([arXiv:1207.1109](http://arxiv.org/abs/1207.1109))
On [[Lagrangian densities]] for [[self-dual higher gauge fields]] (generically thought to be [[non-Lagrangian field theories]]) at the cost of an unusual decoupling [[auxiliary field]]:
* {#Sen20} [[Ashoke Sen]], *Self-dual forms: Action, Hamiltonian and Compactification*, Journal of Physics A: Mathematical and Theoretical, **53** 8 (2020) [[arXiv:1903.12196](https://arxiv.org/abs/1903.12196), [doi:10.1088/1751-8121/ab5423](https://doi.org/10.1088/1751-8121/ab5423)]
On [[D-brane instantons]] and [[moduli stabilization]] in [[type II string theory]]:
* Sergei Alexandrov, Atakan Hilmi Fırat, Manki Kim, [[Ashoke Sen]], [[Bogdan Stefański]], *D-instanton Induced Superpotential*, J. High Energ. Phys. **2022** 90 (2022) [[arXiv:2204.02981](https://arxiv.org/abs/2204.02981), <a href="https://doi.org/10.1007/JHEP07(2022)090">doi:10.1007/JHEP07(2022)090</a>]
and in 0B string theory using methods from [[string field theory]]:
* Joydeep Chakravarty, [[Ashoke Sen]], *Normalization of D instanton amplitudes in two dimensional type 0B string theory* [[arXiv:2207.07138](https://arxiv.org/abs/2207.07138)]
reviewed in
* [[Ashoke Sen]], *D-instanton amplitudes in string theory*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Sen2023)]
## Related entries
* [[Sen's conjecture]]
* [[electric-magnetic duality]]/[[S-duality]]
* [[M-theory lift of gauge enhancement on D6-branes]]
* [[BFSS matrix model]]
category: people
|
Ashvin Vishwanath
|
https://ncatlab.org/nlab/source/Ashvin+Vishwanath
|
* [institute page](https://www.physics.harvard.edu/people/facpages/vishwanath)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Ashvin_Vishwanath)
## Selected writings
On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy|topological]]) [[entanglement entropy]] of subregions:
* [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], *Quasiparticle statistics and braiding from ground-state entanglement*, Phys. Rev. B **85** (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$
* [[Yi Zhang]], [[Tarun Grover]], [[Ashvin Vishwanath]], *General procedure for determining braiding and statistics of anyons using entanglement interferometry*, Phys. Rev. B **91** (2015) 035127 $[$[arXiv:1412.0677](https://arxiv.org/abs/1412.0677), [doi:10.1103/PhysRevB.91.035127](https://doi.org/10.1103/PhysRevB.91.035127)$]$
On [[topological semi-metals]]:
* [[Ari M. Turner]], [[Ashvin Vishwanath]], Part I of: *Beyond Band Insulators: Topology of Semi-metals and Interacting Phases*, in: *Topological Insulators*, Contemporary Concepts of Condensed Matter Science **6** (2013) 293-324 $[$[arXiv:1301.0330](https://arxiv.org/abs/1301.0330), [ISBN:978-0-444-63314-9](https://www.sciencedirect.com/bookseries/contemporary-concepts-of-condensed-matter-science/vol/6/suppl/C)$]$
* N. P. Armitage, [[Eugene Mele]], [[Ashvin Vishwanath]], *Weyl and Dirac semimetals in three-dimensional solids*, Rev. Mod. Phys. **90** 015001 (2018) $[$[doi:10.1103/RevModPhys.90.015001](https://doi.org/10.1103/RevModPhys.90.015001)$]$
category: people
|
Ashwin S. Pande
|
https://ncatlab.org/nlab/source/Ashwin+S.+Pande
|
* [institute page](https://ahduni.edu.in/academics/schools-centres/school-of-arts-and-sciences/faculty/ashwin-pande/)
## Selected writings
On [[topological T-duality]] for non-free torus actions (physically: [[KK-monopoles]]):
* {#Pande06} [[Ashwin S. Pande]], _Topological T-duality and Kaluza-Klein Monopoles_, Adv. Theor. Math. Phys. **12** (2007) 185-215 [[arXiv:math-ph/0612034](https://arxiv.org/abs/math-ph/0612034), [doi:10.4310/ATMP.2008.v12.n1.a3](https://dx.doi.org/10.4310/ATMP.2008.v12.n1.a3)]
On [[topological T-duality]] in relation to [[T-folds]]:
* [[Peter Bouwknegt]], [[Ashwin S. Pande]], *Topological T-duality and T-folds*, Advances in Theoretical and Mathematical Physics **13** 5 (2009) [[arXiv:0810.4374](https://arxiv.org/abs/0810.4374), [doi:10.4310/ATMP.2009.v13.n5.a6](https://dx.doi.org/10.4310/ATMP.2009.v13.n5.a6)]
category: people
[[!redirects Ashwin Pande]]
|
Asif Equbal
|
https://ncatlab.org/nlab/source/Asif+Equbal
|
member of [[CQTS]] at NYU Abu Dhabi
* [institute page](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/asif-equbal.html)
* [personal page](https://www.asifequbal.com/)
* [GoogleScholar page](https://scholar.google.com.hk/citations?user=q8KQ9gUAAAAJ&hl=en)
## Selected writings
On [[Floquet theory]] in [[nuclear magnetic resonance]]:
* Konstantin L. Ivanov, Kaustubh R. Mote, Matthias Ernst, [[Asif Equbal]], Perunthiruthy K. Madhu, *Floquet theory in magnetic resonance: Formalism and applications*, Progress in Nuclear Magnetic Resonance Spectroscopy, **126**–**127** (2021) 17-58 [[doi:10.1016/j.pnmrs.2021.05.002](https://doi.org/10.1016/j.pnmrs.2021.05.002)]
On Dynamic Nuclear Polarization in [[nuclear magnetic resonance]]:
* [[Asif Equbal]] et al., *Role of electron spin dynamics and coupling network in designing dynamic nuclear polarization*, Progress in Nuclear Magnetic Resonance Spectroscopy **126**–**127** (2021) 1-16 [[doi:10.1016/j.pnmrs.2021.05.003](https://doi.org/10.1016/j.pnmrs.2021.05.003)]
On [[nuclear magnetic resonance]] and related spin technologies for the use of [[quantum computation]]:
* [[Asif Equbal]], *Molecular spin qubits for future quantum technology*, talk at [[CQTS]] (Nov 2022) [slides: [[Equbal-CQTS-Nov2022.pdf:file]], video: [rec](https://nyu.zoom.us/rec/play/YTjIGL-Bevb1H44UuL-ZimXdxph5cffddWpP3H4ZsuRT2xu3OrnTbC0NZLsKedUGwS68DJ8onVFPAETb.rreoi7Wt6uXFyaYN?continueMode=true&_x_zm_rtaid=e0VPIMlfT9KlVd_wiaOq6A.1669794985784.11f8cd37091ebf6bdd2a878668e26cd6&_x_zm_rhtaid=404)]
On [[nuclear magnetic resonance]] on the [[nitrogen-vacancy center in diamond]] and related systems in [[quantum sensing]]:
* Santiago Bussandri, Daphna Shimon, [[Asif Equbal]], Yuhang Ren, Susumu Takahashi, [[Chandrasekhar Ramanathan]], Songi Han, *P1 center electron spin clusters are prevalent in type Ib diamond* [[arXiv:2311.05396](https://arxiv.org/abs/2311.05396)]
## Related entries
* [[nuclear magnetic resonance]]
* [[quantum computing]]
category: people
|
Assaf Libman
|
https://ncatlab.org/nlab/source/Assaf+Libman
|
* [webpage](http://homepages.abdn.ac.uk/a.libman/pages/home_page/)
## related $n$Lab entries
* [[tower]]
category: people
|
assembly map
|
https://ncatlab.org/nlab/source/assembly+map
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _analytic assembly map_ is a natural [[morphism]] from $G$-[[equivariant cohomology|equivariant]] [[topological K-theory]] to the [[operator K-theory]] of a corresponding [[crossed product C*-algebra]].
More generally in equivariant [[KK-theory]] this is called the _Kasparov descent map_ and is of the form
$$
KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)
$$
where on the left we have $G$-equivariant KK-theory and on the right ordinary [[KK-theory]] of [[crossed product C*-algebras]] (which by the discussion there are models for the [[groupoid convolution algebras]] of $G$-[[action groupoids]]).
(recalled as [Blackadar, theorem 20.6.2](#Blackadar))
## Properties
The [[Baum-Connes conjecture]] states that under some conditions the analytic assembly map is in fact an [[isomorphism]]. The [[Novikov conjecture]] makes statements about it being an [[injection]]. The _[[Green-Julg theorem]]_ states that under some (milder) conditions the Kasparov desent map is an [[isomorphism]].
## Related concepts
* [[Novikov conjecture]]
* [[Baum-Connes conjecture]]
## References
The construction goes back to
* [[Gennady Kasparov]], _The index of invariant elliptic operators, K-theory, and Lie group representations_. Dokl. Akad. Nauk. USSR, vol. 268, (1983), 533-537.
An introduction is in
* [[Alain Valette]], _Introduction to the Baum-Connes conjecture_ ([pdf](http://www.univ-orleans.fr/mapmo/membres/chatterji/Valette.pdf))
A textbook account is in
* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_
{#Blackadar}
See also
* Markus Land, _The Analytical Assembly Map and Index Theory_, ([arXiv:1306.5657](http://arxiv.org/abs/1306.5657))
{#Land13}
[[!redirects assembly maps]]
[[!redirects Baum-Connes assembly map]]
[[!redirects Baum-Connes assembly maps]]
[[!redirects analytic assembly map]]
[[!redirects analytic assembly maps]]
|
ASSet
|
https://ncatlab.org/nlab/source/ASSet
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
The [[category]] $sSet_+$ of **augmented simplicial sets** is the category of [[presheaf|presheaves]] on the _augmented_ version of the [[simplex category]] $\Delta_a$:
$$
sSet_+ := Set^{\Delta^{op}_a}
\,.
$$
This is the category whose [[object]]s are [[augmented simplicial set]]s and whose [[morphism]]s are the evident morphisms between these.
## Related concepts
* [[sSet]]
* **asSet**
## Applications
The [[join of simplicial sets]] is most naturally defined via a construction on augmented simplicial sets.
[[!redirects asSet]]
category: category
|
Assia Mahboubi
|
https://ncatlab.org/nlab/source/Assia+Mahboubi
|
* [website](https://people.rennes.inria.fr/Assia.Mahboubi/)
## Publications
* [[Sophie Bernard]], [[Cyril Cohen]], [[Assia Mahboubi]], [[Pierre-Yves Strub]], *Unsolvability of the Quintic Formalized in Dependent Type Theory*, ITP 2021 - 12th International Conference on Interactive Theorem Proving, Jun 2021, Rome / Virtual, France ([hal:hal-03136002](https://hal.inria.fr/hal-03136002))
## Talks
* [[Assia Mahboubi]], *Continuity in dependent type theory*, [[Homotopy Type Theory Electronic Seminar Talks]], 31 March 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Mahboubi-2022-03-31-HoTTEST.pdf))
category: people
|
assignment > history
|
https://ncatlab.org/nlab/source/assignment+%3E+history
|
< [[assignment]]
|
assignment operator
|
https://ncatlab.org/nlab/source/assignment+operator
|
# Contents
* table of contents
{: toc}
## Idea
A symbol in [[mathematics]] and [[computer science]] to indicate that a particular variable is being initialized or assigned a value or that a particular symbol is being defined.
One sometimes distinguishes between assignment operators which allow reassignment, with what are known as [[single assignment operators]], which do not allow reassignment.
The assignment operator in purely [[functional programming]] languages like [[Haskell]] amd [[Agda]] is an example of a single assignment operator. As purely functional programming languages can be represented in [[type theory]], and every [[foundations of mathematics]] could also be represented in type theory, the assignment operators used in [[definitions]] in mathematics, such as $\coloneqq$, are single assignment operators; see [[definition]] for more details.
## Related concepts
* [[functional programming]]
* [[equality]]
[[!include mathematical statements --- contents]]
## References
* Wikipedia, _<a href="https://en.wikipedia.org/wiki/Assignment_(computer_science)">Assignment (computer science)</a>_
[[!redirects assignment]]
[[!redirects assignment operators]]
|
associahedron
|
https://ncatlab.org/nlab/source/associahedron
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _associahedra_ or _Stasheff polytopes_ $\{K_n\}$ are [[CW complex]]es that naturally arrange themselves into a topological [[operad]] that resolves the standard associative operad: an [[A-infinity-operad]].
The vertices of $K_n$ correspond to ways in which one can bracket a product of $n$ variables. The edges correspond to rebracketings, the faces relate different sequences of rebracketings that lead to the same result, and so on.
The associahedra were introduced by Jim Stasheff in order to describe [[topological space]]s equipped with a multiplication operation that is associative up to every higher coherent homotopy.
## Definition
Here is the rough idea, copied, for the moment, verbatim from Markl94 [p. 26] (http://arxiv.org/PS_cache/hep-th/pdf/9411/9411208v1.pdf#page=26) (for more details see references below):
For $n \geq 1$ the **associahedron** $K_n$ is an $(n-2)$-dimensional polyhedron whose $i$-dimensional cells are, for $0 \leq i \leq n-2$, indexed by all (meaningful) insertions of $(n-i-2)$ pairs of brackets between $n$ independent indeterminates, with suitably defined incidence maps.
[[simplicial set|Simplicially]] ${ }$ [[subdivision|subdivided]] associahedra (complete with simplicial [[operad|operadic]] structure) can be presented efficiently in terms of an abstract [[bar construction]]. Let $\mathcal{O}: Set/\mathbb{N} \to Set/\mathbb{N}$ be the [[monad]] which takes a [[graded set]] $X$ to the non-permutative [[non-unital operad]] freely generated by $X$, with monad multiplication denoted $m: \mathcal{O}\mathcal{O} \to \mathcal{O}$. Let $t_+$ be the graded set $\{X_n\}_{n \geq 0}$ that is [[empty set|empty]] for $n = 0, 1$ and [[terminal object|terminal]] for $n \geq 2$; this carries a unique non-unital non-permutative operad structure, via a structure map $\alpha: \mathcal{O}t_+ \to t_+$. The bar construction $B(\mathcal{O}, \mathcal{O}, t_+)$ is an (augmented) simplicial graded set (an object in $Set^{\Delta^{op} \times \mathbb{N}}$) whose face maps take the form
$$\ldots \mathcal{O}\mathcal{O}\mathcal{O}t_+
\stackrel{\stackrel{\overset{m\mathcal{O} t_+}{\to}}{\underset{\mathcal{O}m t_+}{\to}}}{\underset{\mathcal{O}\mathcal{O}\alpha}{\to}}
\mathcal{O}\mathcal{O}t_+ \stackrel{\overset{m t_+}{\to}}{\underset{\mathcal{O}\alpha}{\to}} \mathcal{O}t_+ \stackrel{\alpha}{\to} t_+.$$
Intuitively, the (graded set of) $0$-cells $\mathcal{O}t_+$ consists of planar trees where each inner node has two or more incoming edges, with trees graded by number of leaves; the extreme points are binary trees [corresponding to complete binary bracketings of words], whereas other trees are barycenters of higher-dimensional faces of Stasheff polytopes. The construction $B(\mathcal{O}, \mathcal{O}, t_+)$ carries a simplicial (non-permutative non-unital) operad structure, where the [[geometric realization]] of the simplicial set at grade (or [[arity]]) $n$ defines the barycentric subdivision of the Stasheff polytope $K_n$. As the operad structure on $B(\mathcal{O}, \mathcal{O}, t_+)$ is expressed in [[doctrine|finite product logic]] and geometric realization preserves finite products, the (simplicially subdivided) associahedra form in this way the components of a topological operad.
## Loday's realization
[[Jean-Louis Loday]] gave a simple formula for realizing the Stasheff polytopes as a convex hull of integer coordinates in Euclidean space [(Loday 2004)](#Loday04). Let $Y_n$ denote the set of (rooted planar) binary trees with $n+1$ leaves (and hence $n$ internal vertices). For any binary tree $t \in Y_n$, enumerate the leaves by left-to-right order, denoted $\ell_1, \ldots, \ell_{n+1}$, and enumerate the internal vertices as $v_1, \ldots, v_n$ where $v_i$ is the closest common ancestor of $\ell_i$ and $\ell_{i+1}$. Define a vector $M(t) \in \mathbb{R}^n$ whose $i$th coordinate is the product $a_i b_i$ of the number $a_i$ of leaves that are left descendants of $v_i$ and the number $b_i$ of leaves that are right descendants of $v_i$.
+-- {: .un_thm}
###### Theorem (Loday)
The convex hull of the points $\{ M(t) \in \mathbb{R}^n \mid t \in Y_n \}$ is a realization of the Stasheff polytope of dimension $n-1$, and is included in the affine hyperplane $\{(x_1, \ldots, x_n): x_1 + \ldots + x_n = \binom{n}{2}\}$.
=--
## Illustrations
* **$K_1$** is the [[empty set]], a degenerate case not usually considered.
* **$K_2$** is simply the shape of a binary operation:
$$ x \otimes y ,$$
which we interpret here as a single [[point]].
* **$K_3$** is the shape of the usual [[associator]] or associative law
$$ (x \otimes y) \otimes z \to x \otimes (y \otimes z) ,$$
consisting of a single [[interval]].
* {#K4} **$K_4$** The fourth associahedron $K_4$ is the [[pentagon identity|pentagon]] which expresses the different ways a product of four elements may be bracketed
\begin{tikzcd}[column sep = -3.1em, row sep = 10em, every label/.append style = {font = \Large}, font=\Large]
& & (w \otimes x) \otimes (y \otimes z) \arrow[drr, "a_{w, x, y \otimes z}"] & & \\
((w \otimes x) \otimes y) \otimes z \arrow[urr, "a_{w \otimes x, y, z}"] \arrow[dr,"a_{w,x,y} \otimes \mathrm{id}_{z}", swap] & & & & w \otimes (x \otimes (y \otimes z)) \\
& (w \otimes (x \otimes y)) \otimes z \arrow[rr, "a_{w, x \otimes y, z}", swap] & & w \otimes ((x \otimes y) \otimes z) \arrow[ur, "\mathrm{id}_{w} \otimes a_{x,y,z}", swap] &
\end{tikzcd}
One can also think of this as the top-level structure of the 4th [[oriental]]. This controls in particular the _pentagon identity_ in the definition of [[monoidal category]], as discussed there.
* **$K_5$** is the [dual polyhedron](http://en.wikipedia.org/wiki/Dual_polyhedron) to the [triaugmented triangular prism](http://en.wikipedia.org/wiki/Triaugmented_triangular_prism)
+--{: style="text-align:center"}
<img src="https://ncatlab.org/nlab/files/K5associahedron.png" alt="">
<br />
(image from the [Wikimedia Commons](http://commons.wikimedia.org/wiki/File:Polytope_K3.svg))
=--
* One can rotate and explore Stasheff polyhedra in [this interactive associahedron app](https://ltrujello.github.io/associahedron/).
* Illustrations of some polytopes, including $K_5$, can also be found [here](http://irma.math.unistra.fr/~chapoton/galerie.html).
* A template which can be cut out and assembled into a $K_5$ can be found in Appendix B of [ChengLauda2004](#ChengLauda2004).
## Relation to other structures
### Relation to orientals
The above list shows that the first few Stasheff polytopes are nothing but the first few [[oriental]]s. This doesn't remain true as $n$ increases.
The orientals are free **strict** [[omega-category|omega-categories]] on [[simplex]]es as parity complexes. This means that certain interchange cells (e.g., Gray tensorators) show up as thin in the oriental description.
The first place this happens is the sixth oriental: where there are three tensorator squares and six pentagons in Stasheff's $K_5$, the corresponding tensorator squares coming from $O(5)$ are collapsed.
It was when [[Todd Trimble]] made this point to [[Ross Street]] that Street began to think about using associahedra to define weak [[n-category|n-categories]].
### Categorified associahedra
{#Categorification}
There is a [[vertical categorification|categorification]] of associahedra discussed in
* S. Saneblidze, R. Umble, Diagonals on the permutahedra, multiplihedra and associahedra, Homology Homotopy Appl. 6(1) (2004) 363--411.
* {#Forcey12} [[Stefan Forcey]], _Quotients of the multiplihedron as categorified associahedra_, Homology Homotopy Appl. __10__:2 (2008) 227--256 ([Euclid](http://projecteuclid.org/euclid.hha/1251811075))
### Tamari lattice
The associahedron is closely related to a structure known as the _Tamari lattice_, which is especially well-studied in [[combinatorics]]. The Tamari lattice $T_n$ can be defined as the [[poset]] of all parenthesizations of $n+1$ variables with the order generated by rightwards reassociation $(a b)c \le a(b c)$, or equivalently as the poset of all binary trees with $n$ internal nodes (and hence $n+1$ leaves), with the order generated by rightwards tree rotation. (Note the off-by-one offset from the convention for associahedra: the Tamari lattice $T_n$ corresponds to the associahedron $K_{n+1}$.) It was originally introduced by Dov Tamari in his thesis "Monoïdes préordonnés et chaînes de Malcev" (Université de Paris, 1951), around a decade before Jim Stasheff's work.[^Stash]
[^Stash]: [[Jim Stasheff]] comments on this in an essay titled "How I 'met' Dov Tamari" [(Tamari Memorial Festschrift 2012)](#TomariFestschrift), writing that the "so-called Stasheff polytope ... more accurately should be called the Tamari or Tamari-Stasheff polytope".
As the name suggests, the Tamari lattice also carries the structure of a [[lattice]]. This property was originally established by Haya Friedman and Tamari (1967), and later simplified by Samuel Huang and Tamari (1972).
## References
The original articles that define associahedra and in which the operad $K$ that gives $A(\infty)$-topological spaces is implicit are
* [[Jim Stasheff]], _Homotopy associativity of H-spaces I_, Trans. Amer. Math. Soc. 108 (1963), 275--312. ([web](http://www.jstor.org/stable/1993608))
* [[Jim Stasheff]], _Homotopy associativity of H-spaces II_, Trans. Amer. Math. Soc. 108 (1963), 293--312. ([web](http://www.jstor.org/stable/1993609))
A textbook discussion (slightly modified) is in section 1.6 of the book
* [[Martin Markl]], [[Steven Shnider]], [[Jim Stasheff]], _Operads in Algebra, Topology and Physics_ ([web](http://books.google.de/books?id=fMhZjT9lQo0C&pg=PA56&lpg=PA56&dq=Stasheff+associahedra&source=bl&ots=ZuGXjT4zbp&sig=V-taGG2LHS0msHK-PTxmUXXCvEY&hl=de#PPP1,M1))
Loday's original article on the Stasheff polytope is
* {#Loday04} [[Jean-Louis Loday]], Realization of the Stasheff polytope, _Archiv der Mathematik_ 83 (2004), 267-278. ([doi](https://dx.doi.org/10.1007%2Fs00013-004-1026-y))
Further explanations and references are collected at
* [AMS entry on associahedra](http://www.ams.org/featurecolumn/archive/associahedra.html)
* [[Alexander Postnikov]], _Permutohedra, associahedra and beyond_, [math.CO/0507163](http://arxiv.org/abs/math/0507163) [pdf](http://math.mit.edu/~apost/papers/permutohedron.pdf)
The connection to Tamari lattices as well as other developments are in
* {#TomariFestschrift} Folkert Müller-Hoissen, Jean Marcel Pallo, [[Jim Stasheff]] (editors), _Associahedra, Tamari Lattices, and Related Structures: Tamari Memorial Festschrift_, Birkhäuser, 2012. ([google books](https://books.google.fr/books?id=Y01d6g5UemQC&lpg=PP1&pg=PP1#v=onepage&q&f=false))
For a template of $K_5$, see Appendix B of the following.
* {#ChengLauda2004} [[Eugenia Cheng]], [[Aaron Lauda]], _Higher-dimensional categories: an illustrated guidebook_, 2004, [available here](http://eugeniacheng.com/guidebook/).
category: combinatorics
[[!redirects associahedra]]
[[!redirects Stasheff polytope]]
[[!redirects Tamari polytope]]
[[!redirects Tamari-Stasheff polytope]]
|
associahedron > K4
|
https://ncatlab.org/nlab/source/associahedron+%3E+K4
|
<svg xmlns="http://www.w3.org/2000/svg" width="30em" height="20em" viewBox="0 0 480 320">
<desc>Pentagon Identity</desc>
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<foreignObject x="360" y="120" width="120" height="20">$w\otimes (x\otimes(y\otimes z))$</foreignObject>
<foreignObject x="60" y="300" width="120" height="20">$(w\otimes (x\otimes y))\otimes z$</foreignObject>
<foreignObject x="320" y="300" width="120" height="20">$w\otimes ((x\otimes y)\otimes z)$</foreignObject>
<foreignObject x="80" y="50" width="60" height="24">$a_{w\otimes x,y,z}$</foreignObject>
<foreignObject x="350" y="50" width="60" height="24">$a_{w,x,y\otimes z}$</foreignObject>
<foreignObject x="10" y="210" width="70" height="24">$a_{w,x,y}\otimes 1_{z}$</foreignObject>
<foreignObject x="410" y="210" width="70" height="24">$1_w\otimes a_{x,y,z}$</foreignObject>
<foreignObject x="215" y="290" width="60" height="24">$ a_{w,x\otimes y,z}$</foreignObject>
</g>
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|
associated bundle
|
https://ncatlab.org/nlab/source/associated+bundle
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### Traditional
Given a right [[principal bundle|principal]] $G$-bundle $\pi: P\to X$ and a left $G$-[[action]] on some $F$, all in a sufficiently strong category $C$ (such as [[Top]]), one can form the [[quotient object]] $P \times_G F = (P\times F)/{\sim}$, where $P \times F$ is a [[product]] and $\sim$ is the smallest [[congruence]] such that (using [[generalized element]]s) $(p g,f)\sim (p,g f)$; there is a canonical projection $P\times_G F\to X$ where the class of $(p,f)$ is mapped to $\pi(p)\in X$, hence making $P\times_G F\to X$ into a fibre bundle with typical fiber $F$, and the transition functions belonging to the action of $G$ on $F$. We say that $P\times_G F\to X$ is the __associated bundle__ to $P\to X$ with fiber $F$.
### In geometric homotopy theory
{#InGeometricHomotopyTheory}
In the context of [[higher topos theory]] there is an elegant and powerful definition and construction of associated bundles. We discuss here some basics and how this recovers the traditional definition. For more see at _[[associated infinity-bundle]]_ and at _[[geometry of physics -- representations and associated bundles]]_.
At _[[geometry of physics -- principal bundles]]_ in the section _[Smooth principal bundles via smooth groupoids](geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids)_ is discussed how smooth [[principal bundles]] for a [[Lie group]] $G$ over a [[smooth manifold]] $X$ are equivalently the [[homotopy fibers]] of morphisms of [[smooth groupoids]] ([[smooth stacks]]) of the form
$$
X \stackrel{}{\longrightarrow} \mathbf{B}G
\,.
$$
Now given an [[action]] $\rho$ of $G$ on some [[smooth manifold]] $V$, and regarding this action via its [[action groupoid]] projection $p_\rho \colon V//G \to \mathbf{B}G$ as discussed [above](#ActionsOf1Groups), then we may consider these two morphisms into $\mathbf{B}G$ jointly
$$
\array{
&& V//G
\\
&& \downarrow^{\mathrlap{p_\rho}}
\\
X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
}
$$
and so it is natural to construct their [[homotopy fiber product]].
We now discuss that this is equivalently the [[associated bundle]] which is associated to the principal bundle $P \to X$ via the action $\rho$.
+-- {: .num_prop}
###### Proposition
For $G$ a [[smooth group]] (e.g. a [[Lie group]]), $X$ a [[smooth manifold]], $P \to X$ a smooth $G$-[[principal bundle]] over $X$ and $\rho$ a smooth [[action]] of $G$ on some [[smooth manifold]] $V$, then the [[associated bundle|associated]] $V$-[[fiber bundle]] $P \times_G V\to X$ is equivalently (regarded as a [[smooth groupoid]]) the [[homotopy pullback]] of the [[action groupoid]]-projection $p_\rho \colon V//G \to \mathbf{B}G$ along a morphism $g \colon X\to\mathbf{B}G$ which [[modulating morphism|modulates]] $P$
$$
\array{
P\times_G V &\longrightarrow& V//G
\\
\downarrow && \downarrow^{\mathrlap{p_\rho}}
\\
X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
}
\,.
$$
=--
+-- {: .proof}
###### Proof
By the discussion at _[[geometry of physics -- principal bundles]]_ in the section _[Smooth principal bundles via smooth groupoids](geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids)_, the morphism $g$ of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an [[open cover]] $\{U_i \to X\}$ over wich $P$ trivialize and choosing a trivialization, by the [[zig-zag]]
$$
\array{
C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet
\\
\downarrow^{\mathrlap{\simeq_{lwe}}}
\\
X
}
$$
where the top morphism is the [[Cech cohomology|Cech cocycle]] of the given local trivialization regarded as a morphism out of the [[Cech groupoid]] of the given cover.
Moreover, by [this proposition](geometry+of+physics+--+representations+and+associated+bundles#MapFromActionGroupoidOnSetBackToBG) the morphism $(p_\rho)_\bullet$ is a global fibration of pre-smooth groupoids, hence, by the discussion at _[[geometry of physics -- smooth homotopy types]]_, the homotopy pullback in question is equivalently computed as the ordinary pullback of pre-smooth groupoids of $(p_\rho)_\bullet$ along this $g_\bullet$
$$
\array{
C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet &\longrightarrow& (V//G)_\bullet
\\
\downarrow && \downarrow^{\mathrlap{(p_\rho)_\bullet}}
\\
C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}&
(\mathbf{B}G)_\bullet
\\
\downarrow^{\mathrlap{\simeq_{lwe}}}
\\
X
}
\,.
$$
This pullback is computed componentwise. Hence it is the pre-smooth groupoid whose morphisms are pairs consisting of a morphism $(x,i)\to (x,j)$ in the Cech groupoid as well as a morphism $s \stackrel{g}{\to} \rho(s)(g)$ in the action groupoid, such that the group label $g$ of the latter equals the cocycle $g_{i j}(x)$ of the cocycle on the former. Schematically:
$$
C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet
=
\left\{
((x,i),s) \stackrel{g_{i j}(x)}{\longrightarrow} ((x,j),\rho(s)(g))
\right\}
\,.
$$
This means that the pullback groupoid has at most one morphism between every ordered pair of objects. Accordingly this groupoid is [[equivalence of groupoids]] equivalent to the [[quotient]] of its space of objects by the [[equivalence relation]] induced by its morphisms:
$$
\cdots \simeq
\left(
\underset{i}{\coprod} U_i \times V
\right)/_\sim
\,.
$$
This is a traditional description of the associated bundle in question.
=--
## Examples
* [[adjoint bundle]]
* [[tractor bundle]]
## Related concepts
* [[associated infinity-bundle]]
[[!redirects associated bundles]]
## References
* [[Norman Steenrod]], _The topology of fibre bundles_, Princeton Mathematical Series __14__, 1951. viii+224 pp. [MR39258](http://www.ams.org/mathscinet-getitem?mr=39258); reprinted 1994
* [[Dale Husemöller]], _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. __20__, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.)
* [[Glen Bredon]], Section II.2 of: _[[Introduction to compact transformation groups]]_, Academic Press 1972 ([ISBN:9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))
[[!redirects associated bundles]]
[[!redirects associated fiber bundle]]
[[!redirects associated fiber bundles]]
|
associated graded object
|
https://ncatlab.org/nlab/source/associated+graded+object
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For
$$
\cdots
\hookrightarrow
X_{(n)}
\hookrightarrow X_{(n+1)} \hookrightarrow \cdots \hookrightarrow X
$$
a [[filtered object]] in an [[abelian category]] $\mathcal{C}$, the _associated graded object_ $Gr(X)$ is the [[graded object]] which in degree $n$ is the [[cokernel]] of the $n$th inclusion, fitting into a [[short exact sequence]]
$$
0 \to X_{(n-1)} \to X_{(n)} \to Gr_n(X) \to 0
\,,
$$
hence the [[quotient]] of the $n$th layer of $X$ by the next lower one:
$$
Gr_n(X) := X_{(n)}/X_{(n-1)}
\,,
$$
## Examples
* For $\mathcal{A}$ an [[abelian category]] and $C_{\bullet, \bullet}$ a [[double complex]] in $\mathcal{A}$, let $X = Tot(C)$ be the corresponding [[total complex]]. This is naturally filtered by either row-degree or by column-degree. The corresponding associated graded complex is what the terms in the _[[spectral sequence of a filtered complex]]_ compute.
* [[associated graded vector space]]
## Related concepts
[[!include filtered objects -- contents]]
## References
Discussion of the universal property of the associated graded construction on [mathoverflow](http://mathoverflow.net/questions/263/what-is-the-universal-property-of-associated-graded)
[[!redirects associated graded]]
[[!redirects associated graded objects]]
[[!redirects associated graded space]]
[[!redirects associated graded spaces]]
|
associated graded ring
|
https://ncatlab.org/nlab/source/associated+graded+ring
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[filtered ring]] is a [[filtered object]] in the [[category]] [[Ring]] of [[rings]].
The **associated graded ring** to a filtered ring is the corresponding _[[associated graded object]]_.
## Properties
A version of [[PBW theorem]] states that if a [[Lie algebra]] $g$ over a [[field]] $k$ is [[flat module|flat]] as a $k$-[[module]] over a commutative ground ring $k\supset \mathbb{Q}$ containing [[rational numbers|rationals]], then the associated graded ring $Gr U(g)$ is [[isomorphism|isomorphic]] to the [[symmetric algebra]] $Sym(g)$ of the underlying $k$-module of $g$.
## Related concepts
[[!include filtered objects -- contents]]
[[!redirects associated graded rings]]
[[!redirects associated graded algebra]]
[[!redirects associated graded algebras]]
|
associated graded vector space
|
https://ncatlab.org/nlab/source/associated+graded+vector+space
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An _associated graded vector space_ is an [[associated graded object]] in a [[category]] [[Vect]] of [[vector spaces]].
## Examples
* [[weight systems are associated graded of Vassiliev invariants]]
## Related concepts
[[!include filtered objects -- contents]]
[[!redirects associated graded vector spaces]]
|
associated infinity-bundle
|
https://ncatlab.org/nlab/source/associated+infinity-bundle
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
An **associated $\infty$-bundle** $E \to X$ is a [[fiber bundle]] in an [[(∞,1)-topos]] $\mathbf{H}$ with typical fiber $F \in \mathbf{H}$ that is classified by a [[cocycle]] $X \to \mathbf{B}\underline{Aut}(F)$ with coefficients in the [[delooping]] of the [[internalization|internal]] [[automorphism ∞-group]] of $F$. We say this is _associated to_ the corresponding $\underline{Aut}(F)$-[[principal ∞-bundle]].
More generally there should be notions of associated $\infty$-bundles whose fibers are objects in an [[(∞,n)-topos]] over $\mathbf{H}$ for some $n \gt 1$.
## Definition
Let $\mathbf{H}$ be an [[(∞,1)-topos]].
+-- {: .num_def}
###### Definition
For $V,X \in \mathbf{H}$ two objects, say a _$V$-[[fiber ∞-bundle]] over $X$_ is a [[morphism]] $E \to X$ (an object in the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$) such that there exists an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] $U \to X$ and an [[(∞,1)-pullback]] square
$$
\array{
U \times V &\to& E
\\
\downarrow && \downarrow
\\
U &\to& X
}
\,.
$$
=--
+-- {: .num_def}
###### Definition
Let $G \in Grp(\mathbf{H})$ be an [[∞-group]] equipped with an
[[∞-action]] $\rho$ on $V$. Then for $P \to X$ a $G$-[[principal ∞-bundle]] over $X$, the _$\rho$-associated $\infty$-bundle_ is
$$
P \times_G V \to X
\,,
$$
where $P \times_G V := (P \times V)//G$ is the homotopy quotient of the diagonal $G$-action.
=--
+-- {: .num_remark}
###### Remark
Below in _[Properties](#Properties)_ we see that every $\rho$-associated $\infty$-bundle is a $V$-fiber $\infty$-bundle and that every $V$-fiber $\infty$-bundle arises as associated to an $\mathbf{Aut}(V)$-[[principal ∞-bundle]]
=--
## Properties
{#Properties}
### General
+-- {: .num_prop #Classification}
###### Proposition
For $V \in \mathbf{H}$, write $\mathbf{Aut}(V) \in Grp(\mathbf{H})$ for the internal [[automorphism ∞-group]] of $V$. This comes with a canonical action on $V$. Then the operation of sending an $\mathbf{Aut}$-[[principal ∞-bundle]] $P \to X$ to the associated $P \times_G V \to X$ establishes an equivalence
$$
H^1(X, \mathbf{Aut}(V)) \simeq \{V-fiber\;\infty-bundles\}
\,.
$$
=--
More specifically, if $\rho$ is an [[∞-action]] of $G$ on some $V \in \mathbf{H}$, then under the [[equivalence of (∞,1)-categories]]
$$
G Act \simeq \mathbf{H}_{/\mathbf{B}G}
$$
it corresponds to a [[fiber sequence]]
$$
\array{
V &\to& V//G
\\
&& \downarrow^{\mathrlap{\overline{\rho}}}
\\
&& \mathbf{B}G
}
$$
in $\mathbf{H}$. This is the **universal $\rho$-associated $V$-bundle** in that for $P \to X$ any $G$-[[principal ∞-bundle]] modulated by $g \colon X \to \mathbf{B}G$ we have a natural equivalence
$$
P \times_G V \simeq g^* \overline{\rho}
\,.
$$
This is discussed in ([NSS, section I 4.1](#NSS)).
### Presentation in simplicial presheaves
In ([Wendt](#Wendt)), section 5.5, a [[presentable (infinity,1)-category|presentation]] of the general situation for [[n-localic (infinity,1)-topos|1-localic]] [[(∞,1)-toposes]] is given in terms of the [[model structure on simplicial presheaves]] (as discussed at [[models for ∞-stack (∞,1)-toposes]]) .
Under this presentation we have:
+-- {: .num_prop}
###### Proposition
The universal $F$-$\infty$-bundle $\mathbf{E} F \to \mathbf{B}Aut(F)$ is presented by the [[bar construction]]
$$
F \to B(*, Aut(F), F) \to B(*, Aut(F), *)
\,.
$$
=--
Compare [[universal principal ∞-bundle]].
## Examples
### Fibrations of topological spaces / simplicial sets
For the special case that $\mathbf{H} = $ [[∞Grpd]] and using the [[presentable (∞,1)-category|presentation]] by the [[model structure on topological spaces]]/[[model structure on simplicial sets]] the classification theorem \ref{Classification} reduces to the classical statement of ([Stasheff](#Stasheff), [May](#May)).
### $\infty$-Gerbes
In the case that the fiber $F$ is the [[delooping]] $F = \mathbf{B}G$ of an [[∞-group]] object $G$, the $\underline{Aut}(\mathbf{B}G)$-associated $\infty$-bundles are called **$G$-[[∞-gerbes]]**. See there for more details.
## Related concepts
* [[action]], [[∞-action]]
* [[representation]], [[∞-representation]]
* [[principal bundle]] / [[torsor]] / [[associated bundle]]
* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]
* [[principal 3-bundle]] / [[2-gerbe]] / [[bundle 2-gerbe]]
* [[principal ∞-bundle]] / [[∞-gerbe]] / **associated $\infty$-bundle**
* [[vector bundle]]
* [[(∞,1)-vector bundle]]
## References
{#References}
Early work on associated $\infty$-bundles takes place in the $(\infty,1)$-topos [[∞Grpd]] $\simeq$ [[Top]]. In
* [[Jim Stasheff]], _A classification theorem for fiber spaces_ , Topology 2 (1963) 239-246
{#Stasheff}
* [[Jim Stasheff]], _H-spaces and classifying spaces: foundations and recent developments_. Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), pp. 247–272. MR0321079 (47 #9612)
a classification of [[fibrations]] of [[CW-complexes]] with given CW-complex fiber in terms of maps into a classifying CW-complex is given.
In
* [[Daniel Gottlieb]], _The total space of universal fibrations._ Pacific J. Math. Volume 46, Number 2 (1973), 415-417.
the total space of the universal $F$-[[fiber ∞-bundle]] in the pointed context is identified with $\mathbf{B}Aut_*(F)$ (the pointed [[automorphism ∞-group]]).
A generalization or more systematic account of the classification theory is then given in
* [[Peter May]], _Classifying Spaces and Fibrations_ Mem. Amer. Math. Soc. 155 (1975) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/Classifying.pdf))
{#May}
This has been reproven in various guises, such as the statement of [[univalence]] in the [[model]] [[sSet]] for [[homotopy type theory]]. See the references at _[[univalence]]_ for more on this.
Generalizations with extra structure on the fibers are discussed in
* Claudio Pacati, Petar Pavesic, Renzo Piccinini, _On the classification of $\mathcal{F}$-fibrations_, Topology and its applications 87 (1998) ([pdf](http://www.fmf.uni-lj.si/~pavesic/RESEARCH/On%20the%20classification%20of%20F-fibrations.pdf))
Consideration of associated $\infty$-bundles / [[fiber sequences]] in general [[n-localic (infinity,1)-topos|1-localic]] [[(∞,1)-toposes]] [[presentable (infinity,1)-category|presented]] by a [[model structure on simplicial presheaves]] (which subsumes the above case for the trivial site) is discussed in
* [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_ , Journal of Homotopy and Related Structures 6(1), 2011, pp. 1--38. ([arXiv](http://arxiv.org/abs/1009.2930)) ([published version](http://tcms.org.ge/Journals/JHRS/volumes/2011/volume6-1.htm))
{#Wendt}
Related discussion on the behaviour of [[fiber sequences]] under left [[Bousfield localization of model categories]] is in
* [[Matthias Wendt]], _Fibre sequences and localization of simplicial sheaves_ ([pdf](http://home.mathematik.uni-freiburg.de/arithmetische-geometrie/preprints/wendt-flocal.pdf))
Similar considerations and results are in
* Martin Blomgren, [[Wojciech Chacholski]], _On the classification of fibrations_ ([arXiv:1206.4443](http://arxiv.org/abs/1206.4443))
{#BlomgrenChacholski}
With the advent of [[(∞,1)-topos theory]] all these statements and their generalizations follow from the existence of [[object classifiers]] in an [[(∞,1)-topos]]. For the classical case in [[∞Grpd]] $\simeq$ [[Top]]${}^\circ$ [[sSet]]${}^\circ$ this is discussed in
* _[object classifier in ∞Grpd](http://ncatlab.org/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos#ObjectClassifierInInfinityGroupoid)_,
which reproduces the classical results ([Stasheff](#Stasheff), [May](#May)).
For general [[(∞,1)-toposes]] the classification of associated $\infty$-bundles is discussed in section I 4.1 of
* [[schreiber:Principal ∞-bundles -- theory, presentations and applications]]
Models in [[rational homotopy theory]] of classifying spaces for homotopy types $Aut(F)$ go back to [[Sullivan]]'s remarks on the [[automorphism L-infinity algebra]]. Further developments are reviewed and developed in
* [[Andrey Lazarev]], _Models for classifying spaces and derived deformation theory_ ([arXiv:1209.3866](http://arxiv.org/abs/1209.3866))
[[!redirects associated infinity-bundle]]
[[!redirects associated infinity-bundles]]
[[!redirects associated ∞-bundle]]
[[!redirects associated ∞-bundles]]
[[!redirects universal associated infinity-bundle]]
[[!redirects universal associated infinity-bundles]]
[[!redirects universal associated ∞-bundle]]
[[!redirects universal associated ∞-bundles]]
|
associative 3-form
|
https://ncatlab.org/nlab/source/associative+3-form
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
On the [[Cartesian space]] $\mathbb{R}^7$ the _associative 3-form_ is the [[differential n-form|differential 3-form]] $\omega \in \Omega^3(\mathbb{R}^7)$ which is constant and whose value at the origin on three vectors is
$$
\omega(u,v,w) \coloneqq \langle u, v \times w\rangle
\,,
$$
where
* $\langle -,-\rangle$ is the canonical [[bilinear form]] on $\mathbb{R}^7$;
* $(-)\times (-)$ is the [[cross product]] in $\mathbb{R}^7$.
This are the structure constants of the unit [[octonions]].
The Hodge dual of the associative 3-form is sometimes called the _co-associative 4-form_.
## Properties
### Relation to $G_2$
The [[group]] of [[linear map|linear]] [[diffeomorphisms]] of $\mathbb{R}^7$ which preserve this form is the [[exceptional Lie group]] [[G2]].
## Related concepts
[[!include special holonomy table]]
* [[associative submanifold]]
* [[octonions]]
* [[Hitchin functional]]
## References
(...)
[[!redirects associative 3-forms]]
[[!redirects coassociative 3-form]]
[[!redirects coassociative 3-forms]]
[[!redirects co-associative 3-form]]
[[!redirects co-associative 3-forms]]
[[!redirects coassociative 4-form]]
[[!redirects coassociative 4-forms]]
[[!redirects co-associative 4-form]]
[[!redirects co-associative 4-forms]]
|
associative dialgebra
|
https://ncatlab.org/nlab/source/associative+dialgebra
|
__Associative dialgebra__ is an algebraic structure introduced by Loday and Pirashvili to formalize the structure for the universal enveloping of a [[Leibniz algebra]].
* [[Jean-Louis Loday]], T. Pirashvili, _Universal enveloping algebras of Leibniz algebras and (co)homology_, Math. Ann. __296__:1 (1993) 139--158
Associative dialgebras are algebras over a Koszul quadratic operad whose Koszul dual is the operad of [[dendriform dialgebra]]s. In the standard presentation,
associative dialgebras have two algebraic operations.
category: algebra
|
associative H-space
|
https://ncatlab.org/nlab/source/associative+H-space
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
=--
=--
## Contents ##
* table of contents
{:toc}
## Idea
In [[dependent type theory]], an *associative [[H-space]]* (cf. [BCFR23](#BCFR23)) or *$A_3$-space* is a naive translation of the notion of [[monoid]]. If the underlying type is an [[h-set]] (such as if the type theory is [[extensional type theory|extensional]]), then it yields a correct notion of monoid. Otherwise, it is only a "partly-coherent" notion of "homotopy monoid": a type-theoretic version of the notion of $A_3$-[[An-space|space]] from homotopy theory.
## Definition
Similar to the definition of [[H-spaces]], there are coherent and non-coherent versions of associative H-spaces:
### Non-coherent associative H-spaces
A **non-coherent associative H-space** or **non-coherent $A_3$-space** consists of
* A type $A$,
* A basepoint $e:A$
* A binary operation $\mu : A \to A \to A$
* A left unitor
$$\lambda:\prod_{x:A} \mu(e,x)=x$$
* A right unitor
$$\rho:\prod_{x:A} \mu(x,e)=x$$
* An asssociator
$$\alpha:\prod_{x:A} \prod_{y:A} \prod_{z:A} \mu(\mu(x, y),z)=\mu(x,\mu(y,z))$$
### Coherent associative H-spaces
A **coherent associative H-space** or **coherent $A_3$-space** is a non-coherent associative H-space $A$ which additionally has the coherence condition
$$\mu_{\lambda \rho}:\lambda(e) = \rho(e)$$
since $\lambda(e)$ and $\rho(e)$ are elements of the identity type $\mu(e, e) = e$.
## Homomorphisms of associative H-spaces
There are also two different notions of homomorphisms between associative H-spaces, depending on whether the H-space is coherent or not.
### Homomorphisms of non-coherent associative H-spaces
A **homomorphism of non-coherent associative H-spaces** between two non-coherent associative H-spaces $A$ and $B$ is a function $\phi:A \to B$ such that
* The basepoint is preserved, i.e. there is a specified identification
$$\phi_e:\phi(e_A) = e_B$$
* The binary operation is preserved, i.e. there is a specified dependent function
$$\phi_\mu:\prod_{x:A} \prod_{y:A} \phi(\mu_A(x, y)) = \mu_B(\phi(x),\phi(y))$$
* The left unitor is preserved, i.e. there is a specified homotopy identifying the concatenation of identifications
$$\phi_\mu(e_A, x):\phi(\mu_A(e_A,x)) = \mu_B(\phi(e_A),\phi(x))$$
$$\mathrm{ap}_{\lambda y:B.\mu_B(y,\phi(x))}(\phi_e):\mu_B(\phi(e_A),\phi(x)) = \mu_B(e_B,\phi(x))$$
$$\lambda_B(\phi(x)):\mu_B(e_B,\phi(x)) = \phi(x)$$
with the image $\mathrm{ap}_{\phi}(\lambda_A(x)):\phi(\mu_A(e_A,x)) = \phi(x)$ of the left unitor of $A$.
$$\phi_\lambda:\prod_{x:A} \phi_\mu(e_A, x) \bullet \mathrm{ap}_{\lambda y:B.\mu_B(y,\phi(x))}(\phi_e) \bullet \lambda_B(\phi(x)) = \mathrm{ap}_{\phi}(\lambda_A(x))$$
* Similarly, the right unitor is preserved, i.e. there is a specified homotopy identifying the concatenation of identifications
$$\phi_\mu(x, e_A):\phi(\mu_A(x,e_A)) = \mu_B(\phi(x),\phi(e_A))$$
$$\mathrm{ap}_{\lambda y:B.\mu_B(\phi(x),y)}(\phi_e):\mu_B(\phi(x),\phi(e_A)) = \mu_B(\phi(x),e_B)$$
$$\rho_B(\phi(x)):\mu_B(\phi(x),e_B) = \phi(x)$$
with the image $\mathrm{ap}_{\phi}(\rho_A(x)):\phi(\mu_A(x,e_A)) = \phi(x)$ of the right unitor of $A$.
$$\phi_\rho:\prod_{x:A} \phi_\mu(x, e_A) \bullet \mathrm{ap}_{\lambda y:B.\mu_B(\phi(x),y)}(\phi_e) \bullet \rho_B(\phi(x)) = \mathrm{ap}_{\phi}(\rho_A(x))$$
* The associator is preserved in an analogous way.
### Homomorphisms of coherent associative H-spaces
A **homomorphism of coherent associative H-spaces** between two coherent associative H-spaces $A$ and $B$ is a homomorphism of non-coherent associative H-spaces $\phi:A \to B$ in which the coherence condition is preserved.
## Examples
* The [[integers]] are an associative H-space under addition, as are the [[natural numbers type|natural numbers]]. More generally, as noted above, any [[h-set]] monoid is an associative H-space, and is coherent.
* Every [[loop space]] type $\Omega_x(X) \equiv (x=_X x)$ is naturally an associative H-space, with [[path]] concatenation as the operation. In this case, the operation is in fact coherent, and indeed every loop space is a $\infty$-group (although this is difficult to express in type theory).
* The type of endofunctions $A \to A$ has the structure of an associative H-space, with basepoint $id_A$, operation function composition. Once again the operation is coherent, so this is an $\infty$-monoid (inverses need not exist).
* The type of coherent H-space structures on a central type is a coherent associative H-space.
## Related concepts
* [[higher algebra]]
* [[H-space]]
* [[monoid]]
* [[ring]]
* [[H-category]]
* [[An-space]]
* [[grouplike A3-space]]
## References
* {#BCFR23} [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]
[[!redirects associative H-space]]
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|
associative magma
|
https://ncatlab.org/nlab/source/associative+magma
|
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###Context###
#### Algebra
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[[!include higher algebra - contents]]
=--
=--
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#Contents#
* table of contents
{: toc}
## Definition
A [[magma]] $(S,\cdot)$ is called _associative_ if it satisfies the [[associativity]] condition, saying that for all $x,y,z \in S$ then the [[equation]]
$$
(x \cdot y) \cdot z
=
x \cdot (y \cdot z)
$$
holds.
## Examples
Examples include [[semigroups]]/[[monoids]], [[rings]], [[associative algebras]], etc.
## Related concepts
* [[unital magma]]
* [[commutative magma]]
* [[nonassociative ring]], [[nonassociative algebra]]
* [[alternative algebra]]
[[!redirects associative magmas]]
[[!redirects associative operation]]
[[!redirects associative operations]]
|
associative n-category
|
https://ncatlab.org/nlab/source/associative+n-category
|
[[!redirects associative n-categories]]
+-- {: .rightHandSide}
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### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of *associative $n$-categories* (ANCs, [Dorn 2018](#Dorn18), [Reutter & Vicary 2019](#ReutterVicary19)) is a [[semi-strict infinity-category|semi-strict]] [[algebraic definition of higher categories|algebraic model]] for [[higher category theory]] based on the [[geometric shape for higher structures|geometric shape]] of [[globular sets]] with strictly [[associativity|associative]] [[composition]]: All higher [[coherence law|coherence laws]] in associative $n$-categories are strict, except for weak versions of higher [[exchange laws]].
Due to the tight control over [[coherence]]-issues, the theory of (free) associative $n$-categories lends itself to [[formal proof]], for implementation of corresponding [[proof assistants]] see *[[homotopy.io]]* (and its precursor *[[Globular]]*).
Composition operations in associative $n$-categories naturally have stratified-geometric semantics in [[manifold diagram|manifold diagrams]], which makes them an instance of [[geometric n-category|geometrical $n$-categories]]. In particular, the weak coherence laws of an associative $n$-category can be thought of in terms of a notion of homotopy between composites. This is similar to the case of a [[Gray-category]], which is strictly associative and unital, but which has a weak exchange law. In this sense, ANCs can be seen as a generalization of Gray categories.
## Examples
* An associative 0-category is a [[set]]
* An associative 1-category is an [[biased definition|unbiased]] [[1-category]]
* An associative 2-category is an [[biased definition|unbiased]] [[strict 2-category]]
* An associative 3-category is an [[biased definition|unbiased]] [[Gray-category|Gray 3-category]]
A separate notion of 'free' associative n-categories has been developed. These are instances of [[geometric computad|geometric computads]].
## Remarks
* It is conjectured that every [[weak n-category]] is weakly equivalent to an associative $n$-category with strict units. A (proof-wise) potentially more realistic version of this conjecture concerns the case of free associative $n$-categories, and is (as of 2023) a topic of active research.
* A related conjecture is [[Simpson's conjecture]], which states that fully weak higher categories are (in an appropriate sense) equivalent to weakly unital higher categories.
* The underlying idea of this line of modelling higher structures can be traced back to work on [[Gray-category|Gray 3-categories]] and [[surface diagram|surface diagrams]].
## Related concepts
* [[manifold-diagrammatic n-category]]
* [[semistrict n-category]]
* [[homotopy.io]], [[Globular]]
* [[manifold diagram]]
* [[surface diagram]]
* [[globular set]]
* [[Gray-category]]
* [[Simpson's conjecture]]
## References
* [[Christoph Dorn]], _Associative $n$-categories_, talk at _[103rd Peripatetic Seminar on Sheaves and Logic](https://www.math.muni.cz/~loregianf/PSSL103/PSSL103.php)_ ([pdf](https://www.math.muni.cz/~loregianf/PSSL103/slides/Dorn.pdf)).
* {#Dorn18} [[Christoph Dorn]], _Associative $n$-categories_, PhD thesis ([arXiv:1812.10586](https://arxiv.org/abs/1812.10586)).
* {#ReutterVicary19} [[David Reutter]], [[Jamie Vicary]], _High-level methods for homotopy construction in associative $n$-categories_, LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer ScienceJune **62** (2019) 1–13 [[arXiv:1902.03831](https://arxiv.org/abs/1902.03831), [doi:10.1109/LICS52264.2021.9470575](https://doi.org/10.1109/LICS52264.2021.9470575)]
* [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], *Zigzag normalisation for associative $n$-categories*, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [[arXiv:2205.08952](https://arxiv.org/abs/2205.08952), [doi:10.1145/3531130.3533352](https://dl.acm.org/doi/10.1145/3531130.3533352)]
[[!redirects associative n-categories]]
|
associative operad
|
https://ncatlab.org/nlab/source/associative+operad
|
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#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The _associative operad_ $Assoc$ is the [[operad]] whose [[algebras over an operad|algebras]] are [[monoid|monoids]]; i.e. objects equipped with an [[associative]] and [[unital]] binary operation.
One might also consider the non-unital version, whose algebras are objects equipped with a binary operation but not with a [[unit]].
## Definition
### As a $Vect$-operad
The _associative operad_, denoted $Assoc$ or $Ass$, is often taken to be the [[Vect]]-[[operad]] whose [[algebra over an operad|algebras]] are precisely [[associative unital algebras]].
### As a $Set$-operad
As a [[Set]]-enriched [[planar operad]], $Assoc$ is the operad that has precisely one single $n$-ary operation for each $n$. Accordingly, $Assoc$ in this sense is the [[terminal object]] in the [[category]] of [[planar operads]].
(Here the unique 0-ary operation is the [[unit]]. Hence the non-unital version of $Assoc$ has a single operation in each positive arity and none in arity 0.)
As a [[Set]]-enriched [[symmetric operad]] $Assoc$ has (the set underlying) the [[symmetric group]] $\Sigma_n$ in each degree, with the [[action]] being the action of $\Sigma_n$ on itself by multiplication from one side.
Similarly, as a _planar_ [[dendroidal set]], $Assoc$ is the [[presheaf]] that assigns the singleton to every [[planar tree]] (hence also the [[terminal object]] in the category of dendroidal sets).
But, by the above, as an [[symmetric operad|symmetric]] [[dendroidal set]], $Assoc$ is not the terminal object.
## Properties
### Resolution
The relative [[Boardman-Vogt resolution]] $W([0,1],I_* \to Assoc)$ of $Assoc$ in [[Top]] is [[Jim Stasheff]]'s version of the [[A-∞ operad]] whose [[algebra over an operad|algebras]] are [[A-∞ algebras]].
### Relation to planar operads
A [[planar operad]] may be identified with a [[symmetric operad]] that is equiped with a map to the associative operad. See at _[[planar operad]]_ for details.
## Related concepts
* **associative operad**
* [[commutative operad]]
* [[Lie operad]]
## References
In the context of [[higher algebra]] of [[(infinity,1)-operads]], the associative operad is discussed in section 4.1.1 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
{#Lurie}
[[!redirects Ass]]
[[!redirects Assoc]]
|
associative quasigroup
|
https://ncatlab.org/nlab/source/associative+quasigroup
|
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#### Algebra
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#### Group Theory
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#### Representation theory
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=--
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=--
#Contents#
* table of contents
{:toc}
## Definitions
A __(left/right/two-sided) [[associative]] [[quasigroup]]__ is a (left/right/two-sided) [[quasigroup]] $(G, \cdot,\backslash,/)$ where $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for all $a$, $b$, and $c$ in $G$.
In every associative quasigroup $G$, $a/a = b\backslash b$ for every $a$ and $b$ in $G$. This is because for every $a$ and $b$ in $G$, $b \cdot a = b \cdot (a/a) \cdot a = b \cdot (b\backslash b) \cdot a$. Left dividing both sides by $b$ and right dividing both sides by $a$ results in $a/a = b\backslash b$. This means in particular that $a/a = a\backslash a$, which means that every associative quasigroup is a [[possibly empty loop]]. Thus, an associative quasigroup is a associative possibly empty loop, or a __possibly empty group__. In particular, there is an additional definition of an associative quasigroup in terms of the left and right divisions alone, without any semigroup operation at all:
A __left associative quasigroup__ is a [[set]] $G$ with a binary operation $(-)\backslash(-):G \times G \to G$ (a [[magma]]) such that:
* For all $a$ and $b$ in $G$, $a\backslash a=b\backslash b$
* For all $a$ in $G$, $(a\backslash (a\backslash a))\backslash (a\backslash a)=a$
* For all $a$, $b$, and $c$ in $G$, $(a\backslash b)\backslash c= b\backslash ((a\backslash (a\backslash a))\backslash c)$.
For any element $a$ in $G$, the element $a\backslash a$ is called a __right identity element__, and the element $a\backslash (a\backslash a)$ is called the __right inverse element__ of $a$. For all elements $a$ and $b$ in $G$, __left multiplication__ of $a$ and $b$ is defined as $(a\backslash (a\backslash a))\backslash b$.
A __right associative quasigroup__ is a [[set]] $G$ with a binary operation $(-)/(-):G \times G \to G$ such that:
* For all $a$ and $b$ in $G$, $a/a=b/b$
* For all $a$ in $G$, $(a/a)/((a/a)/a)=a$
* for all $a$, $b$, and $c$ in $G$, $a/(b/c)=(a/((c/c)/c)/b$
For any element $a$ in a $G$, the element $a/a$ is called a __left identity element__, and the element $(a/a)/a$ is called the __left inverse element__ of $a$. For all elements $a$ and $b$, __right multiplication__ of $a$ and $b$ is defined as $a/((b/b)/b)$.
An __associative quasigroup__ is a possibly empty left and right group as defined above such that the following are true:
* left and right identity elements are equal (i.e. $a/a = a \backslash a$) for all $a$ in $G$
* left and right inverse elements are equal (i.e. $(a/a)/a = a\backslash (a\backslash a)$) for all $a$ in $G$
* left and right multiplications are equal (i.e. $a/((b/b)/b) = (a\backslash (a\backslash a))\backslash b$) for all $a$ and $b$ in $G$.
This definition [first appeared on the heap article](https://ncatlab.org/nlab/revision/diff/heap/13) and is due to [[Toby Bartels]].
## Pseudo-torsors
Every left associative quasigroup $G$ has a [[pseudo-torsor]] $t_G:G^3 \to G$ defined as $t_G(x,y,z) = x \cdot (y \backslash z)$. Every right associative quasigroup $H$ has a [[pseudo-torsor]] $t_H:H^3 \to H$ defined as $t_H(x,y,z) = (x / y) \cdot z$. This means every associative quasigroup has two pseudo-torsors. If the (left or right) associative quasigroup is inhabited, then those pseudo-torsors are actually [[torsors]] or [[heaps]].
## Category of associative quasigroups
An __associative quasigroup homomorphism__ is a [[semigroup homomorphism]] between associative quasigroups that preserves left and right quotients. Associative quasigroup homomorphisms are the [[morphisms]] in the [[category]] of associative quasigroups $AssocQuasiGrp$.
As the category of associative quasigroups is a concrete category, there is a [[forgetful functor]] $U:AssocQuasiGrp \to Set$. $U$ has a [[left adjoint]], the __free associative quasigroup__ [[free functor|functor]] $F:Set \to AssocQuasiGrp$.
The __empty associative quasigroup__ $0$ whose underlying set is the [[empty set]] is the [[initial object|initial associative quasigroup]], and is [[strict initial object|strictly initial]]. The __trivial associative quasigroup__ $1$ whose underlying set is the [[singleton]] is the [[terminal object|terminal associative quasigroup]].
The __direct product__ $G \times H$ of associative quasigroups $G$ and $H$ is the [[cartesian product]] of sets $U(G) \times U(H)$ with an associative quasigroup structure defined componentwise by
$$
(g_1, h_1) \cdot_{G \times H} (g_2, h_2) = (g_1 \cdot_G g_2, h_1 \cdot_H h_2)
$$
$$
(g_1, h_1) /_{G \times H} (g_2, h_2) = (g_1 /_G g_2, h_1 /_H h_2)
$$
$$
(g_1, h_1) \backslash_{G \times H} (g_2, h_2) = (g_1 \backslash_G g_2, h_1 \backslash_H h_2)
$$
for all $(g_1, h_1), (g_2, h_2) \in U(G) \times U(H)$, with [[product projections]] $p_G: G \times H \to G$ $p_H: G \times H \to H$ where $p_G(g, h) = g$ and $p_H(g,h) = h$ for all $(g,h)\in G \times H$ The direct product of associative quasigroups is thus the [[cartesian product]] in $AssocQuasiGrp$. The [[endofunctor]] $P(G) = G \times 1$ is the [[identity functor]] on $AssocQuasiGrp$ while the endofunctor $P(G) = G \times 0$ is a [[constant functor]] that sends every associative quasigroup to $0$.
An __associative subquasigroup__ of an associative quasigroup $G$ is an associative quasigroup $H$ with an associative quasigroup [[monomorphism]]
$$
H \hookrightarrow G
\,.
$$
The empty associative is the initial associative subquasigroup of $G$.
Let $G$ and $H$ be associative quasigroups and let $f:G \to H$ be an associative quasigroup homomorphism. Then given an element $h \in H$, there is an associative subquasigroup
$$
i: I \hookrightarrow G
\,.
$$
such that $g \in I$ if and only if $f(g) = h$. $I$ is called the __[[fiber]]__ of $f$ over $h$.
Because $AssocQuasiGrp$ has a terminal object, cartesian products, and fibers, it is a [[finitely complete category]].
Every associative quasigroup $G$ and associative subquasigroup $H \hookrightarrow G$ has a __set of left [[ideal in a semigroup|ideals]]__ $G H$ in $G$ and a __set of right ideals__ $H G$ in $G$.
## Examples
* Every [[group]] is an associative quasigroup.
* The empty associative quasigroup is an associative quasigroup that is not a group.
## Related concepts
* [[quasigroup]]
* [[commutative quasigroup]]
* [[loop (algebra)|loop]]
* [[group]]
* [[n-ary group]]
[[!include oidification - table]]
## References
* [The Group With No Elements](https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html#c059798) at the [n-category café](https://golem.ph.utexas.edu/category/)
[[!redirects associative quasigroups]]
[[!redirects possibly empty group]]
[[!redirects possibly empty groups]]
|
associative quasigroupoid
|
https://ncatlab.org/nlab/source/associative+quasigroupoid
|
+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
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#### Algebra
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#### Categorification
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#Contents#
* table of contents
{:toc}
## Idea
Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], an associative quasigroupoid should be the oidification of an [[associative quasigroup]].
## Definition
An __associative quasigroupoid__ is a [[magmoid]] $Q$ such that for every [[diagram]]
$$
a\underset{\quad f \quad}{\to}b\underset{\quad g \quad}{\to}c\underset{\quad h \quad}{\to}d
\,,
$$
$$
h \circ (g \circ f) = (h \circ g) \circ f
\,,
$$
for every [[span]]
$$
\array{
&& s
\\
& {}^{f}\swarrow
&& \searrow^{g}
\\
x
&&&&
y
}
$$
in $Q$, there exists morphisms $i:x\to y$ and $j:y \to x$ such that $i \circ f = g$ and $j \circ g = f$, and for every [[cospan]]
$$
\array{
&& a &&&& b
\\
&
&& {}_{f}\searrow
&
& \swarrow_g
&&
\\
&&&&
c
&&&&
}
$$
in $Q$, there exists morphisms $d:a\to b$ and $e:b \to a$ such that $g \circ d = f$ and $f \circ e = g$.
## Examples
* Every [[groupoid]] is an associative quasigroupoid.
* A one-object associative quasigroupoid is an [[associative quasigroup]].
* An associative quasigroupoid [[enriched magmoid|enriched]] in [[truth values]] is an [[equivalence relation]].
## Related concepts
[[!include oidification - table]]
|
associative ring spectrum
|
https://ncatlab.org/nlab/source/associative+ring+spectrum
|
#Definition#
An **associative ring spectrum**, or [[A-infinity ring]], is an [[algebra in an (infinity,1)-category|monoid object in]] the [[stable (infinity,1)-category of spectra]] with its [[smash product of spectra]] [[monoidal (infinity,1)-category|monoidal structure]].
Equivalently, this should be the same as an ordinary [[monoid]] with respect to the [[symmetric monoidal smash product of spectra]].
#Remarks#
* A commutative associative ring spectrum is a [[commutative ring spectrum]], or [[E-infinity ring]].
#References#
the $(\infty,1)$-categorical description of associative ring spectra is in section 4.3 of
* [[Jacob Lurie]], [[higher algebra|Noncommutative algebra]]
|
associative submanifold
|
https://ncatlab.org/nlab/source/associative+submanifold
|
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###Context###
#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
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* table of contents
{:toc}
## Definition
A [[3-manifold|3-dimensional]] [[submanifold]] $\Sigma_3 \hookrightarrow X_7$ of a [[G2-manifold]] $X_7$ is called _associative_ if it is a [[calibrated submanifold]], hence if the canonical 3-form $\phi$ of the $G_2$-manifold (tangent space-wise the [[associative 3-form]]) restricts to the [[volume form]] on $\Sigma_3$.
Accordingly a [[4-manifold|4-dimensional]] [[submanifold]] $\Sigma_4 \hookrightarrow X_7$ is called _coassociative_ if it is a [[calibrated submanifold]] with respect to the coassociative 4-form $\star_X \phi$.
## Application in string theory
In [[M-theory on G2-manifolds]],
* associative 3-manifolds appear as [[supersymmetric cycles]] relevant for [[membrane instantons]] for [[M2-branes]] [[wrapped brane|wrapping them]];
* coassociative 4-manifolds appear as fibers of compactifications dual to [[F-theory on CY4-folds]] ([Gukov-Yau-Zaslow 02](#GukovYauSaslow02)).
## References
* Ian Weiner, _Associative 3-manifolds in $\mathbb{R}^7$_, 2001 ([pdf](https://www.math.hmc.edu/seniorthesis/archives/2001/iweiner/iweiner-2001-thesis.pdf))
* Selman Akbulut, Sema Salur, _Associative submanifolds of a $G_2$-manifold_ ([pdf](http://users.math.msu.edu/users/akbulut/papers/associativeG2.pdf))
* Damien Gayet, _Smooth moduli spaces of associative submanifolds_ ([arXiv:1011.1744](http://arxiv.org/abs/1011.1744))
On [[associative submanifolds]] of the [[7-sphere]]:
* [[Jason Lotay]], _Associative Submanifolds of the 7-Sphere_, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 ([arXiv:1006.0361](https://arxiv.org/abs/1006.0361))
Discussion of [[wrapped branes]] on associative 3-cycles is in
* [[Katrin Becker]], [[Melanie Becker]], D.R.Morrison, [[Hirosi Ooguri]], Y.Oz, Z.Yin, _Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds_, Nucl.Phys.B480:225-238,1996 ([arXiv:hep-th/9608116](http://arxiv.org/abs/hep-th/9608116))
Further discussion specifically of [[M5-branes]] [[wrapped brane|wrapped]] on associative 3-cycle which are either the [[3-sphere]] or a [[hyperbolic 3-manifold]] is in
* [[Bobby Acharya]], [[Jerome Gauntlett]], Nakwoo Kim, _Fivebranes Wrapped On Associative Three-Cycles_, Phys.Rev. D63 (2001) 106003 ([arXiv:hep-th/0011190](http://arxiv.org/abs/hep-th/0011190))
* {#GukovYauZaslow02} [[Sergei Gukov]], [[Shing-Tung Yau]], [[Eric Zaslow]], _Duality and Fibrations on $G_2$ Manifolds_ ([arXiv:hep-th/0203217](http://arxiv.org/abs/hep-th/0203217))
[[!redirects associative submanifolds]]
[[!redirects associative 3-manifold]]
[[!redirects associative 3-manifolds]]
[[!redirects associative 3-cycle]]
[[!redirects associative 3-cycles]]
|
associative unital algebra
|
https://ncatlab.org/nlab/source/associative+unital+algebra
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Similar to the way [[modules]] generalize [[abelian groups]] by adding the operation of taking non-integer multiples, an $R$-algebra can be thought of as a generalization of a ring $S$, where the operation of taking integer multiples (seen as iterated addition) has been extended to taking arbitrary multiples with coefficients in $R$. In the trivial case, a $\mathbb{Z}$-algebra is simply a ring.
## Definition
### Over ordinary rings
{#OverOrdinaryRings}
For $R$ a [[commutative ring]], an **associative unital $R$-algebra** is equivalently:
* a [[monoid in a monoidal category|monoid]] [[internalization|internal to]] the [[category]] [[Mod|$R Mod$]] of $R$-[[modules]] equipped with the [[tensor product of modules]] $\otimes$;
* a [[pointed object|pointed]] single-[[object]] [[enriched category|category enriched over]] $(R Mod, \otimes)$;
* a pointed $R$-[[algebroid]] with a single object;
* an $R$-[[module]] $A$ equipped with $R$-[[linear maps]] $p \colon A \otimes A \longrightarrow A$ and $i \colon R \to A$ satisfying [[associativity]] and [[unitality]];
* a [[ring]] $A$ [[under category|under]] $R$ such that the corresponding map $R \to A$ lands in the [[center]] of $A$.
If there is no danger for confusion, one often says simply 'associative algebra', or even only '[[algebra]]'.
More generally:
* a (merely) **associative algebra** need not have a [[unit]] $i \colon R \to A$; that is, it is a [[semigroup]] instead of a [[monoid]];
* an [[ring over a ring|$R$-ring]] is a [[monoid object]] [[internalization|in]] the category [[Bimod|$R BiMod$]] of $R$-[[bimodules]] equipped with, crucially, the [[tensor product of bimodules]].
Less generally, a **[[commutative algebra]]** (where [[associativity]] and [[unitality]] are usually assumed) is a [[commutative monoid in a symmetric monoidal category|commutative monoid objecy]] [[internalization|in]] [[Mod|$R Mod$]].
For a given ring the algebras form a category, [[Alg]], and the commutative algebras a subcategory, [[CommAlg]].
### Over semi-rings
Note that everywhere rings can be replaced by [[semi-rings]] in the previous paragraph. For example a commutative associative unital $\mathbb{Q}^{+}$-algebra is nothing more than a commutative semi-ring $R$ with a [[semi-ring homomorphism]] $\mathbb{Q}^{+} \rightarrow R$.
### Over monoids in a monoidal category
{#OverMonoidsInAMonoidalCategory}
+-- {: .num_defn #MonoidsInMonoidalCategory}
###### Definition
Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, then a **[[monoid in a monoidal category|monoid internal to]]** $(\mathcal{C}, \otimes, 1)$ is
1. an [[object]] $A \in \mathcal{C}$;
1. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the _[[unit]]_)
1. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the _product_);
such that
1. ([[associativity]]) the following [[commuting diagram|diagram commutes]]
$$
\array{
(A\otimes A) \otimes A
&\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}&
A \otimes (A \otimes A)
&\overset{id \otimes \mu}{\longrightarrow}&
A \otimes A
\\
{}^{\mathllap{\mu \otimes id}}\downarrow
&& &&
\downarrow^{\mathrlap{\mu}}
\\
A \otimes A
&\longrightarrow&
&\overset{\mu}{\longrightarrow}&
A
}
\,,
$$
where $a$ is the associator isomorphism of $\mathcal{C}$;
1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]:
$$
\array{
1 \otimes A
&\overset{e \otimes id}{\longrightarrow}&
A \otimes A
&\overset{id \otimes e}{\longleftarrow}&
A \otimes 1
\\
& {}_{\mathllap{\ell}}\searrow
& \downarrow^{\mathrlap{\mu}} &
& \swarrow_{\mathrlap{r}}
\\
&& A
}
\,,
$$
where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.
Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1, B)$ with symmetric [[braiding]] $\tau$, then a monoid $(A,\mu, e)$ as above is called a **[[commutative monoid in a symmetric monoidal category|commutative monoid in]]** $(\mathcal{C}, \otimes, 1, B)$ if in addition
* (commutativity) the following [[commuting diagram|diagram commutes]]
$$
\array{
A \otimes A
&& \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} &&
A \otimes A
\\
& {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}}
\\
&& A
}
\,.
$$
A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism
$$
f \;\colon\; A_1 \longrightarrow A_2
$$
in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]]
$$
\array{
A_1 \otimes A_1
&\overset{f \otimes f}{\longrightarrow}&
A_2 \otimes A_2
\\
{}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}}
\\
A_1 &\underset{f}{\longrightarrow}& A_2
}
$$
and
$$
\array{
1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1
\\
& {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}}
\\
&& A_2
}
\,.
$$
Write $Mon(\mathcal{C}, \otimes,1)$ for the [[category of monoids]] in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.
=--
+-- {: .num_defn #ModulesInMonoidalCategory}
###### Definition
Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, and given $(A,\mu,e)$ a [[monoid in a monoidal category|monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), then a **left [[module object]]** in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is
1. an [[object]] $N \in \mathcal{C}$;
1. a [[morphism]] $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the _[[action]]_);
such that
1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]:
$$
\array{
1 \otimes N
&\overset{e \otimes id}{\longrightarrow}&
A \otimes N
\\
& {}_{\mathllap{\ell}}\searrow
& \downarrow^{\mathrlap{\rho}}
\\
&& N
}
\,,
$$
where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.
1. (action property) the following [[commuting diagram|diagram commutes]]
$$
\array{
(A\otimes A) \otimes N
&\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}&
A \otimes (A \otimes N)
&\overset{A \otimes \rho}{\longrightarrow}&
A \otimes N
\\
{}^{\mathllap{\mu \otimes N}}\downarrow
&& &&
\downarrow^{\mathrlap{\rho}}
\\
A \otimes N
&\longrightarrow&
&\overset{\rho}{\longrightarrow}&
N
}
\,,
$$
A [[homomorphism]] of left $A$-module objects
$$
(N_1, \rho_1) \longrightarrow (N_2, \rho_2)
$$
is a morphism
$$
f\;\colon\; N_1 \longrightarrow N_2
$$
in $\mathcal{C}$, such that the following [[commuting diagram|diagram commutes]]:
$$
\array{
A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2
\\
{}^{\mathllap{\rho_1}}\downarrow
&&
\downarrow^{\mathrlap{\rho_2}}
\\
N_1 &\underset{f}{\longrightarrow}& N_2
}
\,.
$$
For the resulting **[[category of modules]]** of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write
$$
A Mod(\mathcal{C})
\,.
$$
This is naturally a (pointed) [[topologically enriched category]] itself.
=--
+-- {: .num_defn #TensorProductOfModulesOverCommutativeMonoidObject}
###### Definition
Given a (pointed) [[topologically enriched category|topological]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$, given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-[[module objects]] (def.\ref{MonoidsInMonoidalCategory}), then the **[[tensor product of modules]]** $N_1 \otimes_A N_2$ is, if it exists, the [[coequalizer]]
$$
N_1 \otimes A \otimes N_2
\underoverset
{\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}}
{\overset{N_1 \otimes \rho_2}{\longrightarrow}}
{\phantom{AAAA}}
N_1 \otimes N_1
\overset{coequ}{\longrightarrow}
N_1 \otimes_A N_2
$$
=--
+-- {: .num_prop #MonoidalCategoryOfModules}
###### Proposition
Given a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}), and given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}). If all [[coequalizers]] exist in $\mathcal{C}$, then the [[tensor product of modules]] $\otimes_A$ from def. \ref{TensorProductOfModulesOverCommutativeMonoidObject} makes the [[category of modules]] $A Mod(\mathcal{C})$ into a [[symmetric monoidal category]], $(A Mod, \otimes_A, A)$ with [[tensor unit]] the object $A$ itself.
=--
+-- {: .num_defn #AAlgebra}
###### Definition
Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ as in prop. \ref{MonoidalCategoryOfModules}, then a [[monoid in a monoidal category|monoid]] $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. \ref{MonoidsInMonoidalCategory}) is called an **$A$-[[associative algebra|algebra]]**.
=--
+-- {: .num_prop }
###### Proposition
Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ in a [[monoidal category]] $(\mathcal{C},\otimes, 1)$ as in prop. \ref{MonoidalCategoryOfModules}, and an $A$-algebra $(E,\mu,e)$ (def. \ref{AAlgebra}), then there is an [[equivalence of categories]]
$$
A Alg_{comm}(\mathcal{C})
\coloneqq
CMon(A Mod)
\simeq
CMon(\mathcal{C})^{A/}
$$
between the [[category of commutative monoids]] in $A Mod$ and the [[coslice category]] of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.
=--
(e.g. [EKMM 97, VII lemma 1.3](#EKMM97))
+-- {: .proof}
###### Proof
In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$
$$
\array{
E \otimes A \otimes E
&
\underoverset
{\underset{}{\longrightarrow}}
{\overset{}{\longrightarrow}}
{\phantom{AAA}}
&
E \otimes E
&\overset{coeq}{\longrightarrow}&
E \otimes_A E
\\
&& & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
\\
&& && E
}
\,.
$$
By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.
Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that
$$
(\star)
\;\;\;\;\;
\;\;\;\;\;
\array{
A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E
\\
{}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}}
\\
A &\underset{e_E}{\longrightarrow}& E
}
$$
[[commuting diagram|commutes]]. Moreover it satisfies the unit property
$$
\array{
A \otimes_A E
&\overset{e_A \otimes id}{\longrightarrow}&
E \otimes_A E
\\
& {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
\\
&& E
}
\,.
$$
By forgetting the tensor product over $A$, the latter gives
$$
\array{
A \otimes E
&\overset{e \otimes id}{\longrightarrow}&
E \otimes E
\\
\downarrow && \downarrow^{\mathrlap{}}
\\
A \otimes_A E
&\overset{e_E \otimes id}{\longrightarrow}&
E \otimes_A E
\\
{}^{\mathllap{\simeq}}\downarrow
&&
\downarrow^{\mathrlap{\mu_{E/A}}}
\\
E &=& E
}
\;\;\;\;\;\;\;\;
\simeq
\;\;\;\;\;\;\;\;
\array{
A \otimes E
&\overset{e_E \otimes id}{\longrightarrow}&
E \otimes E
\\
{}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}}
\\
E &\underset{id}{\longrightarrow}& E
}
\,,
$$
where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be [[pasting|pasted]] to the square $(\star)$ above, to yield a [[commuting square]]
$$
\array{
A \otimes A
&\overset{id\otimes e_E}{\longrightarrow}&
A \otimes E
&\overset{e_E \otimes id}{\longrightarrow}&
E \otimes E
\\
{}^{\mathllap{\mu_A}}\downarrow
&&
{}^{\mathllap{\rho}}\downarrow
&&
\downarrow^{\mathrlap{\mu_{E}}}
\\
A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E
}
\;\;\;\;\;\;\;\;\;\;
=
\;\;\;\;\;\;\;\;\;\;
\array{
A \otimes A
&\overset{e_E \otimes e_E}{\longrightarrow}&
E \otimes E
\\
{}^{\mathllap{\mu_A}}\downarrow
&&
\downarrow^{\mathrlap{\mu_E}}
\\
A &\underset{e_E}{\longrightarrow}& E
}
\,.
$$
This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.
Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-[[module]] structure by
$$
\rho
\;\colon\;
A \otimes E
\overset{e_A \otimes id}{\longrightarrow}
E \otimes E
\overset{\mu_E}{\longrightarrow}
E
\,.
$$
By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the [[universal property]] of the [[coequalizer]] gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.
Finally one checks that these two constructions are inverses to each other, up to isomorphism.
=--
## Variants
* A [[cosimplicial algebra]] is a [[cosimplicial object]] in the category of algebras.
* A [[dg-algebra]] is a [[monoid]] not in [[Vect]] but in the category of (co)[[chain complex]]es.
* A [[smooth algebra]] is an associative $\mathbb{R}$-algebra that has not only the usual binary product induced from the product $\mathbb{R}\times \mathbb{R} \to \mathbb{R}$, but has a $n$-ary product operation for every [[smooth function]] $\mathbb{R}^n \to \mathbb{R}$.
This may be understood as a special case of an [[algebra over a Lawvere theory]], here the [[Lawvere theory]] [[CartSp]].
## Examples
* [[function algebra]]
## Properties
### Tannaka duality
[[!include structure on algebras and their module categories - table]]
## Related concepts
* [[noncommutative algebra]]
* [[nonassociative algebra]]
* [[nonunital algebra]]
* [[finitely generated algebra]], [[finitely presented algebra]]
* [[power-associative algebra]]
* [[augmented algebra]]
* [[unitisation of C*-algebras]]
* [[differential algebra]]
* [[differential graded algebra]], [[A-infinity algebra]]
## References
See most references on *[[algebra]]*.
See also:
* Wikipedia, *[Associative algebra](https://en.wikipedia.org/wiki/Associative_algebra)*
Discussion in the generality of [[brave new algebra]]:
* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))
[[!redirects associative unital algebra]]
[[!redirects associative unital algebras]]
[[!redirects unital associative algebra]]
[[!redirects unital associative algebras]]
[[!redirects associative algebra]]
[[!redirects associative algebras]]
[[!redirects unital algebra]]
[[!redirects unital algebras]]
|
associative Yang-Baxter equation
|
https://ncatlab.org/nlab/source/associative+Yang-Baxter+equation
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The __associative Yang-Baxter equation__ (AYBE) is an [[associativity|associative]] analogue of the [[classical Yang-Baxter equation]].
## Definition
If $A$ is a $k$-algebra then, in Leningrad notation, the AYBE for a matrix $r\in A\otimes A$ is the condition
$$
r_{1 3} r_{1 2} - r_{1 2} r_{2 3} + r_{2 3} r_{1 3} = 0
$$
If $r = \sum a_i\otimes b_i$, then the operator $P:A\to A$ given by
$$
P(x) = \sum a_i x b_i
$$
is a Rota-Baxter operator of weight $0$, i.e. $(A,P)$ is a [[Rota-Baxter algebra]] of weight $\lambda = 0$.
The skew-symmetric solutions ($r_{1 2} = - r_{2 1}$) of AYBE give rise to
* an algebra with the trace quadratic Poisson bracket
* double Poisson structures on a free associative algebra
* an anti-Frobenius associative subalgebra of a matrix algebra
## Related concepts
[[!include Yang-Baxter equations -- contents]]
## References
* V.N. Zhelyabin, _Jordan bialgebras of symmetric elements and Lie bialgebras_, Siberian Mathematical Journal __39__ (1998, 261-276, [open access pdf in Russian](http://www.mathnet.ru/links/da921ffaf25317abe8ca0e15e9c91eda/smj275.pdf))
* [[Marcelo Aguiar]], _Infinitesimal Hopf algebras_, Contemporary Mathematics, 267 (2000) 1-29; _Pre-Poisson algebras_, Lett. Math. Phys. 54 (2000) 263-277, [doi](http://dx.doi.org/10.1023/A:1010818119040); _On the associative analog of Lie bialgebras_, Journal of Algebra __244__ (2001, 492-532, [open access pdf](http://www.sciencedirect.com/science/article/pii/S0021869301988775/pdf?md5=322e83568195266edf53989698f3ae78&pid=1-s2.0-S0021869301988775-main.pdf)
* A. Polishchuk, _Classical Yang-Baxter equation and the $A_\infty$-constraint_, Adv. Math. __168__ (2002, 56-95) [open access pdf](http://ac.els-cdn.com/S000187080192047X/1-s2.0-S000187080192047X-main.pdf?_tid=39dd1e78-93f8-11e7-9f5c-00000aab0f01&acdnat=1504808179_dfd4e9b508ad159abe651d8258654b58)
* Chengming Bai, Li Guo, Xiang Ni, _$\mathcal{O}$-operators on associative algebras and associative Yang-Baxter equations_, Pacific J. Math. __256__ (2012) 257-289, [arxiv/0910.3261](http://arxiv.org/abs/0910.3261)
* Travis Schedler, _Poisson algebras and Yang-Baxter equations_, in: Advances in quantum computation (Tyler, TX, 2007) (Providence, RI) (K. Mahdavi and D. Koslover, eds.), Contemp. Math. __482__ (2009) 91--106
arXiv:[math.QA/0612493](https://arxiv.org/abs/math.QA/0612493)
* V. Sokolov, _Classification of constant solutions for associative Yang-Baxter on $gl(3)$_, [arxiv/1212.6421](http://arxiv.org/abs/1212.6421)
* [[A. Odesskii]], [[V. Rubtsov]], V. Sokolov, _Double Poisson brackets on free associative algebras_, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. __592__, Amer. Math. Soc. (2013) 225--239 [doi](https://arxiv.org/abs/1208.2935) [arxiv/1208.2935](https://arxiv.org/abs/1208.2935)
* [[A. Odesskii]], [[V. Rubtsov]], V. Sokolov, _Parameter-dependent associative Yang–Baxter equations and Poisson brackets_, Int. J. Geom. Meth. Mod. Phys. __11__:09, 1460036 (2014) Proc. XXII IFWGP, Univ. of Évora, Portugal, 2013 [doi](https://doi.org/10.1142/S0219887814600366)
category: algebra, physics
[[!redirects AYBE]]
|
associativity
|
https://ncatlab.org/nlab/source/associativity
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Definition
In a [[category]] **associativity** is the condition that the two ways to use binary [[composition]] of [[morphisms]] to compose a sequence of three morphisms are _equal_
$$
h \circ (g \circ f) = (h \circ g) \circ f
\phantom{AAAA}
$$
<center>
<img src="https://ncatlab.org/nlab/files/AssociativityDiagram.png" width="300">
</center>
$$
\array{
c_2 &\overset{ \phantom{AA}g\phantom{AA} }{\longrightarrow}& c_3
\\
{}^{\mathllap{f}}\Big\uparrow
& \searrow^{\mathrlap{h \circ g}} &
\Big\downarrow{}^{\mathrlap{h}}
\\
c_1 &\underset{ (h \circ g) \circ f }{\longrightarrow}& c_4
}
\;\;\;\;=\;\;\;\;
\array{
c_2 &\overset{ \phantom{AA}g\phantom{AA} }{\longrightarrow}& c_3
\\
{}^{\mathllap{f}}\Big\uparrow
& {}^{\mathllap{ g \circ f }}\nearrow &
\Big\downarrow{}^{\mathrlap{h}}
\\
c_1 &\underset{ h \circ (g \circ f) }{\longrightarrow}& c_4
}
$$
If the category has a single [[object]] it is the [[delooping]] $\mathcal{B} A$ a [[monoid]] $A$, and then this condition is the _associativity_ condition on the binary operation of [[monoids]] such as [[groups]], [[rings]], [[algebras]], etc.
More generally, in [[higher category theory]], _associativity_ of composition of morphisms in an [[n-category]] means that the different ways to use binary composition for composing collections of [[k-morphisms]] form a [[contractible]] [[infinity-groupoid]]. This is a [[coherence law]].
For instance the associativity law in an [[A-infinity algebra]] is the special case of associativity in a 1-object [[A-infinity-category]].
## Examples
* See [[associahedron]].
* In a [[monoidal category]] associativity is the statement that the [[associator]] satisfies its [[coherence law]], which is true by a [[coherence theorem]].
## Related concepts
* [[unitality]]
* [[co-associativity]]
* [[commutative monoidal category]]
## References
The [[coherence law]] of associativity is stated in
* [[Hermann Grassmann]], §3 of _[[Ausdehnungslehre]]_, 1844
[[!redirects associativity]]
[[!redirects associativity law]]
[[!redirects associativity laws]]
[[!redirects associative]]
[[!redirects associative law]]
[[!redirects associative laws]]
|
associator
|
https://ncatlab.org/nlab/source/associator
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
* **associator**, [[unitor]], [[Jacobiator]]
* [[pentagonator]], [[hexagonator]]
***
#Contents#
* table of contents
{:toc}
## Idea
In [[algebra]], given any [[non-associative algebra]] $A$, then the trilinear map
$$
[-,-,-] \;-\; A \otimes A \otimes A \longrightarrow A
$$
given on any elements $a,b,c \in A$ by
$$
[a,b,c] \coloneqq (a b) c - a (b c)
$$
is called the _[[associator]]_ (in analogy with the [[commutator]] $[a,b] \coloneqq a b - b a$ ).
An algebra for which the associator vanishes is hence an _[[associative algebra]]_. If the associator is possibly non-vanishing but completely anti-symmetric (in that for any [[permutation]] $\sigma$ of three elements then $[a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3]$ for $\vert \sigma \vert$ the [[signature of a permutation|signature of the permutation]]) then $A$ is called an _[[alternative algebra]]_.
In [[category theory]] and [[higher category theory]] (for [[monoidal categories]], [[bicategories]] and their higher versions) one considers relaxing the [[equation]] that exhibits the vanishing of the associator
$$
(a b) c = a (b c)
$$
to a [[natural equivalence]]
$$
\alpha_{a,b,c} \;\colon\; (a b) c \overset{\simeq}{\longrightarrow} a (b c)
\,.
$$
By slight mismatch with the terminology in [[algebra]], it is then this equivalence which is called _the associator_.
### In Bicategories
In a [[bicategory]] the [[composition]] of [[1-morphism]]s does not satisfy [[associativity]] as an equation, but there are natural _associator_ 2-morphisms
$$
h \circ (g \circ f) \stackrel{\simeq}{\Rightarrow} (h \circ g) \circ f
$$
that satisfy a [[coherence law]] among themselves.
If one thinks of the bicategory as obtained from a [[geometric definition of higher categories|geometrically defined 2-category]] $C$, then the composition operation of 1-morphisms is a _choice_ of 2-[[horn]]-fillers and the associator is a _choice_ of filler of the spheres $\partial \Delta[3] \to C$ formed by these.
### In monoidal categories
By the [[periodic table of higher categories]] a [[monoidal category]] is a pointed [[bicategory]] with a single object, its [[object]]s are the 1-morphisms of the bicategory. Accordingly, here the associator is a [[natural isomorphism]]
$$ a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z) $$
relating the triple [[tensor products]] of these objects.
## Examples
* In a [[tensor category]] of [[representations]] the associator is called the _[[Wigner 6-j symbol]]_.
## Related concepts
* [[commutator]]
* [[coherence]]
[[!redirects associators]]
|
Astra Kolomatskaia
|
https://ncatlab.org/nlab/source/Astra+Kolomatskaia
|
* [website](https://www.math.stonybrook.edu/cards/kolomatskaiaastra.html)
* [github](https://github.com/FrozenWinters)
* [Youtube](https://www.youtube.com/astradiol)
## Selected writings
* [[Matthew Kennedy]], [[Astra Kolomatskaia]], [[Daniel Spivak]] *An infinite quantum Ramsey theorem*, Journal of Operator Theory, Volume 84, Issue 1, Summer 2020 pp. 49-65. ([doi:10.7900/jot.2018dec11.2259](http://dx.doi.org/10.7900/jot.2018dec11.2259), [arXiv:1711.09526](https://arxiv.org/abs/1711.09526))
* [[Astra Kolomatskaia]], [[Michael Shulman]], *Displayed Type Theory and Semi-Simplicial Types* ([arXiv:2311.18781](https://arxiv.org/abs/2311.18781))
## Talks
On [[semi-simplicial types]]:
* [[Astra Kolomatskaia]], *Semi-Simplicial Types*, [[Homotopy Type Theory Electronic Seminar Talks]], 15 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kolomaskaia-2022-12-15-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=fQv2FpeFxew))
category:people
|
astronomy
|
https://ncatlab.org/nlab/source/astronomy
|
#Contents#
* table of contents
{:toc}
## Idea
(...)
## Related entries
* [[galaxy]], [[Miky way]]
* [[cosmology]], [[observable universe]]
## References
See also
* Wikipedia, *[Astronomy](https://en.wikipedia.org/wiki/Astronomy)*
|
Astérisque
|
https://ncatlab.org/nlab/source/Ast%C3%A9risque
|
>Astérisque journal was created in 1973 for the SMF's first centenary. Astérisque is a top-level international journal. It publishes various monographs, renowned seminars or reports of the main international colloquiums ( available in French and English). The articles are chosen for their ability to show a research branch under a new perspective. Each volume deals with only one subject. The whole annual collection covers all the different fields of mathematics. Seven or eight volumes are published in a year. One of them is completely dedicated to Bourbaki Seminar notes.
>The editorial staff committee insists that published works should meet the highest standards as to both contents and presentation. Every manuscript is closely examined by a reporter: results, either summarised or not proved, will not be accepted. The articles are required to either contain some unusual results or to give a new viewpoint on a specific subject.
[web](http://smf4.emath.fr/en/Publications/Asterisque/Presentation/)
|
asymmetric relation
|
https://ncatlab.org/nlab/source/asymmetric+relation
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--
A (binary) [[relation]] $\sim$ on a set $A$ is __asymmetric__ if no two elements are related in both orders:
$$\forall (x, y: A),\; x \sim y \;\Rightarrow\; y \nsim x$$
In the language of the $2$-poset-with-duals [[Rel]] of sets and relations, a relation $R: A \to A$ is __asymmetric__ if it is disjoint from its dual:
$$R \cap R^{op} \subseteq \empty$$
Of course, this containment is in fact an equality.
An asymmetric relation is necessarily [[irreflexive relation|irreflexive]].
[[!redirects asymmetric]]
[[!redirects asymmetric relation]]
[[!redirects asymmetric relations]]
|
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