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baryon current
|
https://ncatlab.org/nlab/source/baryon+current
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[conserved current]] whose [[charge]] is the [[number]] of [[baryons]].
(Not actually conserved, generally, in the [[standard model of particle physics]], due to the [[chiral anomaly]]. See also st _[[baryogenesis]]_.)
## Related concepts
* [[Dirac current]]
## References
### General
(...)
[[!include WZW term of QCD chiral perturbation theory -- references]]
[[!redirects baryon currents]]
|
baryon number
|
https://ncatlab.org/nlab/source/baryon+number
|
## References
See also:
* Wikipedia, _[Baryon number](https://en.m.wikipedia.org/wiki/Baryon_number)
|
baryon-lepton symmetry
|
https://ncatlab.org/nlab/source/baryon-lepton+symmetry
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[nuclear physics]], the term _baryon-lepton symmetry_ ([Gamba-Marshak-Okubo 59](#GambaMarshakOkubo59)) refers to the observation that [[interactions]] involving the [[weak nuclear force]] are close to invariant under joint exchange of the [[light baryons|light]] [[baryons]] ([[protons]], [[neutrons]]) with the "[[light leptons|light]] [[leptons]]" ([[electron]] and its [[neutrino]]), i.e. within the [[first generation of fermions]].
## Related concepts
* [[hadron supersymmetry]]
## References
The original article:
* {#GambaMarshakOkubo59} A. Gamba, [[Robert Marshak]], S. Okubo, _On a symmetry in weak interactions_, PNAS June 1, 1959 45 (6) 881-885 ([doi:10.1073/pnas.45.6.881](https://doi.org/10.1073/pnas.45.6.881))
Review:
* [[Robert Marshak]], _Yukawa Meson, Sakata Model and Baryon-Lepton Symmetry Revisited_, Progress of Theoretical Physics Supplement, Volume 85, May 1985, Pages 61–74 ([dpi:10.1143/PTP.85.61](https://doi.org/10.1143/PTP.85.61))
Further developments:
* Rabindra N. Mohapatra, _From Old Symmetries to New Symmetries: Quarks, Leptons and B-L_, International Journal of Modern Physics AVol. 29, No. 29, 1430066 (2014) ([arXiv:1409.7557](https://arxiv.org/abs/1409.7557))
|
Bas Spitters
|
https://ncatlab.org/nlab/source/Bas+Spitters
|
Bas Spitters is professor for [[formal logic]] and [[computer science]] at Aarhus University and leader of the *Concordium Blockchain Research Center*
* [website](http://www.cs.au.dk/~spitters/)
* [Concordium page](https://cs.au.dk/research/centers/concordium/about)
## Selected writings
On [[constructive mathematics|constructive]] [[integration]] theory:
* [[Bas Spitters]], *Constructive algebraic integration theory*, Annals of Pure and Applied Logic, Volume 137, Issues 1–3, January 2006, Pages 380-390 ([pdf](https://www.cs.au.dk/~spitters/ftop2.pdf), [doi:10.1016/j.apal.2005.05.031](https://doi.org/10.1016/j.apal.2005.05.031))
* [[Bas Spitters]], *Constructive algebraic integration theory without choice*. In [[Thierry Coquand]], [[Henri Lombardi]], Marie-Francoise Roy, Mathematics, Algorithms, Proofs, 2005, Dagstuhl Seminar Proceedings 05021, Internationales Begegnungs- und Forschungszentrum (IBFI), Schloss Dagstuhl, Germany. ([pdf](https://www.cs.au.dk/~spitters/obs.pdf))
On [[constructive analysis]] with exact [[real numbers]] via [[type theory]]:
* [[Herman Geuvers]], [[Milad Niqui]], [[Bas Spitters]], [[Freek Wiedijk]], _Constructive analysis, types and exact real numbers_, Mathematical Structures in Computer Science **17** 01 (2007) 3-36 [[doi:10.1017/S0960129506005834](https://doi.org/10.1017/S0960129506005834)]
and specifically with [[ufias2012:Type classes]] in [[Coq]]:
* {#KrebbersSpitters11} [[Robbert Krebbers]], [[Bas Spitters]], *Type classes for efficient exact real arithmetic in Coq*, Logical Methods in Computer Science, **9** 1 (2013) lmcs:958 [[arXiv:1106.3448](http://arxiv.org/abs/1106.3448/), <a href="https://doi.org/10.2168/LMCS-9(1:1)2013">doi:10.2168/LMCS-9(1:1)2013</a>]
Exposition:
* [[Bas Spitters]], *Verified Implementation of Exact Real Arithmetic in Type Theory*, talk at *Computable Analysis and Rigorous Numeric* (2013) [[pdf](https://users-cs.au.dk/spitters/CARN.pdf), [[Spitters-ExactRealArithmetic.pdf:file]]]
On [[Bohr toposes]]:
* {#HeunenLandsmanSpitters09} [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], *Bohrification of operator algebras and quantum logic*, Synthese **186** 3 (2012) 719-752 [[arXiv:0905.2275](http://arxiv.org/abs/0905.2275), [doi;10.1007/s11229-011-9918-4](https://doi.org/10.1007/s11229-011-9918-4)]
On [[mathematical structures]] formulated in [[dependent type theory]], specifically via [[ufias2012:Type classes]] in [[Coq]]:
* [[Bas Spitters]], Eelis van der Wegen *Type classes for mathematics in type theory*, Mathematical Structures in Computer Science **21** 4 "Interactive Theorem Proving and the Formalisation of Mathematics" (2011) 795-825 [[doi:10.1017/S0960129511000119](https://doi.org/10.1017/S0960129511000119)]
On [[modalities]] in [[homotopy type theory]] ([[modal homotopy type theory]]):
* {#RSS} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], *Modalities in homotopy type theory*, Logical Methods in Computer Science, **16** 1 (2020) [[arXiv:1706.07526](https://arxiv.org/abs/1706.07526), [episciences:6015](https://lmcs.episciences.org/6015), <a href="https://doi.org/10.23638/LMCS-16(1:2)2020">doi:10.23638/LMCS-16(1:2)2020</a>]
On [[software verification]] for [[blockchain]]-technology:
* [[Bas Spitters]], *[Formal verificaton](https://cs.au.dk/research/centers/concordium/research-areas/formal-verification)* at *[Concordium Blockchain Research Center](https://cs.au.dk/research/centers/concordium/)*
* {#ThomsenSpitters20} Søren Eller Thomsen, [[Bas Spitters]], *Formalizing Nakamoto-Style Proof of Stake* [[eprint:2020/917](https://eprint.iacr.org/2020/917)]
category: people
[[!redirects Bas Spitters]]
[[!redirects BasSpitters]]
|
Bas van Fraassen
|
https://ncatlab.org/nlab/source/Bas+van+Fraassen
|
**Bastiaan Cornelis van Fraassen** is a distinguished [[philosophy of science|philosopher of science]].
* [homepage](http://www.basvanfraassen.org/)
* [wikipedia entry](http://en.wikipedia.org/wiki/Bas_van_Fraassen)
## References
* {#vanFraasen80} [[Bas van Fraassen]], _The Scientific Image_ (1980) ([philpapers:VANTSI](https://philpapers.org/rec/VANTSI),\linebreak [doi:10.1093/0198244274.001.0001/acprof-9780198244271](https://www.oxfordscholarship.com/view/10.1093/0198244274.001.0001/acprof-9780198244271))
(on [[constructive empiricism]])
* B. van Fraassen, _Identity in Intensional Logic: Subjective Semantics_ , pp.201-220 in Eco, Santambrogio, Violi (eds.), _Meaning and Mental Representations_ , Indiana UP Bloomington 1988. ([pdf](http://www.princeton.edu/~fraassen/abstract/SubjSemantics.pdf))
* B. C. van Fraassen, _Laws and Symmetry_ , Oxford UP 1989.
category:people
|
base
|
https://ncatlab.org/nlab/source/base
|
# Bases and subbases
* table of contents
{: toc}
## Idea
For many notions of [[structure]], particularly for [[topological concrete category|topological categories]], one can specify a structure by a _base_ or _subbase_ that generate the structure. Besides being convenient ways to specify a structure, they may even be necessary when using weak [[foundations]].
+-- {: .num_remark #basevsbasis}
###### Warning
Sometimes one says '[[basis]]' instead of 'base', but I ([[Toby Bartels]]) think that it\'s safest to save the former term for the generating set of a [[free object]] (or an analogous situation), especially in an [[algebraic category]]. Although a basis and a base can both generate something, they tend to do so in very different ways. (It doesn\'t help that 'bases' is the plural of both, although the pronunciation is different.)
=--
Typically, every structure of an appropriate type is both a base and a subbase for itself, while every base is a subbase. Bases and subbases can also be characterised independently; every subbase generates a base (which tends to be _saturated_ in some way), while every base (saturated or not) generates a complete structure.
## Examples
### Bases for filters
Recall that a __[[filter]]__ on a [[poset]] $L$ is a [[subset]] $F$ of $L$ such that:
1. If $x \leq y$ and $x \in F$, then $y \in F$.
2. Some $x \in F$.
3. If $x, y \in F$, then for some $z \in F$, $z \leq x, y$.
More generally, any subset $F$ satisfying (2,3) is a __filter base__.
Given a filter base $F$ in a poset, we generate a filter $\overline{F}$ by closing under (1); that is, if $F$ is a filter base on a poset $L$, then
$$ \overline{F} \coloneqq \{ y \;|\; \exists x \in F,\; x \leq y \} $$
is a filter on $L$.
If $L$ is a [[meet]]-[[semilattice]], then we can equivalently define a filter as a subset $F$ such that:
1. If $x \leq y$ and $x \in F$, then $y \in F$ (same as before).
2. The [[top element]] $\top \in F$.
3. If $x, y \in F$, then $x \wedge y \in F$.
Now any subset satisfying (2,3) is a __saturated filter base__, and any subset whatsoever is a __filter subbase__.
Given a filter subbase $F$ in a semilattice, we generate a base $\vec{F}$ by closing under (2,3) in the second list; that is, if $F$ is a filter subbase on a semilattice $L$, then
$$ \vec{F} \coloneqq \{ \bigwedge_{i=1}^n x_i \;|\; x_i \in F \} $$
is a filter base on $L$, which in fact is saturated. (Note that $\top \in \vec{F}$ follows when $n = 0$.)
Given a filter subbase $F$ in a semilattice, we can generate a filter by first generating a base $\vec{F}$ and then generating a filter $\overline{\vec{F}}$. Alternatively, we can generate the same filter by closing under (1,2,3) in the first list all at once. That is, if $F$ is a filter subbase on a semilattice $L$, then
$$ \overline{F} \coloneqq \{ y \;|\; \exists x_1,\ldots,x_n \in F,\; \forall z \leq x_1,\ldots,x_n,\; z \leq y \} $$
is a filter on $L$. Furthermore, this is the same filter as $\overline{\vec{F}}$.
The [[intersection]] of any family of filters on a semilattice $L$ is a filter; that is, being a filter is a [[Moore closure]] property on subsets of $L$. The filter generated by a filter subbase $F$ (which is an arbitrary subset of $L$, remember) is the same as the Moore closure of $F$ under this property, that is the intersection of all filters on $L$ that contain $F$.
Unlike filters and filter bases, the concept of filter subbase does not seem to make sense on an arbitrary poset, but only on a semilattice.
### Bases for topologies
Recall that a [[topological space|topology]] on a [[set]] $X$ is a collection $\mathcal{O}$ of [[subsets]] of $X$ such that:
1. Any [[union]] of elements of $\mathcal{O}$ belongs to $\mathcal{O}$.
2. $X$ itself belongs to $\mathcal{O}$.
3. If $U,V \in \mathcal{O}$, then $U \cap V \in \mathcal{O}$.
More generally, any collection $\mathcal{O}$ satisfying (2,3) is a __saturated topological base__, and any collection whatsoever is a __topological subbase__.
A slightly more complicated but equivalent definition of topology is this:
1. Again, any union of elements of $\mathcal{O}$ belongs to $\mathcal{O}$.
2. $X$ itself is a union of elements of $\mathcal{O}$.
3. If $U,V \in \mathcal{O}$, then $U \cap V$ is contained in a union of elements of $\mathcal{O}$.
Now any collection satisfying (2,3) is a __[[basis for a topology|topological base]]__ (not necessarily saturated).
Given a topological subbase $\mathcal{O}$, we generate a base $\vec{\mathcal{O}}$ by closing under (2,3) in the first list; that is, if $\mathcal{O}$ is a topological subbase on a set $X$, then
$$ \vec{\mathcal{O}} \coloneqq \{ \bigcap_{i=1}^n U_i \;|\; U_i \in \mathcal{O} \} $$
is a topological base on $X$, which in fact is saturated. (Note that $X \in \vec{\mathcal{O}}$ follows when $n = 0$.)
Given a topological base $\mathcal{O}$, we generate a topology $\overline{\mathcal{O}}$ by closing under (1); that is, if $\mathcal{O}$ is a topological base on a set $X$, then
$$ \overline{\mathcal{O}} \coloneqq \{ V \;|\; \forall p \in V,\; \exists U \in \mathcal{O},\; p \in U \subseteq V \} $$
is a topology on $X$.
Given a topological subbase $\mathcal{O}$, we can generate a topology by first generating a base $\vec{\mathcal{O}}$ and then generating a topology $\overline{\vec{\mathcal{O}}}$. Alternatively, we can generate the same topology by closing under (1,2,3) in the second list all at once. That is, if $\mathcal{O}$ is a topological subbase on a set $X$, then
$$ \overline{\mathcal{O}} \coloneqq \{ V \;|\; \forall p \in V,\; \exists U_1, \ldots, U_n \in \mathcal{O},\; p \in \bigcap_{i=1}^n U_i \subseteq V \} $$
is a topology on $\mathcal{O}$. Furthermore, this is the same topology as $\overline{\vec{\mathcal{O}}}$.
As with filters, being a topology is a [[Moore closure]] property, this time on subsets of the [[power set]] $\mathcal{P}X$, and the topology generated by a topological subbase $\mathcal{O}$ is the intersection of all topologies on $X$ that contain $\mathcal{O}$.
Very analogous considerations apply to [[local bases]] for a topology and bases for [[pretopological space|pretopologies]], [[convergence space|convergence structures]], [[gauge space|gauge structures]], [[Cauchy space|Cauchy structures]], etc.
### Bases for uniformities
Uniformities are a little trickier than topologies, at least in the case of subbases. For now, see the page [[uniform space]] for definitions of base and subbase for a uniformity.
### Bases for $\sigma$-algebras
Recall that a $\sigma$-[[sigma-algebra|algebra]] on a [[set]] $X$ is ...
...
### Bases for Grothendieck coverages
See [[basis for a Grothendieck topology]].
## General theory
Is there a general theory of bases? That\'s a good question. I don\'t know!
Obviously this has something to do with [[Moore closures]] (and hence [[monads]]); generating a structure from a subbase is (often) taking a Moore closure. But there\'s some particular property of some closure operators that makes the intermediate concept of base work out.
## References
This paper discusses bases in the generality of algebras over a monad:
* [[Stefan Zetzsche]], [[Alexandra Silva]], [[Matteo Sammartino]]: _Generators and bases for algebras over a monad_ (2020), ([arXiv:010.10223](https://arxiv.org/abs/2010.10223))
[[!redirects base]]
[[!redirects subbase]]
[[!redirects saturated base]]
|
base (infinity,1)-topos
|
https://ncatlab.org/nlab/source/base+%28infinity%2C1%29-topos
|
[[!redirects base (∞,1)-topos]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In ordinary [[topos theory]] it is common to "work over a fixed [[base topos]]" which may or may not be the canonical choice [[Set]].
Similarily, in [[(∞,1)-topos theory]] one may choose to work over a fixed _base $(\infty,1)$-topos_ $\mathbf{B}$ other than [[∞Grpd]].
Basically this amounts to working not with the [[(∞,1)-category]] [[(∞,1)Topos]], but instead in the [[over-(∞,1)-topos]] $(\infty,1)Topos/\mathbf{B}$.
## Related concepts
* [[base topos]]
[[!redirects base (infinity,1)-topos]]
[[!redirects base (infinity,1)-toposes]]
[[!redirects base (∞,1)-toposes]]
|
base change
|
https://ncatlab.org/nlab/source/base+change
|
> This entry is about base change of [[slice categories]]. For base change in [[enriched category theory]] see at [[change of enriching category]].
***
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Topos theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For $f : X \to Y$ a [[morphism]] in a [[category]] $C$ with [[pullbacks]], there is an induced [[functor]]
$$
f^* : C/Y \to C/X
$$
of [[over-categories]]. This is the _base change_ morphism. If $C$ is a [[topos]], then this refines to an [[essential geometric morphism]]
$$
(f_! \dashv f^* \dashv f_*) : C/X \to C/Y
\,.
$$
More generally, such a triple adjunction holds whenever $C$ is [[locally cartesian closed category|locally cartesian closed]], and indeed this [[locally cartesian closed category#DependentProductImpliesLocalCartesinClosure|characterises]] locally cartesian closed categories. The [[duality|dual]] concept is [[cobase change]].
## Definition
### Pullback
For $f : X \to Y$ a [[morphism]] in a [[category]] $C$ with [[pullback]]s, there is an induced [[functor]]
$$
f^* : C/Y \to C/X
$$
of [[over-categories]]. It is on objects given by [[pullback]]/[[fiber product]] along $f$
$$
(p : K \to Y)
\mapsto
\left(
\array{
X \times_Y K &\to & K
\\
{}^{\mathllap{p^*}}\downarrow && \downarrow
\\
X &\stackrel{f}{\to}& Y
}
\right)
\,.
$$
On morphisms, it follows from the universal property of pullback
$$
\left\lbrace
\array{
K &&\stackrel{g}{\to}&& K'
\\
& {}_p \searrow && \swarrow_{p'}
\\
&& Y
}
\right\rbrace
\mapsto
\left\lbrace
\array{
X \times_Y K &&\stackrel{g^*}{\to}&& X \times_Y K'
\\
& {}_{p^*} \searrow && \swarrow_{p'^*}
\\
&& X
}
\right\rbrace
$$
by observing that this square commutes
$$
\array{
&&&& X \times_Y K
\\&
&&
{}^{p^*}\swarrow
&& \searrow^{g \circ p_K}
\\
&& X &&&& K'
\\
&
&& {}_f\searrow
&
& \swarrow_{p'}
&&
\\
&&&&
Y
&&&&
}
\,.
$$
### In a fibered category
The concept of base change generalises from this case to other [[fibered category|fibred categories]].
### Base change geometric morphisms
{#GeometricMorphism}
+-- {: .num_prop #BaseChangeIsEssentialGeometricMorphism}
###### Proposition
For $\mathbf{H}$ a [[topos]] (or [[(∞,1)-topos]], etc.) $f : X \to Y$ a [[morphism]] in $\mathbf{H}$, then base change induces an [[essential geometric morphism]] between over-toposes/[[over-(∞,1)-topos]]es
$$
(\sum_f \dashv f^* \dashv \prod_f) : \mathbf{H}/X
\stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
\mathbf{H}/Y
$$
where $f_!$ is given by postcomposition with $f$ and $f^*$ by [[pullback]] along $f$.
=--
+-- {: .proof}
###### Proof
That we have [[adjoint functor]]s/[[adjoint (∞,1)-functor]]s $(f_! \dashv f^*)$ follows directly from the universal property of the pullback. The fact that $f^*$ has a further [[right adjoint]] is due to the fact that it preserves all small [[colimit]]s/[[(∞,1)-colimit]]s by the fact that in a topos we have [[universal colimits]] and then by the [[adjoint functor theorem]]/[[adjoint (∞,1)-functor theorem]].
=--
+-- {: .num_remark}
###### Remark
The ([[comonad|co-]])[[monads]] induced by the [[adjoint triple]] in prop. \ref{BaseChangeIsEssentialGeometricMorphism} have special names in some contexts:
* $f_\ast f^\ast$ is also called the [[function monad]] (or "[[reader monad]]", see at _[[monad (in computer science)]]_).
* $f_! f^\ast$ is also called the "[[writer comonad]]" (in computer science)
* in [[modal type theory]] $f^\ast f_\ast$ is _[[necessity]]_ while $f^\ast f_!$ is _[[possibility]]_.
=--
+-- {: .num_prop}
###### Proposition
Here $f^\ast$ is a [[cartesian closed functor]], hence base change of toposes constitutes a cartesian [[Wirthmüller context]].
=--
See at _[[cartesian closed functor]]_ for the proof.
+-- {: .num_prop}
###### Proposition
$f^*$ is a [[logical functor]]. Hence $(f^* \dashv f_*)$ is also an [[atomic geometric morphism]].
=--
This appears for instance as ([MacLaneMoerdijk, theorem IV.7.2](#MacLaneMoerdijk)).
+-- {: .proof}
###### Proof
By prop. \ref{BaseChangeIsEssentialGeometricMorphism} $f^*$ is a [[right adjoint]] and hence preserves all [[limit]]s, in particular [[finite limit]]s.
Notice that the [[subobject classifier]] of an [[over topos]] $\mathbf{H}/X$ is $(p_2 : \Omega_{\mathbf{H}} \times X \to X)$. This [[product]] is preserved by the [[pullback]] by which $f^*$ acts, hence $f^*$ preserves the subobject classifier.
To show that $f^*$ is logical it therefore remains to show that it also preserves [[exponential object]]s. (...)
=--
+-- {: .num_defn}
###### Definition
A (necessarily essential and atomic) geometric morphism of the form $(f^* \dashv \prod_f)$ is called the **base change geometric morphism** along $f$.
The [[right adjoint]] $f_* = \prod_f$ is also called the [[dependent product]] relative to $f$.
The [[left adjoint]] $f_! = \sum_f$ is also called the [[dependent sum]] relative to $f$.
In the case $Y = *$ is the [[terminal object]], the base change geometric morphism is also called an **[[etale geometric morphism]]**. See there for more details
=--
## Properties
+-- {: .num_prop}
###### Proposition
If $\mathcal{C}$ is a [[locally cartesian closed category]] then for every morphism $f \colon X \to Y$ in $\mathcal{C}$ the [[inverse image]] $f^* \colon \mathcal{C}_{/Y} \to \mathcal{C}_{/X}$ of the base change is a [[cartesian closed functor]].
=--
See at _[cartesian closed functor -- Examples](cartesian%20closed%20functor#Examples)_ for a proof.
## Examples
### Along $\mathbf{B}H \to \mathbf{B}G$
{#AlongDeloopingsOfGroupHomomorphisms}
For $\mathbf{H}$ an [[(∞,1)-topos]] and $G$ an group object in $\mathbf{H}$ (an [[∞-group]]), then the [[slice (∞,1)-topos]] over its [[delooping]] may be identified with the [[(∞,1)-category]] of $G$-[[∞-actions]] (see there for more):
$$
Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
\,.
$$
Under this identification, then left and right base change long a morphism of the form $\mathbf{B}H \to \mathbf{B}G$ (corresponding to an [[∞-group]] homomorphism $H \to G$) corresponds to forming [[induced representations]] and [[coinduced representations]], respectively.
### Along $\ast \to \mathbf{B}G$
{#AlongPointInclusionIntoBG}
As the special case of the [above](#AlongDeloopingsOfGroupHomomorphisms) for $H = 1$ the trivial group we obtain the following:
+-- {: .num_prop #CyclicLoopSpace}
###### Proposition
Let $\mathbf{H}$ be any [[(∞,1)-topos]] and let $G$ be a group object in $\mathbf{H}$ (an [[∞-group]]). Then the base change along the canonical point inclusion
$$
i \;\colon\; \ast \to \mathbf{B}G
$$
into the [[delooping]] of $G$ takes the following form:
There is a pair of [[adjoint ∞-functors]] of the form
$$
\mathbf{H}
\underoverset
{ \underset{i_\ast \simeq [G,-]/G}{\longrightarrow}}
{ \overset{i^\ast \simeq hofib}{\longleftarrow}}
{\bot}
\mathbf{H}_{/\mathbf{B}G}
\,,
$$
where
* $hofib$ denotes the operation of taking the [[homotopy fiber]] of a map to $\mathbf{B}G$ over the canonical basepoint;
* $[G,-]$ denotes the [[internal hom]] in $\mathbf{H}$;
* $[G,-]/G$ denotes the [[homotopy quotient]] by the [[conjugation action|conjugation]] [[∞-action]] for $G$ equipped with its canonical [[∞-action]] by left multiplication and the argument
regarded as equipped with its trivial $G$-$\infty$-action
(for $G = S^1$ then this is the [[cyclic loop space]] construction).
Hence for
* $\hat X \to X$ a $G$-[[principal ∞-bundle]]
* $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]]
then there is a [[natural equivalence]]
$$
\underset{
\text{original} \atop \text{fluxes}
}{
\underbrace{
\mathbf{H}(\hat X\;,\; A)
}
}
\;\;
\underoverset
{\underset{oxidation}{\longleftarrow}}
{\overset{reduction}{\longrightarrow}}
{\simeq}
\;\;
\underset{
\text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} }
}{
\underbrace{
\mathbf{H}(X \;,\; [G,A]/G)
}
}
$$
given by
$$
\left(
\hat X \longrightarrow A
\right)
\;\;\;
\leftrightarrow
\;\;\;
\left(
\array{
X && \longrightarrow && [G,A]/G
\\
& \searrow && \swarrow
\\
&& \mathbf{B}G
}
\right)
$$
=--
+-- {: .proof #DimensionalReductionAbstractly}
###### Proof
The statement that $i^\ast \simeq hofib$ follows immediately by the definitions. What we need to see is that the [[dependent product]] along $i$ is given as claimed.
To that end, first observe that the [[conjugation action]] on $[G,X]$ is the [[internal hom]] in
the [[(∞,1)-category]] of $G$-[[∞-actions]] $Act_G(\mathbf{H})$.
Under the [[equivalence of (∞,1)-categories]]
$$
Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
$$
(from [NSS 12](https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications)) then $G$ with its canonical [[∞-action]] is $(\ast \to \mathbf{B}G)$
and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.
Hence
$$
[G,X]/G
\simeq
[\ast, X \times \mathbf{B}G]_{\mathbf{B}G}
\;\;\;\;\;
\in
\mathbf{H}_{/\mathbf{B}G}
\,.
$$
So far this is the very definition of what $[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}$ is to mean in the first place.
But now since the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ is itself [[cartesian closed (infinity,1)-category|cartesian closed]], via
$$
E \times_{\mathbf{B}G}(-)
\;\;\;
\dashv
\;\;\;
[E,-]_{\mathbf{B}G}
$$
it is immediate that
there is the following sequence of [[natural equivalences]]
$$
\begin{aligned}
\mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G)
& \simeq
\mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G})
\\
& \simeq
\mathbf{H}_{/\mathbf{B}G}(
Y \times_{\mathbf{B}G} \ast,
\underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
)
\\
& \simeq
\mathbf{H}(
\underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}},
X
)
\\
& \simeq
\mathbf{H}(hofib(Y),X)
\end{aligned}
$$
Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$ denotes the [[base change]] along it.
=--
See also at _[[double dimensional reduction]]_ for more on this.
### Along $V/G \to \mathbf{B}G$
More generally:
+-- {: .num_prop #RightBaseChangeAlongUniversalFiberBundleProjection}
###### Proposition
Let $\mathbf{H}$ be an [[(∞,1)-topos]] and $G \in Grp(\mathbf{H})$ an [[∞-group]].
Let moreover $V \in \mathbf{H}$ be an object equipped with a $G$-[[∞-action]] $\rho$, equivalently (by the discussion there) a [[homotopy fiber sequence]] of the form
$$
\array{
V
\\
\downarrow
\\
V/G & \overset{p_\rho}{\longrightarrow}& \mathbf{B}G
}
$$
Then
1. pullback along $p_\rho$ is the operation that assigns to a morphism $c \colon X \to \mathbf{B}G$ the $V$-[[fiber ∞-bundle]] which is [[associated ∞-bundle|associated]] via $\rho$ to the $G$-[[principal ∞-bundle]] $P_c$ classified by $c$:
$$
(p_\rho)^\ast \;\colon\; c \mapsto P_c \times_G V
$$
1. the right base change along $p_\rho$ is given on objects of the form $X \times (V/G)$ by
$$
(p_\rho)_\ast \;\colon\; X \times (V/G) \;\mapsto\; [V,X]/G
$$
=--
+-- {: .proof}
###### Proof
The first statement is [NSS 12, prop. 4.6](https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications).
The second statement follows as in the proof of prop. \ref{CyclicLoopSpace}: Let
$$
\left(
\array{
Y
\\
\downarrow^{\mathrlap{c}}
\\
\mathbf{B}G
}
\right)
\;\in\;
\mathbf{H}_{/\mathbf{B}G}
$$
be any object, then there is the following sequence of [[natural equivalences]]
$$
\begin{aligned}
\mathbf{H}_{/\mathbf{B}G}(Y, [V,X]/G)
& \simeq
\mathbf{H}_{/\mathbf{B}G}(Y, [V/G, X \times \mathbf{B}G]_{\mathbf{B}G})
\\
& \simeq
\mathbf{H}_{/\mathbf{B}G}(
Y \times_{\mathbf{B}G} (V/G),
\underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
)
\\
& \simeq
\mathbf{H}_{/\mathbf{B}G}
(
(p_\rho)_!( P_c \times_G (V/G) ),
p^\ast X
)
\\
& \simeq
\mathbf{H}_{/(V/G)}
(
P_c \times_G V, (p_\rho)^\ast p^\ast X
)
\\
&
\simeq
\mathbf{H}_{/(V/G)}(P_c \times_G V, X \times (V/G))
\end{aligned}
$$
where again
$$
p \colon \mathbf{B}G \to \ast
\,.
$$
=--
+-- {: .num_example #SymmetricPowers}
###### Example
**(symmetric powers)**
Let
$$
G = \Sigma(n) \in Grp(Set) \hookrightarrow Grp(\infty Grpd) \overset{LConst}{\longrightarrow} \mathbf{H}
$$
be the [[symmetric group]] on $n$ elements, and
$$
V = \{1, \cdots, n\} \in Set \hookrightarrow \infty Grpd \overset{LConst}{\longrightarrow} \mathbf{H}
$$
the $n$-element [[set]] ([[h-set]]) equipped with the canonical $\Sigma(n)$-[[action]]. Then prop. \ref{RightBaseChangeAlongUniversalFiberBundleProjection} says that right base change of any $p_\rho^\ast p^\ast X$ along
$$
\{1, \cdots, n\}/\Sigma(n)
\longrightarrow
\mathbf{B}\Sigma(n)
$$
is equivalently the $n$th [[symmetric power]] of $X$
$$
[\{1,\cdots, n\},X]/\Sigma(n)
\simeq
(X^n)/\Sigma(n)
\,.
$$
=--
## Related concepts
* **base change**
* [[dependent sum]], [[dependent product]]
* [[dependent sum type]], [[dependent product type]]
* [[necessity]], [[possibility]], [[reader monad]], [[writer comonad]]
* [[proper base change theorem]]
* Base change geometric morphisms may be interpreted in terms of [[fiber integration]]. See [[integral transforms on sheaves]] for more on this.
* [[change of enriching category]]
[[!include notions of pullback -- contents]]
## References
A general discussion that applies (also) to [[enriched categories]] and [[internal categories]] is in
* [[Dominic Verity]], _Enriched categories, internal categories and change of base_ Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html))
Discussion in the context of [[topos theory]] is around example A.4.1.2 of
* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_
and around theorem IV.7.2 in
* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
Discussion in the context of [[(infinity,1)-topos theory]] is in section 6.3.5 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
See also
* A. Carboni, G. Kelly, R. Wood, _A 2-categorical approach to change of base and geometric morphisms I_ ([numdam](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_1/CTGDC_1991__32_1_47_0/CTGDC_1991__32_1_47_0.pdf))
[[!redirects change of base]]
[[!redirects changes of base]]
[[!redirects base changes]]
[[!redirects base change geometric morphism]]
[[!redirects base change geometric morphisms]]
|
base change spectral sequence
|
https://ncatlab.org/nlab/source/base+change+spectral+sequence
|
For $R$ a [[ring]] write $R$[[Mod]] for its category of [[modules]].
Given a [[homomorphism]] of [[ring]] $f : R_1 \to R_2$ and an $R_2$-[[module]] $N$ there are composites of [[base change]] along $f$ with the [[hom-functor]] and the [[tensor product]] functor
$$
R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab
$$
$$
R_1 Mod \stackrel{Hom_{R_1 Mod}(-,R_2)}{\to}
R_2 Mod
\stackrel{Hom_{R_2}(-,N)}{\to}
Ab
\,.
$$
The [[derived functors]] of $Hom_{R_2}(-,N)$ and $\otimes_{R_2} N$ are the [[Ext]]- and the [[Tor]]-functors, respectively, so the [[Grothendieck spectral sequence]] applied to these composites is the _base change spectral sequence_ for these.
|
base of enrichment
|
https://ncatlab.org/nlab/source/base+of+enrichment
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[enriched category theory]], a *base of enrichment* is some kind of [[category theory|category theoretic]] [[structure]] $V$ -- most often a [[monoidal category]], but potentially something more general like a [[bicategory]], [[virtual double category]], or [[skew-monoidal category]] -- over which one intends to consider $V$-[[enriched categories]].
Frequently, but not always, $V$ will be a [[Bénabou cosmos]], which provides sufficient infrastructure to carry out enriched versions of most of the standard category theoretic constructions, for example of [[enriched functor categories]], [[tensor products of enriched categories]], [[enriched presheaves|enriched]] [[presheaf categories]], [[Eilenberg-Moore categories]], specific [[weighted limits]] and [[weighted colimits]], and so on.
In this context, "change of base" or "change of base of enrichment", refers to a [[2-functor]] $V\text{-}Cat \longrightarrow W\text{-}Cat$ between ([[very large category|very large]]) [[categories of enriched categories]] that is induced by a [[lax monoidal functor]] $V \to W$ between bases of enrichment. See also the section *[Base change](enriched+category#BaseChange)* at *[[enriched category]]*.
## References
* {#Kelly82} [[Max Kelly]], p. 9 of: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136 [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]
* [[Francis Borceux]], §6.4 of: _[[Handbook of Categorical Algebra]] Vol 2: Categories and Structures_, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]
[[!redirects base of enrichment]]
[[!redirects bases of enrichment]]
[[!redirects base for enrichment]]
[[!redirects bases for enrichment]]
|
base topos
|
https://ncatlab.org/nlab/source/base+topos
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Every [[sheaf topos]] $\mathcal{E}$ of sheaves with values in [[Set]] is canonically and essentially uniquely equipped with its [[global section]] [[geometric morphism]] $\Gamma : \mathcal{E} \to Set$. So in particular for $\mathcal{E} \to \mathcal{F}$ any other [[geometric morphism]], we have necessarily a [[diagram]]
$$
\array{
\mathcal{E} &&\to&& \mathcal{F}
\\
& \searrow &\swArrow_{\simeq}& \swarrow
\\
&& Set
}
$$
in the [[2-category]] [[Topos]].
Accordingly, if $\mathcal{E}$ and $\mathcal{F}$ are both equipped with [[geometric morphism]] to some other topos $\mathcal{S}$, it makes sense to restrict attention to those geometric morphisms between them that do form commuting triangles as before
$$
\array{
\mathcal{E} &&\to&& \mathcal{F}
\\
& \searrow &\swArrow_{\simeq}& \swarrow
\\
&& \mathcal{S}
}
$$
but now over the new _base topos_ $\mathcal{S}$. This is a [[morphism]] in the [[slice 2-category]] [[Topos]]$/\mathcal{S}$.
One can develop essentially all of [[topos theory]] in $Topos/\mathcal{S}$ instead of in $Topos$ itself.
To some extent it is also possible to speak of a base topos entirely [[internal logic|internally]] to a given topos. See for instance ([AwodeyKishida](#AwodeyKishida)).
## Constructions
+-- {: .num_defn}
###### Definition
To $\mathcal{S}$ itself we associate the $\mathcal{S}$-[[indexed category]] (the canonical self-indexing) $\mathbb{S}$ given by
$$
\mathbb{S}^I = \mathcal{S}/I
\,.
$$
To $p : \mathcal{E} \to \mathcal{S}$ a topos over a base $\mathcal{S}$, we associate the $\mathcal{S}$-[[indexed category]]
$$
\mathbb{E} : \mathcal{S}^{op} \to Cat
$$
which sends an object $I \in \mathcal{S}$ to the [[over-topos]] of $\mathcal{E}$ over the [[inverse image]] of $I$ under the [[geometric morphism]] $p$
$$
\mathcal{E}^I := \mathcal{E}/p^*(I)
\,.
$$
=--
+-- {: .num_prop}
###### Proposition
The [[geometric morphism]] $p : \mathcal{E} \to \mathcal{S}$ induces an $\mathcal{S}$-indexed geometric morphism (hence a geometric morphism [[internalization|internal to]] the [[slice 2-category]] [[Topos]]$/\mathcal{S}$)
$$
\mathbb{p} : \mathbb{E} \to \mathbb{S}
\,.
$$
=--
By the discussion at [[indexed category]].
## Related concepts
* [[base (∞,1)-topos]]
* [[internal site]]
* [[internal sheaves]]
* [[bounded geometric morphism]]
* [[unbounded topos]]
## References
The general notion of base toposes is the topic of section B3 of
* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
An internal description of base toposes in the context of [[modal logic]] appears in
* {#AwodeyKishida} [[Steve Awodey]], Kohei Kishida, _Topology and modality: the topological interpretation of first-order modal logic_ ([pdf](http://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf))
[[!redirects base toposes]]
[[!redirects topos over a base]]
[[!redirects toposes over a base]]
|
Basic Bundle Theory and K-Cohomology Invariants
|
https://ncatlab.org/nlab/source/Basic+Bundle+Theory+and+K-Cohomology+Invariants
|
This page collects material related to the textbook
* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]],
_Basic Bundle Theory and K-Cohomology Invariants_,
Lecture Notes in Physics, Springer 2008
[doi:10.1007/978-3-540-74956-1](https://doi.org/10.1007/978-3-540-74956-1)
[pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf)
on [[fiber bundles]] ([[principal bundles]], [[vector bundles]], [[classifying space]], [[characteristic classes]]), [[topological K-theory]], [[twisted K-theory]], [[equivariant K-theory]], [[twisted ad-equivariant K-theory]] and applications to the [[K-theory classification of D-brane charge]].
category: reference
|
basic complex line bundle on the 2-sphere
|
https://ncatlab.org/nlab/source/basic+complex+line+bundle+on+the+2-sphere
|
[[!redirects basic line bundle on the 2-sphere]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _basic line bundle on the 2-sphere_ is the [[complex line bundle]] on the [[2-sphere]] whose [[first Chern class]] is a generator $\pm 1 \in \mathbb{Z} \,\simeq\, H^2(S^2, \mathbb{Z})$, equivalently the [[tautological line bundle]] on the [[Riemann sphere]] regarded as [[complex projective space|complex projective 1-space]].
This is the [[pullback bundle]] of the map $S^2 \to B U(1) \simeq B^2 \mathbb{Z}$ to the [[classifying space]]/[[Eilenberg-MacLane space]] which itself represents a generator of the [[homotopy group]] $\pi_2(S^2) \simeq \mathbb{Z}$.
Beware that this basic line bundle is sometimes called the "canonical line bundle on the 2-sphere", but it is _not_ [[isomorphism|isomorphic]] to what in complex geometry is called the [[canonical bundle]] of the 2-sphere regarded as a [[Riemann surface]]. Instead it is "one half" of the latter, its [[theta characteristic]]. See also at _[[geometric quantization of the 2-sphere]]_.
The basic line bundle is the canonically [[associated bundle]] to basic [[circle principal bundle]]: the [[complex Hopf fibration]]. Another name for it is the [[tautological line bundle]] for the complex [[projective space|projective line]] $\mathbb{P}^1(\mathbb{C})$ (the [[Riemann sphere]]), namely the map $\mathbb{C}^2 \setminus \{(0, 0)\} \to \mathbb{P}^1(\mathbb{C})$ mapping $(x, y)$ to $[x; y]$.
## Definition
The [[classifying space]] for [[circle principal bundles]], or equivalently (via forming [[associated bundles]]) that of [[complex line bundles]] is $B U(1)$, which as a [[Grassmannian]] is the infinite [[complex projective space]] $\mathbb{C}P^\infty$. The [[homotopy type]] of this space is that of the [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$. This means that $K(\mathbb{Z},2)$ is in particular [[path-connected topological space|path-connected]] and has second [[homotopy group]] the [[integers]]: $\pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z}$.
It being the [[classifying space]] for complex line bundles means that
$$
\left\{
\array{
\text{isomorphism classes of}
\\
\text{complex line bundles}
\\
\text{on}\,\, S^2
}
\right\}
\;\simeq\;
\left\{
\array{
\text{continuous functions}
\\
S^2 \longrightarrow K(\mathbb{Z},2)
\\
\text{up to homotopy}
}
\right\}
\;\simeq\;
\pi_2(K(\mathbb{Z},2))
\;\simeq\;
\mathbb{Z}
\,.
$$
The ([[isomorphism class]]) of the complex line bundle which corresponds to $+1 \in \mathbb{Z}$ under this sequence of [[isomorphisms]] is called the _basic complex line bundle on the 2-sphere_.
Hence the basic complex line bundle on the 2-sphere is [[generalized the|the]] [[pullback bundle]] of the [[universal complex line bundle]] on $B U(1)$ along the map $S^2 \to B U(1)$ which represents the element $1 \in \mathbb{Z} \simeq \pi_2(B U(1))$. If the [[classifying space]] $B U(1)$ is represented by the infintie [[complex projective space]] $\mathbb{C}P^\infty$ with its canonical [[CW-complex]] structure ([this prop.](complex+projective+space#CellComplexStructureOnComplexProjectiveSpace)), then this map is represented by the canonical cell incusion $S^2 \hookrightarrow\mathbb{C}P^\infty$.
Notice that there is a non-trivial [[automorphism]] of $\mathbb{Z}$ as an [[abelian group]] given by $n \mapsto -n$. This means that there is an ambiguity in the definition of the basic line bundle on the 2-sphere.
## Properties
+-- {: .num_lemma #ClutchingConstructionOfBasicLineBundle}
###### Lemma
**([[clutching construction]] of the basic line bundle)
Under the [[clutching construction]] of [[vector bundles]] on the [[2-sphere]], the basic complex line bundle on the 2-sphere is given by the [[transition function]]
$$
\mathbb{C} \supset \, S^1 \longrightarrow GL(1,\mathbb{C}) \, \subset \mathbb{C}
$$
from the [[Euclidean space|Euclidean]] [[circle]] $S^1 \subset \mathbb{R}^2 \simeq \mathbb{C}$ to the complex [[general linear group]] in 1-dimension, which is $GL(1,\mathbb{C}) \simeq \mathbb{C} \setminus \{0\}$ given simply by
$$
z \mapsto z
\,,
$$
Alternatively, due to the sign ambiguity in the definition of the basic bundle, its clutching transition function is given by
$$
z \mapsto - z
\,.
$$
=--
+-- {: .proof}
###### Proof
Under the [[clutching construction]] the [[isomorphism class]] of a complex line bundle corresponds to the [[homotopy class]] of its clutching transition function
$$
S^1 \to GL(1, \mathbb{C}) \simeq \mathbb{C} \setminus \{0\}
$$
hence to an element of the [[fundamental group]] $\pi_1(\mathbb{C} \setminus \{0\}) \simeq \mathbb{Z}$. Hence by definition, the basic bundle has clutching transition function corresponding to $\pm 1 \in \mathbb{Z} \simeq [S^1, GL(1,\mathbb{Z})]$ and this element is represented by the function $z \mapsto \pm z$.
=--
+-- {: .num_prop #TensorRelationForBasicLineBundleOn2Sphere}
###### Proposition
**(fundamental tensor/sum relation of the basic complex line bundle)**
Under [[direct sum of vector bundles]] $\oplus_{S^2}$ and [[tensor product of vector bundles]] $\otimes_{S^2}$, the basic line bundle on the 2-sphere $H \to S^2$ satisfies the following relation
$$
H \oplus_{S^2} H
\;\simeq\;
\left(
H \otimes_{S^2} H
\right)
\oplus_{S^2}
1_{S^2}
$$
(where $1_{S^2}$ denotes the [[trivial vector bundle]] [[complex line bundle]] on the 2-sphere).
=--
(e.g ([Hatcher, Example 1.13](#Hatcher)))
+-- {: .proof}
###### Proof
Via the [[clutching construction]] there is a single [[transition function]] of the form
$$
S^1 \longrightarrow GL(n,\mathbb{C})
$$
that characterizes all the bundles involved. With $S^1 \hookrightarrow \mathbb{C}$ identified with the [[topological subspace]] of [[complex numbers]] of unit [[absolute value]], the standard choice for these functions is
* for the [[trivial vector bundle|trivial]] [[line bundle]] $1_{S^2}$ we may choose
$f_1 \colon z \mapsto \left( 1 \right)$;
* for the basic line bundle we may choose (by lemma \ref{ClutchingConstructionOfBasicLineBundle})
$f_H \colon z \mapsto \left( z\right)$
This yields
* for $H \otimes H \oplus 1_{S^2}$ the clutching function
$z \mapsto \left( \array{ z^2 & 0 \\ 0 & 1 }\right)$
* for $H \oplus H$ the clutching function
$z \mapsto \left( \array{ z & 0 \\ 0 & z } \right)$.
Since the complex [[general linear group]] $Gl(n,\mathbb{C})$ is [[path-connected topological space|path-connected]] (by [this prop.](general+linear+group#ConnectednessOfGeneralLinearGroup)), there exists a [[continuous function]]
$$
\gamma \colon [0,1] \longrightarrow GL(2,\mathbb{C})
$$
connecting the identity matrix on $\mathbb{C}^2$ with the one that swaps the two entries, i.e.
with $\gamma(0) = \left( \array{ 1 & 0 \\ 0 & 1 } \right)$
and $\gamma(1) = \left( \array{ 0 & 1 \\ 1 & 0 } \right)$
Therefore the function
$$
\array{
S^1 \times [0,1] &\overset{}{\longrightarrow}& GL(2,\mathbb{C})
\\
(z,t)
&\overset{\phantom{AA}}{\longrightarrow}&
f_{H \oplus 1}(z) \cdot \gamma(t) \cdot f_{1 \oplus H}(z) \cdot \gamma(t)
}
$$
(with [[matrix multiplication]] on the right) is a [[left homotopy]] from $f_{H \oplus H}$ to $f_{H \otimes H \oplus 1}$.
=--
+-- {: .num_remark}
###### Remark
**([[fundamental product theorem in topological K-theory]])**
Under the map
$$
Vect(S^2)_{/\sim} \longrightarrow K(X)
$$
that sends [[complex vector bundles]] to their class in the [[topological K-theory]] ring $K(X)$, the fundamental tensor/sum relation of prop. \ref{TensorRelationForBasicLineBundleOn2Sphere} says that the K-theory class $H$ of the basic line bundle in $K(X)$ satisfies the relation
$$
\begin{aligned}
(H - 1)^2
& =
H^2 + 1 - \underset{= H^2 + 1}{\underbrace{2 H}}
\\
= & 0
\end{aligned}
$$
in $K(X)$.
(Notice that $H-1$ is the image of $[H]$ in the [[reduced K-theory]] $\tilde K(X)$ of $S^2$ under the splitting $K(X) \simeq \tilde K(X) \oplus \mathbb{Z}$ (by [this prop.](topological+K-theory#KGrupDirectSummandReducedKGroup)).)
It follows that there is a [[ring homomorphism]] of the form
$$
\array{
\mathbb{Z}[h]/\left( (h-1)^2 \right)
&\overset{}{\longrightarrow}&
K(S^2)
\\
h &\overset{\phantom{AAA}}{\mapsto}& H
}
$$
from the [[polynomial ring]] in one abstract generator, [[quotient ring|quotiented]] by this relation, to the [[topological K-theory]] ring.
It turns out that this homomorphism is in fact an [[isomorphism]], hence that the relation $(H-1)^2 = 0$ from prop. \ref{TensorRelationForBasicLineBundleOn2Sphere} is the _only_ relation satisfied by the basic complex line bundle in topological K-theory.
More generally, for $X$ a [[topological space]], then there is a composite ring homomorphism
$$
\array{
K(X) \otimes \mathbb{Z}[h]/((h-1)^2)
& \longrightarrow &
K(X) \times K(S^2)
& \longrightarrow &
K(X \times S^2)
\\
(E, h)
&\overset{\phantom{AAA} }{\mapsto}&
(E,H)
&\overset{\phantom{AAA}}{\mapsto}&
(\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H)
}
$$
to the topological K-theory ring of the [[product topological space]] $X \times S^2$, where the second map is the [[external tensor product]] of vector bundles.
This composite is an [[isomorphism]] if $X$ is a [[compact Hausdorff space]] (for $X = \ast$ the [[point space]] this reduces to the previous statement).
This is called the _[[fundamental product theorem in topological K-theory]]_. It is the main ingredient in the [[proof]] of [[Bott periodicity]] in complex topological K-theory.
=--
## Related concepts
* [[universal complex line bundle]]
* [[tautological line bundle]], [[equivariant tautological line bundle]]
## References
* {#Hatcher} [[Allen Hatcher]], _Vector bundles and K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))
* {#Wirthmuller12} [[Klaus Wirthmüller]], _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])
[[!redirects basic line bundles on the 2-sphere]]
|
basic differential form
|
https://ncatlab.org/nlab/source/basic+differential+form
|
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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
* table of contents
{:toc}
## Definition
### For general submersions
Given a [[submersion]] $p\colon E\to B$, one may ask: which [[differential forms]] on $E$ are [[pullback of differential forms|pullbacks]] of [[differential forms]] on $B$?
If the [[fibers]] of $p$ are [[connected topological space|connected]] (otherwise the characterization given below is valid only locally in $E$), the answer is provided by the notion of a __basic form__: a differential form $\omega$ is basic if the following two conditions are met:
* The [[tensor contraction|contraction]] of $\omega$ with any $p$-[[vertical vector field]] is [[zero]].
* The [[Lie derivative]] of $\omega$ with respect to any $p$-[[vertical vector field]] is [[zero]].
Using [[Cartan's magic formula]], in the presence of the first condition, the second condition can be replaced by the following one:
* The [[tensor contraction|contraction]] of $d\omega$ (where $d$ is the [[de Rham differential]]) with any $p$-[[vertical vector field]] is zero.
### Via pullback to smooth principal bundles
Given a [[Lie group]] $G$ and a [[smooth manifold|smooth]] $G$-[[principal bundle]] $P \overset{p}{\longrightarrow} X$ over a base [[smooth manifold]] $X$, a [[differential form]] $\omega \in \Omega^\bullet\big( P \big)$ on the total space $P$ is called _basic_ if it is the [[pullback of differential forms]] along the [[bundle]] projection $p$ of a differential form $\beta \in \Omega^\bullet\big( X \big)$ of the base manifold
$$
\omega \;=\; p^\ast(\beta)
\,.
$$
### In terms of Cartan calculus
Equivalently, if $\mathfrak{g} \simeq T_e G$ denotes the [[Lie algebra]] of $G$, and for $v \in \mathfrak{g}$ we write
$$
\hat v
\;\colon\;
P
\overset{ (e,v), (-,0) }{\hookrightarrow}
T G \times T P
\simeq
T ( G \times P )
\overset{ \;\;\; d \rho \;\;\; }{\longrightarrow}
T P
$$
for the [[vector field]] on $P$ which is the [[derivative]] of the $G$-[[action]] $G \times P \overset{\rho}{\to} P$ along $v$, then differential form $\omega$ is _basic_ precisely if
1. it is annihilated by the contraction with $\hat v$
$$
\iota_{\hat v} \omega = 0
$$
1. it is annihilated by the [[Lie derivative]] along $\hat v$:
$$
\mathcal{L}_{\hat v}\omega
\;=\;
[d_{dR}, \iota_{\hat v}] \omega
\;=\;
0
$$
(where the first equality holds generally by [[Cartan's magic formula]], we are displaying it just for emphasis)
for all $v \in \mathfrak{g}$.
In this form the definition of basic forms makes sense more generally whenever a [[Cartan calculus]] is given, not necessarily exhibited by smooth vector fields on actual manifolds. This more general concept of basic differential forms appears notably in the construction of the Weil model for [[equivariant de Rham cohomology]].
## Related concepts
* [[horizontal differential form]]
[[!redirects basic differential forms]]
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basic Fraenkel model
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https://ncatlab.org/nlab/source/basic+Fraenkel+model
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* table of contents
{: toc}
## Idea
The _basic Fraenkel model_ is a model of the [[set theory]] [[ZFA]] that doesn't satisfy the [[axiom of choice]]. It was one of the first examples of a [[permutation model]] of set theory. In the basic Fraenkel model, we have a countable set of atoms $A$, and we take the full permutation group $G$ of $A$. The normal filter of subgroups is generated by the stabilizers of finite subsets of $A$.
The [[second Fraenkel model]] is similar, but uses the [[countable set|countable]] group $(\mathbb{Z}/2\mathbb{Z})^\mathbb{N}$.
## Description
### Material set theory
In the language of [[material set theory]], this is just a [[Fraenkel-Mostowski model]] given by a countable set of atoms $A$ with the full permutation group $G$ acting on it. The normal filter of subgroups is generated by the [[stabilizers]] of [[finite]] subsets of $A$, ie. groups of the form
$$
Stab (K) = \{ g \in G: \forall x \in K, gx = x\},
$$
where $K$ is a finite subset of $A$.
### Structural set theory
We can take the same group $G$ as above, with topology generated by the stabilizers $Stab (K)$. We can then form the [[topos]] of [[category of continuous G sets|continuous G-sets]], whose objects are [[sets]] with a continuous [[action]] of $G$ (with the set given the discrete topology), and morphisms are the $G$-equivariant maps. This topos is also known as the [[Schanuel topos]].
The [[internal logic]] of this topos can be identified with the standard logic of the material model of ZFA constructed above, if we interpret the universal quantifiers in ZFA suitably (eg. via [[stack semantics]], or other semantics as mentioned in [[Fraenkel-Mostowski models]])
## Properties
It is clear that the set of atoms in the Fraenkel-Mostowski model cannot be linearly ordered. So the ordering principle (that every set can be linearly ordered) fails. Consequently, the [[axiom of choice]], [[prime ideal theorem#boolean_pit_and_the_ultrafilter_principle|boolean prime ideal theorem]] etc. all fail. In fact, even countable choice fails.
We can consider the object $\mathcal{A}$ consisting of the set of atoms with the obvious action of $G$. It can be shown that this does not biject with any finite set (ie. a set that is a finite coproduct of the terminal object), but any injection $\mathcal{A} \to \mathcal{A}$ must be a surjection (this is true both as an external statement or as a statement in the [[internal logic]]). So this is an example of a [[Dedekind-finite]] but not [[finite]] set.
## Variations
We can consider the case where we have more atoms, or allow stabilizers of larger subsets. In different cases, weak versions of Choice may hold. For example, if we have $\aleph_1$ many atoms, and take the normal filter generated by stabilizers of countable subsets, then the axiom of choice holds for any well-ordered family of sets. Details can be found in the book [Consequences of the Axiom of Choice](#consequences).
## Related concepts
* [[Fraenkel-Mostowski model]]
* [[G-set]]
* [[topos of G-sets]]
* [[second Fraenkel model]]
* [[Schanuel topos]]
* [[ZFA]]
## References
The model is given by $\mathcal{N}1$ in the book
* {#consequences} Paul Howard, Jean E. Rubin, _Consequences of the Axiom of Choice_, AMS Mathematical Surveys and Monographs 59
The various properties of the model are listed in detail. Variations of the model are listed as $\mathcal{N}12$ and $\mathcal{N}16$.
[[!redirects basic Fraenkel model of ZFA]]
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basic gerbe
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https://ncatlab.org/nlab/source/basic+gerbe
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+-- {: .rightHandSide}
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###Context###
#### Differential cohomology
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[[!include differential cohomology - contents]]
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\tableofcontents
## Idea
With some abuse of terminology, by the "basic gerbe" over a [[Lie group]] $G$ authors mean, following [Meinrenken 2003](#Meinrenken03), a $U(1)$-[[bundle gerbe]] (typically understood [[connection on a bundle gerbe|with 2-connection]]) on the [[underlying]] [[topological space]] (or underlying [[smooth manifold]]) of $G$, whose [[Dixmier-Douady class]] is a generator (whence "basic") of the [[ordinary cohomology]]-[[cohomology group|group]] $H^3(G;\mathbb{Z})$.
For that to make verbatim sense we need for instance that $H^3(G;\mathbb{Z}) \,\simeq\, \mathbb{Z}$, which is the case notably for [[simply connected topological space|simply connected]] [[compact Lie groups|compact]] [[simple Lie groups]] $G$, such as [[SU(n)]] for $n \geq 2$ or [[Spin(n)]] for $n \geq 3$ (but the notion has been considered more generally).
If this bundle gerbe is considered [[connection on a bundle gerbe|with 2-connection]], then the [[curvature]] 3-form is typically understood to be a [[de Rham cohomology|de Rham representative]] of $H^3(G;\mathbb{R})$ by a [[differential 3-form]] which is $G$-bi-invariant (i.e. invariant under [[pullback of differential forms]] along the action of $G$ on itself by multiplication.)
With due care, the basic bundle gerbe itself has a multiplicative structure covering that on $G$ and making it a [[2-group]], as such equivalent to the [[string 2-group]] ([Waldorf 2012](#Waldorf12)).
With bundle gerbes understood as a [[Kalb-Ramond field|Kalb-Ramond]] [[background field]] for a [[string]] (its background [[B-field]]), the basic gerbes define the [[worldsheet]] [[sigma-model]] known as the [[Wess-Zumino-Witten model]], which was the original motivation of [Gawędzki & Reis 2002](#GawędzkiReis02). In terms of [[higher prequantum geometry]] the bundle gerbe is the [[prequantum circle 2-bundle]] of the theory ([Fiorenza, Sati & Schreiber 2015](#FiorenzaSatiSchreiber15), [Fiorenza, Rogers & Schreiber 2016](#FiorenzaRogersSchreiber16)).
## References
The original articles:
* {#GawędzkiReis02} [[Krzysztof Gawędzki]], [[Nuno Reis]], *WZW branes and gerbes*, Rev. Math. Phys. **14** (2002) 1281-1334 [[arXiv:hep-th/0205233](https://arxiv.org/abs/hep-th/0205233), [doi:10.1142/S0129055X02001557](https://doi.org/10.1142/S0129055X02001557)]
* {#Meinrenken03} [[Eckhard Meinrenken]], _The basic gerbe over a compact simple Lie group_, Enseign. Math. (2) 49 (2003), no. 3-4, 307-333 [[arXiv:math/0209194](https://arxiv.org/abs/math/0209194), [e-periodica](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::562)]
* [[Krzysztof Gawędzki]], [[Nuno Reis]], _Basic gerbe over non-simply connected compact groups_, Journal of Geometry and Physics **50** 1-4 (2004) 28–55 [[arXiv:0307010](https://arxiv.org/abs/math/0307010), [doi:10.1016/j.geomphys.2003.11.004](http://dx.doi.org/10.1016/j.geomphys.2003.11.004)]
A specific construction for [[unitary groups]] and generalization to groups with [[unitary representations]] on a [[Hilbert space]]:
* [[Michael K. Murray]], [[Danny Stevenson]], _The basic bundle gerbe on unitary groups_, Journal of Geometry and Physics **58** 11 (2008) 1571-1590 [[doi:10.1016/j.geomphys.2008.07.006](https://doi.org/10.1016/j.geomphys.2008.07.006), [arXiv:0804.3464](https://arxiv.org/abs/0804.3464)]
Review:
* [[Christoph Schweigert]], [[Konrad Waldorf]], *Gerbes and Lie Groups*, in: *Developments and Trends in Infinite-Dimensional Lie Theory*, Progress in Mathematics **288** (2011) [[arXiv:0710.5467](https://arxiv.org/abs/0710.5467), [doi:10.1007/978-0-8176-4741-4_10](https://doi.org/10.1007/978-0-8176-4741-4_10)]
As a model for the [[string 2-group]]:
* {#Waldorf12} [[Konrad Waldorf]], *A Construction of String 2-Group Models using a Transgression-Regression Technique*, Contemp. Math. **584** (2012) 99-115 [[arXiv:1201.5052](http://arxiv.org/abs/1201.5052), [doi:10.1090/conm/584/11588](https://doi.org/10.1090/conm/584/11588)]
and as the [[prequantum circle 2-bundle]] of the [[WZW model]] discussed in [[smooth infinity-groupoids]]:
* {#FiorenzaSatiSchreiber15} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §3.4.2 in: *[[schreiber:A higher stacky perspective on Chern-Simons theory]]*, in: *Mathematical Aspects of Quantum Field Theories*, Mathematical Physics Studies, Springer (2015) 153-211 [[arXiv:1301.2580](https://arxiv.org/abs/1301.2580), [doi:10.1007/978-3-319-09949-1_6](https://doi.org/10.1007/978-3-319-09949-1_6)]
* {#FiorenzaRogersSchreiber16} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], §4.1 in: _[[schreiber:Higher geometric prequantum theory|Higher $U(1)$-gerbe connections in geometric prequantization]]_, Rev. Math. Phys. **28** 06 1650012 (2016) [[arXiv:1304.0236](http://arxiv.org/abs/1304.0236), [doi:10.1142/S0129055X16500124](https://doi.org/10.1142/S0129055X16500124)]
[[!redirects basic gerbes]]
[[!redirects basic bundle gerbe]]
[[!redirects basic bundle gerbes]]
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basic ideas of moduli stacks of curves and Gromov-Witten invariants > history
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https://ncatlab.org/nlab/source/basic+ideas+of+moduli+stacks+of+curves+and+Gromov-Witten+invariants+%3E+history
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< [[basic ideas of moduli stacks of curves and Gromov-Witten invariants]]
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basic ideas of moduli stacks of curves and Gromov-Witten theory
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https://ncatlab.org/nlab/source/basic+ideas+of+moduli+stacks+of+curves+and+Gromov-Witten+theory
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+-- {: .standout}
This is a sub-entry of [[Gromov-Witten invariants]]. See there for further background and context.
This entry is supposed to provide an exposition of some basic ideas underlying Gromov-Witten theory.
=--
> **raw material**: notes taken verbatim in some seminar -- needs to be polished
#Contents#
* [part I: basics of moduli stacks of curves](#intropart1)
* [part I: basics of Gromov-Witten theory](#intropart2)
## Intro Part I: basics of moduli stacks of curves {#intropart1}
**question**: what is a **moduli problem**?
* some kind of object;
* a notion of "familiy"/"deformation" of these objects
* some [[equivalence relation]] $\sim$ (possibly trivial)
on the set of such families $S(B)$ over $B$
from this we get a [[functor]] (a [[presheaf]])
$$
F : C^{op} \to Set
$$
$$
B \mapsto S(B)/_\sim
$$
**terminology** an object $M$ is called a [[fine moduli space]] if the corresponding [[representable functor|represented functor]]
$$
Y_M : C^{op} \to Set
$$
$$
x \mapsto Hom(x,M)
$$
is isomorphic to $F$, i.e. if it represents $F$.
**examples** in the [[homotopy category]] of [[topological space]]s we have
$$
\left\{
complex line bundles over B
\right\}/isom
\leftrightarrow
Hom_{Ho(Top)}(B, \mathbb{C}P^\infty)
$$
so $\mathbb{C}P^\infty$ is a [[classifying space]] for complex line bundle. Similarly for higher rank vector bundles and Grassmannians.
The analogue in the algebraic category is
$$
\left\{
line bundles L over B with generating sections s_0,...,s_n \in \Gamma(L) \right\} / isom
\leftrightarrow
Hom(B, \mathbb{P}^n).
$$
And also similarly for higher rank [[vector bundle]]s and Grassmannians.
Despite these examples, in a lot of cases the functors are not representable. We'll see some of these examples below.
Why are [[fine moduli space]]s desireable?
They allow us to study a _single_ family which tells us universal things about all families. Even if you do not care about families or deformations, moduli spaces can help, because perhaps they can tell you something about trivial families, i.e. the objects that you are studying to begin with.
**example** studying the [[cohomology ring]]s of $Gr_n(\mathbb{R}^\infty)$ or $Gr_n(\mathbb{C}^\infty)$, which are the classifying space for higher rank real and complex [[vector bundle]]s gives universal relations (or, rather, the lack thereof!) among [[Chern class]]es, etc.
Let's look at [[elliptic curve]]s (we'll work over $\mathbb{C}$).
the functor of families here is
$$
F : Sch/\mathbb{C}^{op} \to Set
$$
$$
B \mapsto \left\{
\array{
E \\ \downarrow \\ B
}
flat families of elliptic curves
\right\}
$$
**Fact**: there is no [[fine moduli space]] of [[elliptic curve]]s representing this functor
**why not?**
there is something called the [[j-invariant]] which classifies [[elliptic curve]]s up to [[isomorphism]]
let
$$
E : y^2 = x^3 + a x + b
$$
be an [elliptic curve]] given by parameters $a,b$. Then its [[j-invariant]] is the number
$$
j(E) = \frac{2^8 3^3 a^3}{4 a^3 + 27 b^2}
$$
so we might guess that the "$j$-line" $\mathbb{A}^1$ is a [[fine moduli space]] for [[elliptic curve]]s,
i.e. that there is a "universal family" of elliptic curves $C \to \mathbb{A}^1$ such that the fiber over $j \in \mathbb{A}^1$ is the elliptic cuvre with that $j$-invariant.
so that for any family $E \to B$ we'd have a [[pullback]]
$$
\array{
E &\to& C
\\
\downarrow && \downarrow
\\
B &\to& \mathbb{A}^1
}
$$
where $B \to \mathbb{A}^1$ sends a point in $B$ to the [[j-invariant]] of the elliptic curve in the fiber over $B$.
Now consider the family
$$
\chi = (y^2 = x(x-1)(x+\lambda)) \subset \mathbb{A}^1 \times \mathbb{A}_\lambda^1 \times \mathbb{P}^2
$$
$$
\array{
\chi &\to& C
\\
\downarrow && \downarrow
\\
\mathbb{A}^1_\lambda &\to& \mathbb{A}^1
}
$$
$$
j(\chi_\lambda) = s^8 \frac{(\lambda^2 - \lambda + 1)^2}{\lambda^2(\lambda-1)^2}
$$
now send $\mathbb{A}^1_\lambda \to \mathbb{A}^1_{\lambda}$
by dually sending $\lambda \mapsto 1-\lambda$
check that $j(\chi_\lambda) = j(\chi_{1-\lambda})$
this induces in the fibers a map
$\phi : \chi \to \chi$
$$
\phi : (x,y,\lambda) \mapsto (1-x,\pm i y , 1-\lambda)
$$
this maps further to $(x,-y,\lambda)$. But this would have to be the identity for $\mathbb{A}^1$ to be a fine moduli space, which it is not. So $\mathbb{A}^1$ at least is not a fine moduli space, even though it might look like one.
so that gives some computational insight that something goes wrong
This argument does not yet prove that there exists no moduli space of elliptic curves. It merely proves that the "j-line" $\mathbb{A}^1$ can not be the moduli space of elliptic curves.
However, the basic argument can be adapted, if one so desires, to in fact prove that there is no moduli space of elliptic curves.
In fact, below we will see (in an exercise), that the $j$-line _is_ a [[coarse moduli space]], as explained below. Moreover, if a coarse moduli space exists, then it is unique up to canonical isomorphism. Since fine moduli spaces, if they exist, are also coarse moduli spaces, and the $j$-line is a coarse but not a fine moduli space by the above argument, it follows that no fine moduli space exists.
**Abstract argument** Yet another, more abstract way, to see that no fine moduli space can exist it to realize that since elliptic curves have nontivial automorphisms, it is possible to construct families of elliptic curves that are locally trivial families (of the form $U \times E \to U$ for a fixed elliptic curve $U$) but which are glued together from these local pieces using nontrivial automorphisms such that the resulting family $V \to X$ is not globally trivial, i.e. not globally of the form $E \to X \to X$.
With a bit of care this alone can be used to show that a fine moduli space cannot exist. This is often summarized by a slogan of the form
> **Slogan** : nontrivial automorphisms of objects prevent the family-assignment of these objects to be representable by a fine moduli space.
However, one has to be careful with interpreting this slogan correctly. Taken naively, the slogan alone would also seem to imply that, since [[vector space]]s have nontrivial automorphisms, no classifying space for "families of vector spaces", i.e. for [[vector bundle]]s does exist, while of course this does exist (recall also the example further above). But if one interprets the slogan carefully, it does yield a true statement. For more on that see the discusson at [[moduli space]].
**How to "fix" these problems**.
1. add extra structure to the objects under consideration (e.g. add marked points) to make the automorphism groups trivial.
2. instead of looking for representing [[topological space]]s, look for representing [[groupoid]]s / [[stack]]s.
3. use [[coarse moduli space]]s $M \in Sch/\mathbb{C}$ with $\Psi_M : F \to h_M$ such that
a) $F(Spec(\mathbb{C})) \to h_M(Spec \mathbb{C}) = hom(Spec \mathbb{C}, M)$ is a [[bijection]]
b) given $M'$ and $\Psi_{M'} : F \to h_{M'}$ then there exists unique $M \to M'$ such that $\array{ F && \stackrel{\Psi_{M'}}{\to}&& h_{M'} \\ & {}_{\Psi_M}\searrow && \nearrow \\ && h_M} $
So a [[coarse moduli space]] is one that at least has the right underlying set of points as the _right_ [[moduli stack]] has: as long as we don't look at families but just at single things, it does give the right classification.
**exercise** show that the $j$-line $\mathbb{A}^1$ _is_, while not a [[fine moduli space]], a [[coarse moduli space]].
**exercise** Show that if a coarse moduli space exists, then it is unique up to canonical isomorphism.
**fact** there exists a [[coarse moduli space]] $M_{g,n}$ of [[Riemann surface]]s of [[genus]] $g$ with $n$ marked points and a [[fine moduli stack]] $M_{g,n}$ such that for all $g,n$ we have
$M_{g,n}$ is a [[smooth scheme|smooth]] [[Deligne-Mumford stack]] - aka [[orbifold]]
**except** for $(g,n) =$ $(0,0), (0,1), (0,2), (1,0)$ (these are the cases where the [[automorphism group]] is infinite, so in these cases we don't get a [[Deligne-Mumford stack]])
**"Issues"**:
1. $M_{g,n}$ is not proper, meaning: not compact, so we don't have, for example, [[Poincare duality]] on $M_{g,n}$ and no [[integration]] theory. Proper, or even better projective, schemes or stacks are just a lot easier to deal with.
2. Sometimes one wants to study singular curves or families with degeneracies.
Both "issues" can be "resolved" via [[Deligne-Mumford compactification]].
$$
\bar M_{g,n}
$$
which parameterizes at-most-nodal curves, that are connected, of arithmetic [[genus]] $g$, with $n$ smooth marked points, and the group of [[automorphism]]s is finite.
$\bar M_{g,n}$ is a smooth proper [[Deligne-Mumford stack]].
smooth here means smoothness as for [[orbifold]]s.
Deligne and Mumford were able to prove many theorems about the ordinary moduli space of curves by studying instead the compactification. For example they were able to prove that $M_{g,n}$ is irreducible.
#Intro Part II: basics of Gromov-Witten theory{#intropart2}
**Gromov** was looking for invariants of [[symplectic manifold]]s: his idea was to use $J$-holomorphic curves in compact symplectic manifolds to get symplectic invariants
**[[Edward Witten]]** and other physicists studied [[worldsheet]]s of [[string theory|string]]s in some [[spacetime]] [[manifold]] (e.g a [[Calabi-Yau space|Calabi-Yau 3-fold]])
we want to consider now [[genus]] $g$ [[Riemann surface]]s with $n$ marked points mapping into some space $X$
For a fixed
$$
\beta \in H_2(X, \mathbb{Z}),
$$
$$
M_{g,n}(X,\beta)
$$
is the space that parameterizes maps
$$
\Sigma \stackrel{f}{\to} X
$$
where $\Sigma$ is smooth, and has $n$ marked points, and
such that for $[\Sigma]$ the [[fundamental homology class]] of $\Sigma$ we have
$$
f_*[\Sigma] = \beta
\,.
$$
this is a smooth [[Deligne-Mumford stack]]. (Again we must exclude the cases of small $(g,n)$. Excluding these cases, the automorphisms of the surfaces and the automorphisms of the maps are automatically finite.)
similarly, write $\bar M_{g,n}(X,\beta)$ for the same setup but with $\Sigma$ from $\bar M_{g,n}$ as above in part 1 (the DM compactified moduli stack). EXCEPT here we do not require that $\Sigma$ (with its $n$ marked points) has finite automorphism group; we require instead that the MAP has finite automorphism group, which means.......(fill in)
This is a _proper_ (compact) [[Deligne-Mumford stack]] - but _not_ smooth (not even in the sense of smooth stacks!). This is very important. This is what makes the theory difficult/nontrivial/interesting.
we have
$$
\bar M_{g,n}(pt,0) = \bar M_{g,n}
$$
what do [[string theory|string theorists]] want to do?
we have evaluation maps
$$
\bar M_{g,n}(X,\beta) \stackrel{ev_i}{\to} X
$$
where $i$ labels a marked point, and we want morphisms
$$
H^\bullet(X)^{\otimes n}
\stackrel{ev^*_1 \wedge ev^*_2 \wedge \cdots}{\to}
H^*(\bar M_{g,n}(X,\beta))
\stackrel{\int_{[\bar M_{g,n}(x,\beta)]^{virtual}}}{\to}
\mathbb{C}
$$
There should be a virtual fundamental class $[\bar M_{g,n}(x,\beta)]^{virtual}$ that makes the maps above into the [[correlation function]]s of a [[quantum field theory]] (the integral would be the [[path integral]] of the [[worldsheet]] [[sigma-model]]) with state space $H^*(X)$. We'll explain some of what this means below.
This [[virtual fundamental class]] in algebraic geometry was constructed by Li-Tian and Behrend-Fantechi; in symplectic geometry it was done by Li-Tian.
Why do we want to use a "virtual" fundamental class? Because \bar M_{g,n}(x,\beta) is not smooth, the actual fundamental class may not behave very well. **NEEDS ELUCIDATION -- it would be nice if someone could give a more complete explanation of why we need a _[[virtual fundamental class|virtual]]_ fundamental class**
the mathematical structure of GW-theory was elucidated Ruan and then by [[Maxim Kontsevich]] and Manin in 1994.
$$
\array{
\bar M_{g,n}(X,\beta)
\\
\downarrow^\phi
\\
\bar M_{g,n}
}
$$
$$
I_{g,n,\beta} :
H^*(X)^{\otimes}
\to
H^*(\bar M_{g,n}(X,\beta))
\stackrel{\phi_*}{\to}
H^*(\bar M_{g,n})
$$
where the map $\phi_*$ is some kind of "pushforward" or "integration along the fibers"; this map uses the virtual fundamental class
there is also a map
$$
\alpha
:
\bar M_{g_1,n_1+1}
\times
\bar M_{g_2,n_2+1}
\to
\bar M_{g,n}
$$
with $g = g_1 + g_2$ and $n = n_1 + n_2$ obtained by gluing the last marked point of the first curve to the first marked point of the second curve.
so consider now the combination of these two maps
$$
\array{
H^\bullet(X)^{\otimes}
&\stackrel{I_{g,n,\beta} }{\to}&
H^\bullet(\bar M_{g,n}(X,\beta))
\stackrel{\phi_*}{\to}
H^*(\bar M_{g,n})
\\
\downarrow^{- \otimes \Delta \otimes -}
&&
\downarrow^\alpha
\\
H^\bullet(X)^{\otimes n_1}
\otimes H^\bullet(X) \otimes
H^\bullet(X) \otimes H^\bullet(X)^{\otimes n_2}
&&
H^{\bullet}(\bar M_{g_1,n_1+1} \times \bar M_{g_2,n_2+1})
\\
\downarrow^\simeq
&& \downarrow^{\simeq}
\\
H^\bullet(X)^{\otimes n_1 + 1}
\otimes
H^\bullet(X)^{\otimes n_2 + 1}
&\stackrel{\sum_{\beta_1+\beta_2 = \beta}I_{g_1,n_1+1,\beta_1} \otimes I_{g_2,n_2+1,\beta_2}}{\to}&
H^{\bullet}(\bar M_{g_1,n_1+1}) \otimes H^\bullet (\bar M_{g_2,n_2+1}))
}
$$
where the bottom right iso uses the [[Kunneth formula|Künneth formula]]
here $\Delta_a$ is a homogeneous basis of $H^\bullet(X)$
$$
g_{a b} = \int_X \Delta_a \wedge \Delta_b
$$
$$
\Delta = \sum g^{a b} \Delta_a \wedge \Delta_b
$$
where
$$
g^{a b} = (g_{a b})^{-1}
$$
so this diagram above says that this satisfies the [[sewing law]]s that defines a [[quantum field theory]].
There are various other axioms that Gromov-Witten theory must satisfy, but the sewing law above is the most important. The hard part of all of this is constructing the virtual fundamental class, and then proving that this class indeed makes the sewing law and the other axioms (which I have omitted) satisfied.
The above discussion does not yet reveal much of the rich structure of Gromov-Witten invariants. But GW invariants indeed have a very rich and beautiful mathematical structure. Indeed, if we look at just the easiest part of the theory, namely g=0, we are lead to quantum cohomology and Frobenius manifolds.
There is also the mirror symmetry conjecture, which roughly posits that the GW invariants can be found via calculations that a priori seem completely unrelated.
#References#
* M. Kontsevich, Yu. Manin, _Gromov-Witten classes, quantum cohomology, and enumerative geometry_, Comm. Math. Phys. 164 (1994), no. 3, 525--562 ([euclid](http://projecteuclid.org/euclid.cmp/1104270948)).
* [[Yuri Manin]], _Frobenius manifolds, quantum cohomology and moduli spaces_, Amer. Math. Soc., Providence, RI, 1999.
* W. Fulton, R. Pandharipande, _Notes on stable maps and quantum cohomology_, in: Algebraic Geometry-
Santa uz 1995 ed. Kollar, Lazersfeld, Morrison. Proc. Symp. Pure Math. 62, 45–96 (1997)
* J Robbin, D A Salamon, _A construction of the Deligne-Mumford orbifold_, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611--699 ([arxiv](http://arxiv.org/abs/math/0407090); [pdf at JEMS](http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=8&iss=4&rank=3)); corrigendum J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 901--905 ([pdf at JEMS](http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=9&iss=4&rank=11)).
* J Robbin, Y Ruan, D A Salamon, _The moduli space of
regular stable maps_, Math. Z. 259 (2008), no. 3, 525--574 ([doi](http://dx.doi.org/10.1007/s00209-007-0237-x)).
* Martin A. Guest, _From quantum cohomology to integrable systems_, Oxford Graduate Texts in Mathematics, 15. Oxford University Press, Oxford, 2008. xxx+305 pp.
* Joachim Kock, Israel Vainsencher, _An invitation to quantum cohomology. Kontsevich's formula for rational plane curves_, Progress in Mathematics, 249. Birkhäuser Boston, Inc., Boston, MA, 2007. xiv+159 pp.
* Sheldon Katz, _Enumerative geometry and string theory_, Student Math. Library __32__. IAS/Park City AMS & IAS 2006. xiv+206 pp.
* Eleny-Nicoleta Ionel, Thomas H. Parker, _Relative Gromov-Witten invariants_, Ann. of Math. (2) 157 (2003), no. 1, 45--96 ([doi](http://dx.doi.org/10.4007/annals.2003.157.45)).
[[!redirects basic ideas of moduli stacks of curves and Gromov–Witten theory]]
[[!redirects basic ideas of moduli stacks of curves and Gromov--Witten theory]]
[[!redirects basic ideas of moduli stacks of curves and Gromov-Witten invariants]]
[[!redirects basic ideas of moduli stacks of curves and Gromov–Witten invariants]]
[[!redirects basic ideas of moduli stacks of curves and Gromov--Witten invariants]]
|
basic localizer
|
https://ncatlab.org/nlab/source/basic+localizer
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
# Basic localizers
* table of contents
{: toc}
## Idea
By a _basic localizer_ one means a [[localizer]] on the category [[Cat]] of [[categories]], hence a choice of a [[class]] of [[functors]] to be called the _[[weak equivalences]]_, subject to some conditions.
These conditions ensure in particular that a basic localizer always contains the weak equivalences of the [[Thomason model structure]] on [[Cat]] (see [Maltsiniotis 11, 1.2.1](#Maltsiniotis11)), the [[localization]] at which is equivalent to the standard [[homotopy category]]. On the other hand, the standard [[equivalences of categories]], which are the weak equivalences in the [[canonical model structure]] on [[Cat]], do not form a basic localizer.
Hence basic localizers are a tool for [[homotopy theory]] modeled on [[category theory]]. In fact, their introduction by [[Grothendieck]] was motivated from the study of [[test categories]] (see remark \ref{CitationOnTestCategories} below).
## Definition
The definition is due to [[Grothendieck]]:
+-- {: .num_defn #BasicLocalizer}
###### Definition
A **basic localizer** is a [[class]] $W$ of [[morphisms]] in [[Cat]] such that
1. $W$ contains all [[identities]], satisfies the [[2-out-of-3 property]] and, for any functor $i\colon A\to B$ such that there exists a functor $r\colon B\to A$ with $r i=1_A$ and $i r$ in $W$, we have $i$ in $W$ (in the literature this is sometimes called being *weakly saturated*),
1. If $A$ has a [[terminal object]], then the functor $A\to 1$ is in $W$, and
1. Given a commutative triangle in $Cat$:
$$\array{ A & & \overset{u}{\to} & & B\\ & _v \searrow & & \swarrow_w \\ & & C } $$
if each induced functor $v/c \to w/c$ between [[comma categories]] is in $W$, then $u$ is also in $W$.
=--
+-- {: .num_remark}
###### Remark
The term in French is **localisateur fondamental**, which is sometimes translated as **fundamental localizer**.
=--
+-- {: .num_remark}
###### Remark
One can show that basic localizers are stable under [[retracts]]; see Proposition 4.2.4 in ([Cisinski 06](#Cisinski06)). They are also closed under small filtered colimits; see Corollaire 4.2.22 in ([Cisinski 06](#Cisinski06)).
=--
+-- {: .num_remark #CitationOnTestCategories}
###### Remark
In [[Pursuing Stacks]] [[Grothendieck]] wrote about def. \ref{BasicLocalizer} the following:
> These conditions are enough, I quickly checked this night, in order to validify all results developed so far on [[test categories]], [[weak test categories]], [[strict test categories]], weak test functors and test functors (with values in $(Cat)$) (of notably the review in par. 44, page 79–88), provided in the case of test functors we restrict to the case of loc. cit. when each of the categories $i(a)$ has a final object. All this I believe is justification enough for the definition above.
=--
+-- {: .num_remark #weakBasicLocalizer}
###### Remark
Grothendieck's comments above refer in fact to a weaker definition, in which in Definition \ref{BasicLocalizer}, we replace the third condition with an analogue of Quillen's Theorem A, namely: any functor $f\colon A\to B$ such that, for any object $b$ of $B$, the induced functor $A/b\to 1$ is in $W$, is in $W$. This weaker notion is sometimes called a **weak basic localizer**.
=--
## Examples
* The class of *all* functors between small categories is, of course, the maximal basic localizer.
* The class of functors inducing an [[isomorphism]] on [[connected components]] is a basic localizer.
* The class of functors whose [[nerve]] is a [[weak homotopy equivalence]] is a basic localizer. (These are the [[weak equivalences]] in the [[Thomason model structure]].)
* For any [[derivator]] $D$, the class of $D$-equivalences is a basic localizer. This includes all the previous examples.
* The class of [[equivalences of categories]] is *not* a basic localizer (it fails the second condition). (These are the weak equivalences of the [[canonical model structure]].)
## Properties
### Asphericity and local equivalences
If $W$ is a basic localizer, we define the following related classes. We sometimes refer to functors in $W$ as *weak equivalences*.
* A category $A$ is **($W$-)aspherical** if $A\to 1$ is in $W$. Thus the second axiom says exactly that any category with a terminal object is aspherical.
* A functor $u\colon A\to B$ is **($W$-)aspherical** if for all $b\in B$, the comma category $u/b$ is aspherical.
* When the hypotheses of the third axiom are satisfied, we say that $u$ is a **local weak equivalence over $C$**. Thus the third axiom says exactly that every local weak equivalence is a weak equivalence.
+-- {: .num_example}
###### Example
If $W = \pi_0$-equivalences, then a category is aspherical iff it is connected, and a functor is aspherical iff it is [[initial functor|initial]].
=--
+-- {: .num_example}
###### Example
If $W = $ nerve equivalences, then a category is aspherical iff its nerve is contractible, and a functor is aspherical iff it is [[homotopy initial functor|homotopy initial]].
=--
We observe the following.
* A category $A$ is aspherical iff the functor $A\to 1$ is aspherical, since the only comma category involved in the latter assertion is $A$ itself.
* An aspherical functor is a weak equivalence. For if $u\colon A\to B$ is aspherical, then consider the triangle
$$\array{ A & & \overset{u}{\to} & & B\\ & _u \searrow & & \swarrow \\ & & B } $$
The third axiom tells us to consider, for a given $b\in B$, the functor $u/b \to B/b$. But $u/b$ is aspherical by assumption, while $B/b$ is aspherical by the second axiom since it has a terminal object. Thus, by 2-out-of-3, the functor $u/b \to B/b$ is in $W$, and thus by the third axiom $u$ is in $W$.
* If $u$ has a right adjoint, then it is aspherical. For in this case, each category $u/b$ has a terminal object, and thus is aspherical.
* If $I$ denotes the [[interval category]], then for any category $A$ the projection $A\times I\to A$ has a right adjoint, hence is aspherical and thus a weak equivalence. By 2-out-of-3, the two injections $A \rightrightarrows A\times I$ are also weak equivalences, so $A\times I$ is a [[cylinder object]] for $W$. It follows that if we have a natural transformation $f\to g$, then $f$ is in $W$ if and only if $g$ is. Moreover, if $f$ is a "homotopy equivalence" in the sense that it has an "inverse" $g$ such that $f g$ and $g f$ are connected to identities by arbitrary natural zigzags, then $f$ is a weak equivalence.
* In particular, any left or right adjoint is a weak equivalence.
### Self-duality
It is a non-obvious fact that the notion of basic localizer is self-dual.
+-- {: .num_theorem}
###### Theorem
A functor $u : A \to B$ is in a basic localizer $W$ if and only if $u^{op} : A^{op} \to B^{op}$ is in $W$.
=--
+-- {: .proof}
###### Proof
See Proposition 1.1.22 in ([Maltsiniotis 05](#Maltsiniotis05)).
=--
### Cisinski's theorem
Since the definition consists merely of closure conditions, the intersection of any family of basic localizers is again a basic localizer. It follows that there is a unique *smallest* basic localizer. The following was conjectured by [[Grothendieck]] and proven by [[Denis-Charles Cisinski]].
+-- {: .num_theorem}
###### Theorem (Cisinski)
The class of functors whose [[nerve]] is a [[weak homotopy equivalence]] is the smallest basic localizer.
=--
+-- {: .proof}
###### Proof
See Théorème 2.2.11 in ([Cisinski 04](#Cisinski04)). A completely different proof is given in Corollaire 4.2.19 in ([Cisinski 06](#Cisinski06))
=--
Note that this is a larger class than the class of "[[homotopy equivalences]]" considered above. For instance, the category generated by the graph
$$\bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \cdots$$
has a contractible nerve, but its identity functor is not connected to a constant one by any natural zigzag.
+-- {: .num_remark}
###### Remark
One can in fact prove that the class of functors whose nerve is a weak homotopy equivalence is the smallest weak basic localizer, as defined in Remark \ref{weakBasicLocalizer}; see Théorème 6.1.18 in ([Cisinski 06](#Cisinski06)). This refinement is related to the fact that weak homotopy equivalences are determined by the property that they induce equivalences of categories of locally constant presheaves; see Corollary 1.17 in ([Cisinski 09](#Cisinski09)). This is also related to the fact that cohomology with coefficients in local systems determine weak homotopy equivalences; see Théorème 6.5.11 and Scholie 6.5.13 in ([Cisinski 06](#Cisinski06)).
=--
## Related concepts
* [[canonical model structure]], [[Thomason model structure]] on [[Cat]]
* [[Cisinski model structure]]
## References
* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([numdam](http://www.numdam.org/item/AST_2006__308__R1_0/))
{#Cisinski06}
* [[Denis-Charles Cisinski]], _Le localisateur fondamental minimal_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 45 (2004) no. 2, pp. 109-140, [numdam](http://www.numdam.org/item/CTGDC_2004__45_2_109_0/)
{#Cisinski04}
* [[Denis-Charles Cisinski]], _Locally constant functors_, Math. Proc. Camb. Phil. Soc. Volume 147 (2009), pp. 593-614, [arXiv](https://arxiv.org/abs/0803.4342)
{#Cisinski09}
* [[Georges Maltsiniotis]], _La théorie de l'homotopie de Grothendieck _, Astérisque, Volume 301, Soc. Math. France (2005), 146 pages ([numdam](http://www.numdam.org/item/AST_2005__301__R1_0/))
{#Maltsiniotis05}
See also at _[[Cisinski model structure]]_.
* [[Georges Maltsiniotis]], _Homotopical exact squares and derivators_ ([arXiv:1101.4144](http://arxiv.org/abs/1101.4144))
{#Maltsiniotis11}
[[!redirects basic localizor]]
[[!redirects basic localizors]]
[[!redirects basic localizers]]
[[!redirects basic localiser]]
[[!redirects basic localisers]]
[[!redirects fundamental localizer]]
[[!redirects fundamental localizers]]
[[!redirects fundamental localiser]]
[[!redirects fundamental localisers]]
[[!redirects localisateur fondamental]]
|
basics of etale cohomology
|
https://ncatlab.org/nlab/source/basics+of+etale+cohomology
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--
This page goes through some basics of [[étale cohomology]].
#Contents#
* table of contents
{:toc}
## Étale topos
To every [[scheme]] $X$ is assigned a [[site]] which is a geometric analog of the collection of [[étale spaces]] over a [[topological space]]. This is called the [[étale site]] $X_{et}$ of the scheme. The [[category of sheaves]] on that site is called the [[étale topos]] of the scheme. The intrinsic [[cohomology]] of that [[topos]], hence the [[abelian sheaf cohomology]] over the [[étale site]], is the _[[étale cohomology]]_ of $X$.
This section starts with looking at some basic aspects of the [[étale topos]] as such, the basic definitions and the central [[descent theorem]] for characterizing its [[sheaves]]. The [next section](#EtaleCohomology) then genuinely considers the corresponding [[abelian sheaf cohomology]].
Étale cohomology is traditionally motivated by the route by which it was historically discovered, namely as a fix for technical problems encountered with the [[Zariski topology]]. You can find this historical motivation in all textbooks and lectures, see the _[References](#References)_ below.
But [[étale cohomology]] has a more fundamental _raison d'être_ than this. As discussed at _[[étale topos]]_ it is induced in any context in which one has a "[[reduction modality]]". While fundamental, this is actually a simple point of view which leads to a simple characterization of [[étale morphisms]], and this is what we start with now.
### Étale morphisms
Of the many equivalent characterizations of [[étale morphisms]], here we will have use of the following incarnation:
+-- {: .num_defn #EtaleMoprhism}
###### Definition
A morphisms of [[schemes]] is an _[[étale morphism of schemes]]_ if it is
1. [[formally étale morphism of schemes|formally étale]] -- recalled in a [moment](#ExplicitDefinition);
1. [[locally of finite presentation]].
=--
+-- {: .num_remark }
###### Remark
The first condition makes an [[étale morphism of schemes]] be like an [[étale space]] over its [[codomain]]. The second essentially just says demands this has [[finite set|finite]] [[fibers]].
=--
+-- {: .num_defn #EtaleSite}
###### Definition
For $X$ a [[scheme]], its [[étale site]] has a [[objects]] the [[étale morphisms of schemes]] into $X$, as [[morphisms]] the morphisms of schemes [[over category|over]] $X$, and as [[coverings]] the jointly surjective [[étale morphisms of schemes|étale morphisms]] over $X$.
The [[category of sheaves]] on $X_{et}$ is the _[[étale topos]]_ of $X$. The corresponding [[abelian sheaf cohomology]] is its _[[étale cohomology]]_.
=--
The definition of [[formally étale morphisms of schemes|formally étale]] in components goes like this.
+-- {: .num_defn #ExplicitDefinition}
###### Definition
A [[morphism]] of [[commutative rings]] $R \longrightarrow A$
is called _[[formally étale morphisms of schemes|formally étale]]_ if for every ring $B$ and for every [[nilpotent ideal]] $I \subset B$ and for every [[commuting diagram]] of the form
$$
\array{
B/I &\leftarrow& A
\\
\uparrow && \uparrow
\\
B &\leftarrow& R
}
$$
there is a unique diagonal morphism
$$
\array{
B/I &\leftarrow& A
\\
\uparrow &\swarrow& \uparrow
\\
B &\leftarrow& R
}
$$
that makes both triangles commute.
=--
(e.g. [Stacks Project 57.9, 57.12](#StackProject))
+-- {: .num_remark}
###### Remark
So [[Isbell duality|dually]] this means that $Spec(A) \to Spec(R)$ is formally étale if it has the unique [[right lifting property]] against all [[infinitesimal object|infinitesimal extensions]]
$$
\array{
Spec(B_{red}) &\longrightarrow& Spec(A)
\\
\downarrow &\nearrow& \downarrow
\\
Spec(B) &\longrightarrow & Spec(R)
}
\,.
$$
=--
and [[sheafification|locality]] this yields a notion of formally étale morphisms of [[affine varieties]] and of [[schemes]].
It is useful to realize this equivalently but a bit more naturally as follows.
+-- {: .num_defn #ReductionOnRings}
###### Definition
Write $CRing_{fin}$ for the [[category]] of [[finitely generated ring|finitely generated]] [[commutative rings]] and write $CRing_{fin}^{ext}$ for the category of [[infinitesimal ring extensions]].
Write
$$
Red \;\colon\; CRing_{fin}^{ext} \longrightarrow CRing_{fin}
$$
for the [[functor]] which sends an [[infinitesimal ring extension]] to the underlying [[commutative ring]] (in the maximal case this sends a commutative ring to its [[reduced ring]], whence the name of the functor), and write
$$
i \;\colon\; CRing_{fin} \hookrightarrow CRing_{fin}^{ext}
$$
for the [[full subcategory]] inclusion that regards a ring as the trivial infinitesimal extension over itself.
=--
+-- {: .num_prop #DifferentialCohesionModality}
###### Proposition
There is an [[adjoint triple]] of [[idempotent monad|idempotent]] ([[comonad|co]]-)[[monads]]
$$
(Red \dashv \int_{inf} \dashv \flat_{inf}) \;\colon\; PSh((CRing_{fin}^{ext})^{op}) \longrightarrow PSh((CRing_{fin}^{ext})^{op})
$$
where the [[left adjoint]] [[comonad]] $Red$ is given on [[representable functor|representables]] by the [[reduced scheme|reduction]] functor of def. \ref{ReductionOnRings} (followed by the inclusion).
=--
This statement and the following prop. \ref{FormalEtalenessBydeRhamSpace} is a slight paraphrase of an observation due to ([Kontsevich-Rosenberg 04](Q-category#KontsevichRosenbergSpaces)). A closely related adjunction appeared in ([Simpson-Teleman 13](de+Rham+space#SimpsonTeleman)) in the discussion of [[de Rham spaces]]. The general abstract situation of "[[differential cohesion]]" has been discussed in ([Schreiber 13](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos)).
+-- {: .proof}
###### Proof
The functors from def. \ref{ReductionOnRings} form an
[[adjoint pair]] $(Red \dashv i)$ because an extension element
can only map to an extension element; so for $\widehat R \to R$
an [[infinitesimal ring extension]] of $R = Red(\widehat R)$,
and for $S$ a commutative ring with
$i(S) = (S \to S)$ its trivial extension, there is a [[natural isomorphism]]
$$
Hom_{CRing_{fin}^{ext}}(\widehat R, i(S))
\simeq
Hom_{CRing_{fin}}(R,S)
\,.
$$
This exhibits $CRing_{fin}$ as a [[reflective subcategory]]
of $CRing_{fin}^{ext}$.
$$
(Red \dashv i)
\;\colon\;
CRing_{fin}
\stackrel{\overset{Red}{\leftarrow}}{\underset{i}{\hookrightarrow}}
CRing_{fin}^{ext}
\,.
$$
Via [[Kan extension]] this [[adjoint pair]] induces an [[adjoint quadruple]] of [[functors]] on [[categories of presheaves]]
$$
PSh(CRing_{fin}^{op})
\stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast = Red_!}{\leftarrow}}{\stackrel{\overset{Red^\ast}{\hookrightarrow}}{\underset{Red_\ast}{\leftarrow}}}}
PSh((CRing_{fin}^{ext})^{op})
\,.
$$
The [[adjoint triple]] to be shown is obtained from composing these adjoints pairwise.
That $Red$ coincides with the reduction functor on representables is a standard property of [[left Kan extension]] (see [here](Kan+extension#LeftKanOnRepresentables) for details).
=--
+-- {: .num_remark }
###### Remark
These considerations make sense in the general abstract context of
"[[differential cohesion]]" where the [[adjoint triple]] of prop. \ref{DifferentialCohesionModality} would be called:
([[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]).
=--
Due to the [[full subcategory]] inclusion $i_!$ in the proof of prop. \ref{DifferentialCohesionModality} we may equivalently regard presheaves on $(CRing_{fin})^{op}$ (e.g. [[schemes]]) as presheaves on $(CRing_{fin}^{ext})^{op}$ (e.g. [[formal schemes]]). This is what we do implicitly in the following.
+-- {: .num_prop #FormalEtalenessBydeRhamSpace}
###### Proposition
A morphism $f \;\colon\; Spec A \to Spec R$ in $CRing_{fin}^{op} \hookrightarrow PSh(CRing_{fin}^{op})$ is
formally étale, def. \ref{ExplicitDefinition}, precisely if it is $\int_{inf}$-[[modal type|modal]] relative $Spec R$, hence if the [[natural transformation|naturality square]] of the [[infinitesimal shape modality]]-[[unit of a monad|unit]]
$$
\array{
Spec A &\longrightarrow& \int_{inf} Spec A
\\
\downarrow && \downarrow
\\
Spec R &\longrightarrow& \int_{inf} Spec R
}
$$
is a [[pullback]] square.
=--
+-- {: .proof}
###### Proof
Evaluated on $I \hookrightarrow R \to R/I \in CRing_{fin}^{ext}$ any object, by the [[Yoneda lemma]] and the $(Red \dashv \int_{inf})$-[[adjunction]] the naturality square becomes
$$
\array{
CRing(A,B) &\longrightarrow& CRing(A,B/I)
\\
\downarrow && \downarrow
\\
CRing(R,B) &\longrightarrow& CRing(R,B/I)
}
\,.
$$
in [[Set]]. Chasing elements through this shows that this is a [[pullback]] precisely if the condition in def. \ref{ExplicitDefinition} holds.
=--
The basic stability property of [[étale morphisms]], which we need in the following, immediately follows from this characterization:
+-- {: .num_prop #ClosureForFormallyEtale}
###### Proposition
For $\stackrel{f}{\to} \stackrel{g}{\to}$ two composable morphisms, then
1. if $f$ and $g$ are both (formally) étale, then so is their composite $g \circ f$;
1. if $g$ and $ g\circ f$ are (formally) étale, then so is $f$;
1. the [[pullback]] of a (formally) étale morphism along any morphism is again (formally) étale.
=--
+-- {: .proof}
###### Proof
With prop. \ref{FormalEtalenessBydeRhamSpace} this is equivalently the
statement of the [[pasting law]] for [[pullback]] diagrams.
=--
Apart from that, for the proofs in the following we need the following basic facts
+-- {: .num_prop }
###### Proposition
* Every etale morphism is a [[flat morphism]].
* Flat morphism between affines $Spec(B) \to Spec(A)$ is [[faithfully flat]] precisely if it is surjective
=--
We repeatedly use the following example of étale morphisms.
+-- {: .num_prop #OpenImmersionIsEtale}
###### Proposition
Every [[open immersion of schemes]] is an [[étale morphism of schemes]].
In particular a standard open inclusion (a [[cover]] in the [[Zariski topology]]) induced by the [[localization of a commutative ring]]
$$
Spec(R[S^{-1}]) \longrightarrow Spec(R)
$$
is étale.
=--
(e.g. [[The Stacks Project|Stacks Project, lemma 28.37.9]])
+-- {: .proof}
###### Proof
By def. \ref{EtaleMoprhism} we need to check that the map $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is a [[formally étale morphism]] and [[locally of finite presentation]].
The latter is clear, since the very definition of [[localization of a commutative ring]]
$$
R[S^{-1}] = R[s_1^{-1}, \cdots, s_n^{-1}](s_1 s_1^{-1} - 1, \cdots , s_n s_n^{-1} - 1)
$$
exhibits a [[finitely presented algebra]] over $R$.
To see that it is formally étale we need to check that for every [[commutative ring]] $T$ with [[nilpotent ideal]] $J$ we have a [[pullback]] diagram
$$
\array{
Hom(R[S^{-1}], T) &\longrightarrow& Hom(R[S^{-1}],T/J)
\\
\downarrow && \downarrow
\\
Hom(R, T) &\longrightarrow& Hom(R, T/J)
}
\,.
$$
Now by the [[universal property]] of the [[localization of a commutative ring|localization]], a homomorphism $R[S^{-1}] \longrightarrow T$ is a homomorphism $R \longrightarrow T$ which sends all elements in $S \hookrightarrow R$ to invertible elements in $T$. But no element in a [[nilpotent ideal]] can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from $R$ to $T$ such that $S$ is taken to invertibles, which is indeed the top left set.
=--
### Descent theorem and examples of étale sheaves
{#SheafConditionAndExamples}
Since there are "many more" [[étale morphisms of schemes]] than there are [[open immersions of schemes]], a priori the discussion of [[descent]] over the [[étale site]] is more intricate than that in, say, the [[Zariski topology]]. However, the following proposition drastically reduces the types of étale [[covers]] over which [[descent]] has to be checked in addition to the [[open immersions of schemes|open immersions]]. Then the following [[descent theorem]] effectively solves the descent problem over these remaining covers.
+-- {: .num_prop #EtaleDescentDetectedOnOpenImmersionCovers}
###### Proposition
For $X$ a [[scheme]], and $A \in PSh(X_{et})$ a [[presheaf]] on its [[étale site]], def. \ref{EtaleSite},
for checking the [[sheaf]] condition it is sufficient to
check [[descent]] on the following two kinds of [[covers]]
in the [[étale site]]
1. jointly surjective collections of [[open immersions of schemes]];
1. single [[faithfully flat morphisms]] between [[affine schemes]]
(all over $X$).
=--
([Tamme, II Lemma (3.1.1)](#Tamme), [Milne, prop. 6.6](#Milne))
+-- {: .proof}
###### Proof
Suppose given an arbitrary étale [[covering]] $\{X'_i \to X'\}$ over $X$.
We show how to refine it to a more special cover which itslf is the composition of covers of the form as in the statement.
To that end, first choose a cover $\{U'_j \to X'\}$ of $X_i$ by affine [[open immersions of schemes]]. Then pulling back the original cover along that one yields covers
$$
\{X'_i \times_{X'} U'_j \to U'_j\}
$$
of each of the open affines. By pullback stability, prop. \ref{ClosureForFormallyEtale}, these are still étale maps. Now these patches in turn we cover by open affines
$$
\{
\{U'_{i j k} \to X'_i \times_{X'} U'_j \}
\}
$$
leading to covers
$$
\{
U'_{i j k} \to U'_j
\}
$$
by affines.
(Notice here crucially that while the $U'_{i j k}$ are affine open immersions in $X'_i \times_{X'} U'_j$, after this composition with an [[étale morphism]] they no longer need to be open immersions in $U'_j$, all we know is that the map is étale. This is the source of the second condition in the proposition to be shown, as discussed now. )
Since each $U'_j$, being affine, is a [[quasi-compact scheme]], we may find a finite subcover
$$
\{
U'_{j l} \to U'_j
\}
\,.
$$
Composed with the original $\{U'_j \to X'\}$ this yields a refinement of the original cover by open affines.
Hence for checking descent it is sufficient to check it for these two kinds of overs. The latter is by open immersions. For the former, we may factor
$\{U'_{j l} \to U'_j\}$ as a collection of open immersions
$$
\{U'_{j i} \to \coprod U'_{j i}\}
$$
followed by the epimorphism of affines of the form
$$
\{
\coprod U'_{j i} \to U'_j
\}
\,.
$$
Now this is morphism is etale, hence [[flat morphism|flat]], but also surjective. That makes it a [[faithfully flat morphism]].
=--
Therefore we are led to consider [[descent]] along [[faithfully flat morphisms]] of affines. For these the _[[descent theorem]]_ says that they are [[effective epimorphisms]]:
+-- {: .num_defn }
###### Definition
Given a [[commutative ring]] $R$ and an $R$-[[associative algebra]] $A$, hence a [[ring]] [[homomorphism]] $f \colon R \longrightarrow A$, the _[[Amitsur complex]]_ is the [[Moore complex]] of the dual [[Cech nerve]] of $Spec(A) \to Spec(R)$, hence the [[chain complex]]
$$
0 \to R \stackrel{f}{\to} A \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} A \otimes_R A \to A \otimes_R A \otimes_R A \to \cdots
\,.
$$
=--
(See also at _[[Sweedler coring]]_ and at _[[commutative Hopf algebroid]]_ for the same or similar constructions.)
+-- {: .num_prop #DescentTheorem}
###### Proposition
**(descent theorem)**
If $A \to B$ is [[faithfully flat]] then its Amitsur complex is [[exact sequence|exact]].
=--
This is due to ([[Grothendieck]], [[FGA]]1). The following reproduces the proof in low degree following ([Milne, prop. 6.8](#Milne)).
+-- {: .proof}
###### Proof
We show that
$$
0 \to A \stackrel{f}{\longrightarrow} B \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} B \otimes_A B
$$
is an [[exact sequence]] if $f \colon A \longrightarrow B$ is [[faithfully flat]].
First observe that the statement follows if $A \to B$ admits a [[section]] $s \colon B \to A$. Because then we can define a map
$$
k \colon B \otimes_A B \longrightarrow B
$$
$$
k \;\colon\; b_1 \otimes b_2 \mapsto b_1 \cdot f(s(b_2))
\,.
$$
This is such that applied to a [[coboundary]] it yields
$$
k(1 \otimes b - b \otimes 1) = f(s(b)) - b
$$
and hence it exhibits every [[cocycle]] $b$ as a coboundary $b = f(s(b))$.
So the statement is true for the special morphism
$$
B \to B \otimes_A B
$$
$$
b \mapsto b \otimes 1
$$
because that has a section given by the multiplication map.
But now observe that the morphism $B \to B \otimes_A B$ is the [[tensor product]] of the morphism $f$ with $B$ over $A$, hence the [[Amitsur complex]] of this morphism is [[exact sequence|exact]].
Finally, the fact that $A \to B$ is [[faithfully flat]] by assumption, hence that it exhibits $B$ as a [[faithfully flat module]] over $A$, means by definition that the [[Amitsur complex]] for $(A \to B)\otimes_A B$ is exact precisely if that for $A \to B$ is exact.
=--
+-- {: .num_prop #XSchemesRepresentSheaves}
###### Proposition
For $Z \to X$ any [[scheme]] over a [[scheme]] $X$, the induced [[presheaf]] on the [[étale site]]
$$
(U_Y \to X) \mapsto Hom_X(U_Y, Z)
$$
is a [[sheaf]].
=--
This is due to ([[Grothendieck]], [[SGA]]1 exp. XIII 5.3) A review is in ([Tamme, II theorem (3.1.2)](#Tamme), [Milne, 6.2](#Milne)).
+-- {: .proof}
###### Proof
By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers} we are reduced to
showing that the represented presheaf satisfies [[descent]] along collections of open immersions and along surjective maps of affines. For the first this is clear (it is [[Zariski topology]]-descent).
For the second case of a [[faithfully flat]] cover of affines $Spec(B) \to Spec(A)$ it follows with the exactness of the corresponding [[Amitsur complex]], by the [[descent theorem]], prop. \ref{DescentTheorem}.
=--
+-- {: .num_remark}
###### Remark
This map from $X$-schemes to sheaves on $X_{et}$ is not injective, different $X$-schemes may represent the same sheaf on $X_{et}$.
Unique representatives are given by [[étale morphism of schemes|étale schemess]] over $X$.
=--
(e.g. [Tamme, II theorem 3.1](#Tamme))
We consider some examples of [[sheaves of abelian groups]]
induced by prop. \ref{XSchemesRepresentSheaves} from [[group schemes]]
over $X$.
+-- {: .num_example}
###### Example
The [[additive group]] over $X$ is the [[group scheme]]
$$
\mathbb{G}_a \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X
\,.
$$
By the [[universal property]] of the [[pullback]], the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment
$$
\begin{aligned}
(\mathbb{G}_a)_X(U_X \to X) & =
Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X)
\\
& =
Hom(U_X, Spec(\mathbb{Z}[t]))
\\
& =
Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X}))
\\
& = \Gamma(U_X, \mathcal{O}_{U_X})
\end{aligned}
\,.
$$
=--
+-- {: .num_remark}
###### Remark
In other words, the sheaf represented by the [[additive group]] is the [[abelian sheaf]] underlying the [[structure sheaf]] of $X$, and in particular the structure sheaf is indeed an étale sheaf.
=--
Similarly one finds:
+-- {: .num_example}
###### Example
The [[multiplicative group]] over $X$
$$
\mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X
$$
represents the sheaf $(\mathbb{G}_m)_X$ given by
$$
(\mathbb{G}_m)_X(U_X)
\mapsto
\Gamma(U_X, \mathcal{O}_{U_X})^\times
\,.
$$
=--
(e.g. [Tamme, II, 3](#Tamme))
### Base change and sheaf cohomology
{#BaseChange}
+-- {: .num_defn #BaseChangeOnSites}
###### Definition
For $f \colon X \longrightarrow Y$ a [[homomorphism]] of [[schemes]], there is induced a [[functor]] on the [[categories]] underlying the [[étale site]]
$$
f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}
$$
given by sending an [[object]] $U_Y \to Y$ to the [[fiber product]]/[[pullback]] along $f$
$$
f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X)
\,.
$$
=--
+-- {: .num_prop #DirectAndInverseImageAlongMapOfBases}
###### Proposition
The morphism in def. \ref{BaseChangeOnSites} is a [[morphism of sites]]
and hence induces a [[geometric morphism]] between the étale toposes
$$
(f^\ast \dashv f_\ast)
\;\colon\;
Sh(X_{et})
\stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}
Sh(Y_{et})
\,.
$$
Here the [[direct image]] is given on a [[sheaf]] $\mathcal{F} \in Sh(X_{et})$ by
$$
f_\ast \mathcal{F} \;\colon\; (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y)
$$
while the [[inverse image]] is given on a [[sheaf]] $\mathcal{F} \in Sh_(Y_{et})$ by
$$
f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y)
\,.
$$
=--
By the discussion at _[morphisms of sites -- Relation to geometric morphisms](morphism+of+sites#RelationToGeometricMorphisms)_.
See also for instance ([Tamme I 1.4](#Tamme)).
+-- {: .num_prop #DerivedDirectImageByCohomology}
###### Proposition
The $q$th [[derived functor]] $R^q f_\ast$ of the [[direct image]] functor of def. \ref{DirectAndInverseImageAlongMapOfBases} sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]]
$$
(U_Y \to Y)
\mapsto
H^q(X \times_Y U_Y, \mathcal{F})
\,,
$$
where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$ ([[étale cohomology]]).
=--
(e.g. [Tamme, I (3.7.1), II (1.3.4)](#Tamme), [Milne, 12.1](#Milne)).
+-- {: .proof}
###### Proof
We have a [[commuting diagram]]
$$
\array{
Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y))
\\
\uparrow^{\mathrlap{inc}} && \downarrow^{L}
\\
Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y))
}
\,,
$$
where the right vertical morphism is [[sheafification]]. Because $(-) \circ f^{-1}$ and $L$ are both [[exact functors]] it follows that for $\mathcal{F} \to I^\bullet $ an [[injective resolution]] that
$$
\begin{aligned}
R^p f_\ast(\mathcal{F})
& :\simeq
H^p( f_\ast I)
\\
& = H^p(L I^\bullet(f^{-1}(-)))
\\
& = L (H^p(I^\bullet)(f^{-1}(-)))
\end{aligned}
$$
=--
+-- {: .num_remark #GrothendieckAndLeraySpectralSequence}
###### Remark
For $O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z$
two composable [[morphisms of sites]],
the [[Grothendieck spectral sequence]] for the corresponding [[direct images]] is of the form
$$
E^{p,q}_2 = R^p g_\ast(R^q f_\ast(\mathcal{F}))
\Rightarrow
E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F})
\,.
$$
For the special case that $S_Z = \ast$ and $g^{-1}$ includes an [[étale morphism of schemes|étale morphism]] $U_Y \to Y$ this yields the [[Leray spectral sequence]]
$$
E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F})
\Rightarrow
E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F})
\,.
$$
=--
## Étale cohomology
{#EtaleCohomology}
With some basic facts about [[sheaves]] on the [[étale site]] in hand, we now consider basics of [[abelian sheaf cohomology]] with [[coefficients]] in some such sheaves.
1. [With coefficients in coherent modules](#WithCoefficientsInCoherentModules)
1. [With coefficients in cyclic groups](#WithCoefficientsInACyclicGroup)
1. [With coefficients in the multiplicative group](#WithCoefficientsInTheMultiplicativeGroup)
This may serve to give a first idea of the nature of [[étale cohomology]]. An outlook on the deep structurual theorems about [[étale cohomology]] is in the next section [below](#MainTheorems).
### With coefficients in coherent modules
{#WithCoefficientsInCoherentModules}
+-- {: .num_prop }
###### Proposition
For $X$ a [[scheme]] and $N$ a ([[flat module|flat]]) [[quasicoherent module]] over its [[structure sheaf]] $\mathcal{O}_X$, then this induces an [[abelian sheaf]] on the [[étale site]] by
$$
N_{et} \;\colon\; (U_X \to X)
\mapsto
\Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_Y})
\,.
$$
=--
(e.g. [Tamme, II 3.2.1](#Tamme))
+-- {: .proof}
###### Proof
By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers}
it is sufficient to test the [[sheaf]] condition on open affine covers
and on singleton covers by faithfully flat morphisms of affines.
For the first case we have a sheaf since this is just the sheaf condition in the [[Zariski topology]]. For the second case the corresponding Cech complexes are the [[Amitsur complexes]] of a faithfully flat $A \to B$ [[tensor product|tensored]] with $N$. By the [[descent theorem]], prop. \ref{DescentTheorem} this is exact, hence verifies the sheaf condition.
=--
We consider now the étale [[abelian sheaf cohomology]] with coefficients in such coherent modules.
+-- {: .num_remark }
###### Remark
A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every étale [[sheaf]] is a Zarsiki sheaf, but not necessarily conversely.
=--
+-- {: .num_remark #LerayForInclusionOfZariskiIntoEtale}
###### Remark
For $X$ a [[scheme]], the inclusion
$$
\epsilon \;\colon\; X_{Zar} \longrightarrow X_{et}
$$
of the [[Zariski site]] into the [[étale site]] is indeed a [[morphism of sites]]. Hence there is a [[Leray spectral sequence]], remark \ref{GrothendieckAndLeraySpectralSequence}, which computes étale cohomology in terms of Zarsiki cohomology
$$
E^{p,q}_2 = H^p(X_{Zar}, R^q \epsilon^\ast \mathcal{F})
\Rightarrow
E^{p+q} = H^{p+q}(X_{et}, \mathcal{F})
\,.
$$
=--
This is originally due to ([[Grothendieck]], [[SGA]] 4 (Chapter VII, p355)). Reviews include ([Tamme, II 1.3](#Tamme)).
+-- {: .num_prop #CohomologyWithCoeffsInCoherentModules}
###### Proposition
For $N$ a [[quasi-coherent sheaf]] of $\mathcal{O}_X$-[[modules]] and $N_{et}$ the induced étale sheaf (by the discussion at [étale topos -- Quasicohetent sheaves](etale+topos#QuasiCoherentModules)),
then the [[edge morphism]]
$$
H^p_{Zar}(X, N)
\longrightarrow
H^p_{et}(X,N_{et})
$$
of the [[Leray spectral sequence]] of remark \ref{LerayForInclusionOfZariskiIntoEtale} is an [[isomorphism]] for all $p$, identifying the [[abelian sheaf cohomology]] on the [[Zariski site]] with [[coefficients]] in $N$ with the étale cohomology with coefficients in $N_{et}$.
Moreover, for $X$ affine we have
$$
H^p_{et}(X, N_{et}) \simeq 0
\,.
$$
=--
This is due to ([[Grothendieck]], [[FGA]] 1). See also for instance ([Tamme, II (4.1.2)](#Tamme)).
+-- {: .proof}
###### Proof
By the discussion at _[[edge morphism]]_ it suffices to show that
$$
R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0
\,.
$$
By prop \ref{DerivedDirectImageByCohomology}, $R^q \epsilon_\ast N$ is the [[sheaf]] on the [[Zariski topology]] which is the [[sheafification]] of the [[presheaf]] given by
$$
U \mapsto H^q(X_{et}|U, N)
\,,
$$
hence it is sufficient that this vanishes, or rather, by locality ([[sheafification]]) it suffices to show this vanishes for $X = U = Spec(A)$ an affine [[algebraic variety]].
By the existence of [cofinal affine étale covers](etale+site#CofinalAffineCovers) the [[full subcategory]] $X_{et}^{a} \hookrightarrow X_{at}$ on the étale maps with affien domains, equipped with the induced [[coverage]], is a [[dense subsite]]. Therefore it suffices to show the statement there.
Moreover, by the finiteness condition on [[étale morphisms]] every cover of $X_{et}^{a}$ may be refined by a finite cover, hence by an affine covering map
$$
Spec(B) \longrightarrow Spec(A)
\,.
$$
It follows (by a discussion such as e.g. at [[Sweedler coring]]) that the corresponding [[Cech cohomology]] complex
$$
N_{et}(Spec(A)) \to C^0(\{Spec(B) \to Spec(A)\}, N_{et}) \to C^1(\{Spec(B) \to Spec(A)\}, N_{et}) \to \cdots
$$
is of the form
$$
0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots
\,.
$$
known as the _[[Amitsur complex]]_ of $A \to B$, tensored with $N$.
Since $A \to B$ is a [[faithfully flat morphism]], it follows again by the [[descent theorem]], prop. \ref{DescentTheorem} that this is [[exact sequence|exact]], hence that the cohomology indeed vanishes.
=--
### With coefficients in cyclic groups
{#WithCoefficientsInACyclicGroup}
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).
=--
+-- {: .num_prop}
###### Proposition
If $X = Spec(A)$ is an affine [[reduced scheme]] of [[characteristic]] a [[prime number]] $p$, then its [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/p\mathbb{Z}$ is
$$
H^q(X, (\mathbb{Z}/p\mathbb{Z})_X)
\simeq
\left\{
\array{
A/(F - id)A & if\; q = 1
\\
0 & if \; q \gt 0
}
\right.
\,.
$$
=--
+-- {: .proof}
###### Proof
Under the given assumptions, the [[Artin-Schreier sequence]] (see there) induces a [[long exact sequence in cohomology]] of the form
$$
\begin{aligned}
0 & \to
H^0(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^0(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^0(X_{et}, \mathcal{O}_X)
\\
& \to
H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^1(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^1(X_{et}, \mathcal{O}_X)
\\
& \to
H^2(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^2(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^2(X_{et}, \mathcal{O}_X) \to \cdots
\end{aligned}
\,,
$$
where $F(-) = (-)^p$ is the [[Frobenius endomorphism]].
By prop. \ref{CohomologyWithCoeffsInCoherentModules}
the terms of the form $H^{p \geq 1}(X, \mathcal{O}_X)$ vanish,
and so from [[exact sequence|exactness]] we find an [[isomorphism]]
$$
H^0(X_{et}, \mathcal{O}_X)/(F-id)(H^0(X_{et}, \mathcal{O}_X))
\stackrel{\simeq}{\to}
H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
\,,
$$
hence the claimed isomorphism
$$
A/(F-id)(A)
\stackrel{\simeq}{\to}
H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
\,.
$$
By the same argument all the higher cohomology groups vanish, as claimed.
=--
### With coefficients in the multiplicative group
{#WithCoefficientsInTheMultiplicativeGroup}
the étale cohomology groups with [[coefficients]] in the [[multiplicative group]] $\mathbb{G}_m$ in the first few degrees go by special names:
* $H^0_{et}(-, \mathbb{G}_m)$: [[group of units]];
* $H^1_{et}(-, \mathbb{G}_m)$: [[Picard group]] ([[Hilbert's theorem 90]], [Tamme, II 4.3.1](#Tamme));
* $H^2_{et}(-, \mathbb{G}_m)$: [[Brauer group]];
## Outlook: The main theorems
{#MainTheorems}
What makes [[étale cohomology]] interesting in a broader context is that is verifies a collection of good structural theorems, which we just list now. In their totality these properties make [[étale cohomology]] (in its incarnation as [[ℓ-adic cohomology]]) qualify as a [[Weil cohomology theory]]. This in turn means that using [[étale cohomology]] one can give a [[proof]] of the [[Weil conjectures]] -- a number of [[conjectures]] about properties of the numbers of points in [[algebraic varieties]], hence of the numbers of solutions to certain [[polynomial]] [[equations]] over certain [[rings]] -- , and this was historically a central motivation for introducing [[étale cohomology]] in the first place.
These theorems are
1. [[proper base change theorem]] ([Milne, section 17](#Milne))
1. [[comparison theorem (étale cohomology)]] ([Milne, section 21](#Milne))
1. [[Künneth formula]] ([Milne, section 22](#Milne))
1. cycle map theorem ([Milne, section 23](#Milne))
1. [[Poincaré duality]] ([Milne, section 24](#Milne))
Together these imply the central ingredient for a proof of the [[Weil conjectures]], a Lefschetz fixed-point formula
+ [Künneth formula](#KünnethFormula) + [cycle map](#CycleMap) + [Poincaré duality](#PoincareDuality) $\Rightarrow$ [[Lefschetz fixed-point formula]] ([Milne, section 25](#Milne))
For more on this see... elsewhere.
## References
{#References}
* [[Günter Tamme]], _[[Introduction to Étale Cohomology]]_
{#Tamme}
* [[James Milne]], _[[Lectures on Étale Cohomology]]_
{#Milne}
* [[The Stacks Project]], _Étale cohomology_ ([pdf](http://stacks.math.columbia.edu/download/etale-cohomology.pdf))
{#StackProject}
[[!redirects basics of étale cohomology]]
|
basis
|
https://ncatlab.org/nlab/source/basis
|
There are different notions in mathematics called _basis_. Generally speaking, these fall into two classes:
* situations where the term _[[base]]_ is also used:
* [[topological base]]
* [[basis for a Grothendieck topology]],
* etc ...;
* the generating [[set]] of a [[free object]]:
* [[basis of a free module]] (see also [[dual basis]] of a [[projective module]])
* [[basis of a vector space]]
* [[basis in functional analysis]] (which is applied more generally than to free objects, but analogously)
* etc ....
Note that the plurals of 'basis' and 'base' are both spelt 'bases' (although their pronunciations differ).
[[!redirects basis]]
[[!redirects bases]]
[[!redirects basises]]
[[!redirects basis element]]
[[!redirects basis elements]]
|
basis for a topology > history
|
https://ncatlab.org/nlab/source/basis+for+a+topology+%3E+history
|
< [[basis for a topology]]
|
basis in functional analysis
|
https://ncatlab.org/nlab/source/basis+in+functional+analysis
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--
# Concepts of basis in functional analysis
* table of contents
{: toc}
## Idea
A _basis in functional analysis_ is a [[linear basis]] that is compatible with the [[topological space|topology]] of the underlying [[topological vector space]]. Therefore this is sometimes also referred to as a "topological basis", but beware that this term is also used for referring to the unrelated concept of a "[[basis for the topology]]".
[[basis|Bases]] in [[linear algebra]] are extremely useful tools for analysing problems. Using a basis, one can often rephrase a complicated abstract problem in concrete terms, perhaps even suitable for a computer to work with. A basis provides a way of describing a [[vector space]] in a way that:
1. Is complete: every point in the space can be described in this fashion.
2. Has no redundancies: the description of a point is unique.
When translated into the language of linear algebra, we recover the key properties of a basis: that it be a spanning set and linearly independent.
In infinite dimensions, having a basis becomes more valuable as the spaces get more complicated. However, the notion of a basis also becomes complex because the question of what makes a description admits different answers depending on whether we want only finite sums, we allow sequences, or we want infinite sums.
## Definitions
+-- {: .num_defn #basis}
Let $V$ be a [[topological vector space]] and $B \subseteq V$ a subset.
1. We say that $B$ is a **[[Hamel basis]]** if:
1. Every element of $v$ is a finite linear combination of elements of $B$,
2. If $v = \sum_{b \in B} \alpha_b b$ then the $\alpha_b$ are unique.
Alternatively, $B$ is [[linearly independent subset|linearly independent]] and $\Span(B) = V$; in other words, the [[span]] of $B$ is $V$ but no [[proper subset]] of $B$ has this property.
2. We say that $B$ is a **topological basis** if:
1. Every element $v \in V$ is a limit of a [[sequence]] or (more generally) a [[net]] of finite linear combinations of elements of $B$,
2. No element of $B$ is a limit of a sequence or net of finite linear combinations of the _other_ elements of $B$.
Alternatively, $B$ is [[total subset|total]] (meaning that its span is [[dense subspace|dense]]) but no proper subset of $B$ is total.
3. We say that $B$ is a **Schauder basis** if:
1. Every element of $v$ is a (possibly infinite) sum of scales of elements of $B$,
2. If $v = \sum_{b \in B} \alpha_b b$ then the $\alpha_b$ are unique.
=--
## Properties
1. In the presence of the [[axiom of choice]], Hamel bases always exist.
2. If $B$ is a topological basis, then $B$ has a dual basis. Since $B \setminus \{b\}$ is not total but $B$ is total, the closure of the span of $B \setminus \{b\}$ must be a codimension $1$ subspace, whence the kernel of a non-trivial continuous linear functional on $V$, say $f_b$. By scaling, this functional can be assumed to satisfy $f_b(b) = 1$. Since $B \setminus \{b\} \subseteq \ker f$, $f(b') = 0$ for all $b' \in B$, $b' \ne b$.
3. If $B$ is a Schauder basis then it is a topological basis and so, as mentioned, has a dual basis. Then the coefficients in the sum $v = \sum \alpha_b b$ must be given by evaluating the dual basis on $v$: $v = \sum f_b(v) b$.
## Examples
1. In $C([0,1],\mathbb{C})$ with the norm ${\|f\|} = \max\{{|f(t)|}\}$:
1. The monomials are linearly independent and have dense span, but do not form a topological basis as there is a sequence of polynomials with no linear term converging to $t$.
2. The trigonometric polynomials do form a topological basis. The dual basis is given by taking the Fourier coefficients of a function. However, it is not a Schauder basis as there are continuous functions which are not the uniform limit of their Fourier series.
3. The following is a Schauder basis. Let $(d_n)$ be the sequence $\{0, 1, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \dots\}$. Define $f_n$ to be the piecewise-linear function with the property that: $f_n(d_n) = 1$ and $f_n(d_k) = 0$ for $k \lt n$, and $f_n$ has the least "breaks". Then $f_n$ forms a Schauder basis for $C([0,1],\mathbb{C})$. This is the classical _Faber-Schader_ basis.
## References
* Enflo, P. (1973). A counterexample to the approximation problem in Banach spaces. _Acta Math._, _130_, 309--317.
* Semadeni, Z. (1982). _Schauder bases in Banach spaces of continuous functions_ (Vol. 918). Lecture Notes in Mathematics. Berlin: Springer-Verlag.
category: functional analysis
[[!redirects basis in functional analysis]]
[[!redirects bases in functional analysis]]
[[!redirects basises in functional analysis]]
[[!redirects Schauder basis]]
[[!redirects Schauder bases]]
[[!redirects Schauder basises]]
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basis of a free module
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https://ncatlab.org/nlab/source/basis+of+a+free+module
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+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# Bases of a free module
* table of contents
{: toc}
## Definition
A **basis** of a [[free module|free]] $R$-[[module]] $M$ (possibly a [[vector space]], see [[basis of a vector space]]) is a linear [[isomorphism]] $B\colon M \to \oplus_{i\in I}R$ to a [[direct sum]] of copies of the ring $R$, regarded as a module over itself.
We see how this is equivalent to the classical definition of a basis as a linearly independent spanning set:
+-- {: .num_lemma}
###### Lemma
A basis for a free $R$-module $M$ determines a unique generating set for $M$ of [[linear independence|linearly independent]] elements of $M$.
=--
+-- {: .proof}
###### Proof
Fix a basis $B$ for $M$ over $R$. Then let $a_i\coloneqq (\delta_{ij})_{j\in I}$ for each $i\in I$. Since $B$ is an isomorphism, each $a_i$ determines a unique element $b_i \coloneqq B^{-1}(a_i)$. Since every element of $M$ is of the form $B^{-1}(x)$ for $x\in \oplus_{i\in I}R$, and since every element of $\oplus_{i\in I}R$ can be written as a finite $R$-linear combination of the $a_i$, this proves that $\{b_i\}_{i\in I}$ generates $M$. To show linear independence, we again apply $B$ and its linearity. The result is immediate.
=--
## Related concepts
* [[free module]], [[free group]], [[free abelian group]]
* [[linear independence]]
[[!redirects bases of a free module]]
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basis of a vector space
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https://ncatlab.org/nlab/source/basis+of+a+vector+space
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+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
\begin{definition}
\label{BasisOfAVectorSpace}
For $\mathbb{K}$ a [[field]] and $V$ a $\mathbb{K}$-[[vector space]], hence a $\mathbb{K}$-[[module]], a *[[linear basis]]* for $V$ is
1. a [[set]] $B$ (of *basis elements*)
1. a $\mathbb{K}$-[[linear map|linear]] [[isomorphism]]
$$
\underset{B}{\oplus}
\mathbb{K}
\xrightarrow{\;\; \simeq \;\;}
V
$$
to $V$ from the $B$-indexed [[direct sum]] (the [[coproduct]] in $\mathbb{K}$[[Vect]]) of copies of $\mathbb{K}$ (canonically regarded as a vector space over itself), also known as the free $\mathbb{K}$-[[linear span]] of $B$, hence the vector space of free $\mathbb{K}$-[[linear combinations]] of elements of $B$.
\end{definition}
Hence if a basis for $V$ exists it means in particular that $V$ is a [[free module]] over $\mathbb{K}$.
\begin{remark}
**(finiteness of linear combinations)**
\linebreak
Beware that a certain finiteness-condition is hidden in Def. \ref{BasisOfAVectorSpace}: Since a [[linear combination]] is defined to be a [[sum]] of [[finite set|finitely]] many vectors, a basis of a vector space must be such that every vector in the space is the (unique) combination of *finitely* many basis elements -- even if there are infinitely many elements in the basis.
More abstractly this is to do with the appearance of the [[coproduct]] $\amalg_b = \oplus_B$ ([[direct sum]]) in Def. \ref{BasisOfAVectorSpace} instead of the [[product]] $\prod_B$. Many vector spaces in practice arise as (subspaces) of products $\prod_W \mathbb{K}$ ([[function spaces]] $W \to \mathbb{K}$), but if $W$ here is not a [[finite set]] then it is not going to be a basis set.
(On the other hand, if $W$ *is* a [[finite set]], then we have a *[[biproduct]]* $\prod_W \,\simeq\, \coprod_W \,\simeq\, \oplus_W$ in the [[additive category]] $\mathbb{K}$[[Vect]], see [there](additive+category#ProductsAreBiproducts).)
\end{remark}
Related to this are the following phenomena:
\begin{remark}
**(basis and dimension of finitely-generated spaces)**
\linebreak
For every *finitely generated* vector space $V$ (Def. \ref{GeneratedVectorSpace}) it is straightforward to *[[constructive mathematics|construct]]* a linear basis, and to see that the [[cardinality]] of all bases is the same finite [[natural number]] $dim(V) \in \mathbb{N}$, called the *[[dimension of a vector space|dimension]]* of the vector space (whence a *[[finite-dimensional vector space]]*).
\end{remark}
On the other hand:
\begin{remark}
**(Hamel-bases of infinite-dimensional vector spaces)**
\linebreak
While the definition \ref{BasisOfAVectorSpace} applies also to not-necessarily finitely generated vector spaces -- such as for instance the space of ([[continuous function|continuous]]) [[functions]] from a non-finite ([[topological space|topological]]) [[space]] to the ([[topological field|topological]]) [[ground field]] -- it turns out to be subtle and somewhat ill-behaved in this generality.
In fact, in practice infinite-dimensional vector spaces tend to appear and to be understood with [[extra structure]] (typically that of [[topological vector spaces]] such as [[Banach spaces]] or [[Hilbert spaces]]) in which cases there are more appropriate notions of linear bases for them (such as that of [[Schauder bases]], which allow infinite-[[linear combinations]] subject to a condition of [[convergence of a sequence]]).
In order to distinguish the plain notion of basis (Def. \ref{BasisOfAVectorSpace}) from these more refined notions, one also speaks of *Hamel bases* here.
(This is in honor of [Hamel 1905 pp. 460](#Hamel1905) who considered this notion for the special case of $V = \mathbb{R}$ the [[real numbers]] regarded as a [[rational vector space]], hence over the [[ground field]] $\mathbb{K} = \mathbb{Q}$ of [[rational numbers]]).
Hence, in principle, also a linear basis of a finitely generated vector space is thus a *Hamel basis*, but rarely called this way unless in the context of infinite-dimensional vector spaces.
\end{remark}
\begin{remark}
**(basis theorem and dimension)**
\linebreak
For an infinitely-generated vector space it is *not* in general possible to *[[constructive mathematics|construct]]* a (Hamel-)basis, but the *existence* of such a basis is nevertheless implied, in [[classical mathematics]], by [[Zorn's lemma]] (essentially a form of the [[axiom of choice]]): This is the content of the *[[basis theorem]]*.
With this classical context understood, it follows that every vector vector space admits a linear basis (even if non-constructible in general) and that each basis is of the same [[cardinality]], then called the [[dimension of a vector space|dimension]] of the vector space.
\end{remark}
## Background definitions
\begin{definition}\label{GeneratedVectorSpace}
**(generated vector space)**
\linebreak
For $V$ a vector space, a [[subset]] $G \subset V$ of its [[underlying]] set is called a *generating set* or *spanning set* if every element of $V$ can be expressed as a [[linear combination]] of elements of $G$ in $V$, hence if the [[linear span]] of $G$ inside $V$ is all of $V$.
The vector space $V$ is called *finitely generated* if it admits a generating set (spanning set) which is a [[finite set]].
\end{definition}
## Related concepts
* [[orthogonal basis]]
* [[basis in functional analysis]]
* [[Schauder basis]]
* [[dual basis]]
* [[basis]]
* [[mutually unbiased bases]]
* [[Gelfand-Tsetlin basis]] (in [[representation theory]])
* [[seminormal basis]] (in [[representation theory of the symmetric group]])
## References
Lecture notes with much conceptual exposition:
* Karen E Smith, *Bases for infinite-dimensional vector spaces* [[pdf](https://dept.math.lsa.umich.edu/~kesmith/infinite.pdf), [[Smith-InfiniteDimBases.pdf:file]]]
Lecture notes with the proofs concisely spelled out:
* [[Christoph Schweigert]], *Basis und Dimension*, §2.4 in: *Lineare Algebra*, lecture notes, Hamburg (2022) [[pdf](https://www.math.uni-hamburg.de/home/schweigert/skripten/laskript.pdf)]
See also:
* Wikipedia, *<a href="https://en.wikipedia.org/wiki/Basis_(linear_algebra)">Basis (linear algebra)</a>*
The original discussion (for [[real numbers|$\mathbb{R}$]] regarded as a [[rational vector space]]) after which *Hamel bases* are named:
* {#Hamel1905} [[Georg Hamel]], pp. 460 of: *Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x + y) = f(x) + f(y)$*, Mathematische Annalen **60** (1905) 459–462 [[doi:10.1007/BF01457624](https://doi.org/10.1007/BF01457624)]
[[!redirects Hamel basis]]
[[!redirects Hamel bases]]
[[!redirects Hamel basises]]
[[!redirects linear basis]]
[[!redirects linear bases]]
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basis theorem
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https://ncatlab.org/nlab/source/basis+theorem
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+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **basis theorem** for [[vector space | vector spaces]] states that every [[vector space]] $V$ admits a [[basis of a vector space|basis]], or in other words is a [[free module]] over its [[ground field]] of scalars. It is a famous classical consequence of the [[axiom of choice]] (and is equivalent to it by [Blass (1984)](#Blass84).
## Statement and proofs
+-- {: .un_theorem}
###### Basis theorem
If $V$ is a [[vector space]] over any [[field]] (or, say, a left vector space over a [[skewfield]]) $K$, then $V$ has a [[basis of a vector space|basis]].
=--
+-- {: .proof}
###### Proof
We apply [[Zorn's lemma]] as follows: consider the [[poset]] consisting of the linearly independent [[subsets]] of $V$, ordered by inclusion (so $S \leq S'$ if and only if $S \subseteq S'$). If $(S_\alpha)$ is a chain in the poset, then
$$S = \bigcup_\alpha S_\alpha$$
is an upper bound, for each $v$ in the span of $S$ belongs to some $S_\alpha$ and is uniquely expressible as a finite linear combination of elements in $S_\alpha$ and in any $S_\beta$ containing $S_\alpha$, hence uniquely expressible as a finite linear combination of elements in $S$. Thus the hypothesis of Zorn's lemma obtains for this poset; therefore this poset has a maximal element, say $B$.
Let $W$ be the span of $B$. If $W$ were a proper subspace of $V$, then for any $v$ in the set-theoretic complement of $W$, $B \cup \{v\}$ is a linearly independent set (for if
$$a_0 v + a_1 w_1 + \ldots + a_n w_n = 0 \qquad (w_1, \ldots, w_n \in B)$$
we must have $a_0 = 0$ -- else we can multiply by $1/a_0$ and express $v$ as a linear combination of the $w_i$, contradicting $\neg(v \in W)$ -- and then the remaining $a_i$ are 0 since the $w_i$ are linearly independent). This contradicts the maximality of $B$. We therefore conclude that $W = V$, and $B$ is a basis for $V$.
=--
I'll write out a proof of the converse, that the axiom of choice follows from the basis theorem, as soon as I've digested it -- Todd.
+-- {: .un_theorem}
###### Equipotence of vector space bases (Steinitz)
Any two bases of a vector space are of the same cardinality.
=--
## Generalisations
Given any linearly independent set $A$ and spanning set $C$, if $A \subseteq C$, then there is a basis $B$ with $A \subseteq B \subseteq C$; the theorem above is the special case where $A = \empty$ and $C = V$. The proof of this more general theorem is a straightforward generalisation of the proof above.
## See also
* [[vector space]]
## References
The original proof of the existence of [[Hamel bases]] (after which the concept was named) was for the case of the [[real numbers]] regarded as a [[rational vector space]] and used (not directly [[Zorn's lemma]] but) the [[well-ordering theorem]]:
* {#Hamel1905} [[Georg Hamel]], pp. 460 in: *Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x + y) = f(x) + f(y)$*, Mathematische Annalen **60** (1905) 459–462 [[doi:10.1007/BF01457624](https://doi.org/10.1007/BF01457624)]
Lecture notes:
* [[Christoph Schweigert]], *Basis und Dimension*, §2.4 in: *Lineare Algebra*, lecture notes, Hamburg (2022) [[pdf](https://www.math.uni-hamburg.de/home/schweigert/skripten/laskript.pdf)]
See also:
* {#Blass84} [[Andreas Blass]], *Existence of bases implies the axiom of choice*, Contemporary Mathematics **31** (1984) 31-33 [[doi:10.1090/conm/031](https://doi.org/10.1090/conm/031) [pdf](https://dept.math.lsa.umich.edu/~ablass/bases-AC.pdf)]
|
Bastian Rieck
|
https://ncatlab.org/nlab/source/Bastian+Rieck
|
* [personal page](https://bastian.rieck.me/)
## Selected writings
On [[topological machine learning]]:
* {#RSL} [[Bastian Rieck]], Filip Sadlo, Heike Leitte, *Topological Machine Learning with Persistence Indicator Functions*, In: *Topological Methods in Data Analysis and Visualization V* TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$[arXiv:1907.13496](https://arxiv.org/abs/1907.13496), [doi:10.1007/978-3-030-43036-8_6](https://doi.org/10.1007/978-3-030-43036-8_6)$]$
* Felix Hensel, Michael Moor, [[Bastian Rieck]], *A Survey of Topological Machine Learning Methods*, Front. Artif. Intell., **26** (2021) $[$[doi:10.3389/frai.2021.681108](https://doi.org/10.3389/frai.2021.681108), [[HenselMoorRieck-TopologicalMachineLearning.pdf:file]]$]$
category: people
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Batalin-Vilkovisky integral
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https://ncatlab.org/nlab/source/Batalin-Vilkovisky+integral
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Similarly to the idea of Wilson efective action where some degrees of freedom are integrated out, in BV-formalism one introduces BV-effective action; using a variant of integration utilizing Lagrangean submanifolds. In this procedure BV integrals are involved.
This is related to solving quantum master equation where the "induction" data are equivalent to the data involved in [[homological perturbation theory]].
* Albert Schwarz, Geometry of BV quantization, CMP 1993
* Losev
* Cattaneo, Felder 2008
[[!redirects BV integral]]
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Batalin-Vilkovisky quantization
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https://ncatlab.org/nlab/source/Batalin-Vilkovisky+quantization
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A version of Koszul type resolutions appear in attempts to do quantizations; namely a general theory leads to singularities and new ghost and antighost fields are used to remove them. The simplest such cohomological method is BRST quantization. Much more general versions, allowing more general singularities is BV-quantization introduced by physicists Batalin and Vilkovisky in late 1980s.
See [[nlab:BV theory|BV "theory" in nlab]]
[[!redirects BV quantization]]
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Batanin omega-category
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https://ncatlab.org/nlab/source/Batanin+omega-category
|
I
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _[[Michael Batanin|Batanin]] $\omega$-category_ is a [[weak ∞-category]] defined as an [[algebra over an operad|algebra]] over a suitable contractible [[globular operad]]. So this is an [[algebraic definition of higher category]].
The definition is similar to that of [[Trimble n-category]] (which is actually a special case of a Batanin $\omega$-category) and similar to the definition of [[Grothendieck-Maltsiniotis infinity-category]].
## Morphisms
When a weak $\infty$-category is modeled as a module over an $O$-operad, morphisms of modules $F : C \to D$ will correspond to _strict_ $\infty$ functors. To get weak $\infty$-functors one has to resolve $C$.
One way to do this is described in ([Garner](#Garner)).
## References
* [[Michael Batanin]], _Monoidal globular categories as a natural environment for the theory of weak $n$-categories_ , Advances in Mathematics 136 (1998), no. 1, 39–103.
* [[Ross Street]], _The role of Michael Batanin's monoidal globular categories_, in _Higher Category Theory_, eds. E. Getzler and M. Kapranov, Contemp. Math. 230, American
Mathematial Society, Providence, Rhode Island, 1998, pp. 99--116. ([pdf](https://www.researchgate.net/publication/2741768_The_role_of_Michael_Batanin's_monoidal_globular_categories/link/02bfe5108f456cad85000000/download))
Work towards establishing the [[homotopy hypothesis]] for Batanin $\omega$-groupoids can be found here:
* [[Clemens Berger]], _A cellular nerve for higher categories_, Advances in Mathematics 169, 118-175 (2002) ([pdf](http://math1.unice.fr/~cberger/nerve.pdf))
{#Berger}
* [[Denis-Charles Cisinski]], _Batanin higher groupoids and
homotopy types_ ([arXiv:math/0604442](http://arxiv.org/abs/math/0604442))
A nice introduction to this subject is:
* [[Eugenia Cheng]], _Batanin omega-groupoids and the homotopy hypothesis_, ([recorded lecture](http://www.fields.utoronto.ca/audio/06-07/highercat/cheng)) from the Fields Institute Workshop on Higher Categories and their Applications, January 10, 2007.
A discussion of weak $\omega$-functors between Batanin $\omega$-categories is in
* [[Richard Garner]], _Homomorphisms of higher categories_ ([arXiv:0810.4450](http://arxiv.org/abs/0810.4450))
An application of Batanin weak $\omega$-groupoids to [[homotopy type theory]] appears in
* [[Benno van den Berg]], [[Richard Garner]], _Types are weak $\omega$-groupoids_ ([arXiv:0812.0298](http://arxiv.org/abs/0812.0298))
A discussion of weak $\omega$-functors between Batanin $\omega$-categories, and all kind of weak $n$-transformations in the spirit of Batanin approach, with an emphasis to the possibility to the existence of the weak $\omega$-category of the weak $\omega$-categories in Batanin's sense appears in
* [[Camell Kachour]], Steps toward the weak higher category of weak higher categories in the globular setting, published in *Categories and General Algebraic Structures with Applications* (2015). ([web](https://www.researchgate.net/publication/283018471_Steps_toward_the_weak_higher_category_of_weak_higher_categories_in_the_globular_setting))
Batanin weak $\omega$-categories were further developed here:
* [[Tom Leinster]], Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series **298**, Cambridge University Press, 2004. ([arXiv:math/0305049](https://arxiv.org/abs/math/0305049))
An approach to Batanin's weak $\omega$-categories that also sets up a definition of [[Trimble n-category|Trimble infinity-category]] and a [[fundamental infinity-groupoid]] construction is here:
* [[Eugenia Cheng]] and [[Tom Leinster]], Weak ∞-categories via terminal coalgebras, ([arXiv:1212.5853](https://arxiv.org/abs/1212.5853)).
[[!redirects Batanin infinity-category]]
[[!redirects Batanin infinity-categories]]
[[!redirects Batanin weak infinity-category]]
[[!redirects Batanin infinity-categories]]
[[!redirects Batanin ∞-category]]
[[!redirects Batanin omega-categories]]
[[!redirects Batanin ∞-categories]]
[[!redirects Batanin omega-groupoid]]
[[!redirects Batanin ∞-groupoid]]
[[!redirects Batanin omega-groupoids]]
[[!redirects Batanin ∞-groupoids]]
[[!redirects Batanin weak omega-category]]
[[!redirects Batanin weak ∞-category]]
[[!redirects Batanin weak omega-categories]]
[[!redirects Batanin weak ∞-categories]]
[[!redirects Batanin weak omega-groupoid]]
[[!redirects Batanin weak ∞-groupoid]]
[[!redirects Batanin weak omega-groupoids]]
[[!redirects Batanin weak ∞-groupoids]]
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Baues > history
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https://ncatlab.org/nlab/source/Baues+%3E+history
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< [[Baues]]
[[!redirects Baues -- history]]
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Baues-Wirsching cohomology
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https://ncatlab.org/nlab/source/Baues-Wirsching+cohomology
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##The development of the cohomology of categories.
The [[cohomology of categories|cohomology of (small) categories]] went through several preliminary stages before the publication by Baues and Wirshing. In fact, the actual method and ideas that they use can be found earlier in an unpublished document by Charles Wells from 1979. That paper traces the theory back to Leech earlier in the 1970s for the related situation of cohomology of monoids. It handles non-Abelian coefficients as well as the Abelian ones that the Baues and Wirsching theory uses.
##Idea
##References
###Early related work
* Jonathan Leech, _$\mathcal{H}$-coextensions of monoids_, vol. 1, Mem. Amer. Math. Soc, no. 157, American
Mathematical Society, 1975.
* [[Charles Wells]], _Extension theories for categories (preliminary report)_, (available from [here](http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf), 1979.)
###Baues-Wirsching cohomology of small categories: Explicit references,
* [[Hans Joachim Baues]], Günther Wirsching, _Cohomology of small categories_, J. Pure Appl. Algebra 38 (1985), no. 2-3, 187--211, [doi](http://dx.doi.org/10.1016/0022-4049%2885%2990008-8), [MR87g:18013](http://www.ams.org/mathscinet-getitem?mr=87g:18013)
* Hans Joachim Baues, Winfried Dreckmann, _The cohomology of homotopy categories and the general linear group_, $K$-Theory __3__ (1989), no. 4, 307--338, [doi](http://dx.doi.org/10.1007/BF00584524), [MR91d:18008](http://www.ams.org/mathscinet-getitem?mr=91d:18008)
* H. J. Baues, [[F. Muro]], _The homotopy category of pseudofunctors and translation cohomology_, J. Pure Appl. Algebra __211__ (2007), no. 3, 821--850, [doi](http://dx.doi.org/10.1016/j.jpaa.2007.04.008) [MR2008g:18009](http://www.ams.org/mathscinet-getitem?mr=2008g:18009)
* [[Hans Joachim Baues]], _On the cohomology of categories, universal Toda brackets and homotopy pairs_, $K$-Theory __11__ (1997), no. 3, 259--285, [doi](http://dx.doi.org/10.1023/A:1007796409912), [MR98h:55020](http://www.ams.org/mathscinet-getitem?mr=98h:55020)
* Petar Pavešić, _Diagram cohomologies using categorical fibrations_, J. Pure Appl. Algebra 112 (1996), no. 1, 73--90, <a href="http://dx.doi.org/10.1016/0022-4049(95)00126-3">[doi]</a>
* [[Teimuraz Pirashvili]], _On the center and Baues-Wirsching cohomology of small categories_, Georgian Math. J. __16__ (2009), no. 1, 131--144, [journal](http://www.reference-global.com/doi/abs/10.1515/GMJ.2009.145)
* [[Mamuka Jibladze]], [[Teimuraz Pirashvili]], _Quillen cohomology and Baues-Wirsching cohomology of algebraic theories_ Cah. Topol. Géom. Différ. Catég. 47 (2006), no. 3, 163--205, [numdam](http://www.numdam.org/item?id=CTGDC_2006__47_3_163_0)
* [[Teimuraz Pirashvili]], María Julia Redondo, _Cohomology of the Grothendieck construction_ Manuscripta Math. __120__ (2006), no. 2, 151--162, [doi](http://dx.doi.org/10.1007/s00229-006-0634-1)
Relation with Andre-Quillen cohomology and $(S,O)$-cohomology of Dwyer-Kan is elucidated in
* [[Hans-Joachim Baues]], David Blanc, _Comparing cohomology obstructions_, [arxiv/1008.1712](http://arxiv.org/abs/1008.1712)
|
Baum-Connes conjecture
|
https://ncatlab.org/nlab/source/Baum-Connes+conjecture
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The Baum-Connes conjecture asserts that under suitable technical conditions, the [[operator K-theory]]/[[KK-theory]] of the [[groupoid convolution algebra]] of an [[action groupoid|action]] [[topological groupoid]] $X//G$ is equivalent to the _$G$-[[equivariant cohomology|equivariant]]_ [[topological K-theory]]/[[equivariant operator K-theory]]/[[equivariant KK-theory]] of $X$. Moreover, it says that this equivalences is exhibited by the _[[analytic assembly map]]_.
(Just the [[injective map|injectivity]] of this map is related to the [[Novikov conjecture]].)
The original version of the Baum-Connes conjecture ([Baum-Connes](#BaumConnes00)) stated for a suitable [[topological group]] $G$ that with $X = E G \simeq \ast$ the point incarnated as the $G$-[[universal principal bundle]] with its free $G$-[[action]] the [[analytic assembly map]] (the $G$-equivariant [[index]] map)
$$
K_\bullet^G(E G) \stackrel{}{\to} K_\bullet(C(\mathbf{B}G))
\,,
$$
from the $G$-[[equivariant K-theory]] of $E G$ to the [[operator K-theory]] of the [[group algebra]], hence the [[groupoid convolution algebra]] of the [[delooping]] groupoid $\mathbf{B}G \simeq \ast // G$, is an [[isomorphism]].
This Baum-Connes conjecture is known to be true for
* [[compact topological groups]],
* [[abelian groups]],
* [[Lie groups]] with finitely many [[connected components]];
* $p$-adic groups,
* adelic groups.
It is not known if the conjecture is true for all [[discrete groups]].
Later the statement was generalized ([Tu 99](#Tu99)) to more general groupoids.
In ([Kasparov 88](#Kasparov88)) the refinement of the [[analytic assembly map]] to [[equivariant cohomology|equivariant]] [[KK-theory]] is given, and called the _descent map_. This is of the form (recalled as [Blackadar, theorem 20.6.2](#Blackadar))
$$
KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)
$$
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]]).
This is an [[isomorphism]] at least for $G$ a [[compact topological group]] and restricted to [[operator K-theory]] (hence to the first argument being $\mathbb{C}$) and for $G$ a [[discrete group]] and restricted to [[K-homology]] (hence ot the second argument being $\mathbb{C}$). In this form this is the **Green-Julg theorem**, see [below](#GreenJulgTheorem).
## Theorems
{#Theorems}
### Green-Rosenberg-Julg theorem
The Green-Rosenberg-Julg theorem identifies [[equivariant K-theory]] with the [[operator K-theory]] of [[crossed product algebras]].
+-- {: .num_theorem #GreenJulgTheorem}
###### Theorem
**(Green-Julg theorem)**
Let $G$ be a [[topological group]] acting on a [[C*-algebra]] $A$.
1. If $G$ is a [[compact topological group]] then the descent map
$$
KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)
$$
is an [[isomorphism]], identifying the [[equivariant operator K-theory]] of $A$ with the ordinary [[operator K-theory]] of the [[crossed product C*-algebra]] $G \ltimes A$.
1. if $G$ is a [[discrete group]] then the descent map
$$
KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})
$$
is an [[isomorphism]], identifying the equivariant [[K-homology]] of $A$ with the ordinary [[K-homology]] of the [[crossed product C*-algebra]] $G \ltimes A$.
=--
This goes back to ([Green 82](#Green82)), ([Julg 81](#Julg81)). A [[KK-theory]]-proof is in ([Echterhoff, theorem 0.2](#Echterhoff)); a textbook account is in ([Blackadar, 11.7, 20.2.7](#Blackadar)). See also around ([Land 13, prop. 41](#Land13)).
## Related theorems
* [[Atiyah-Segal completion theorem]], [[Green-Julg theorem]]
* [[delocalized equivariant cohomology]]
[[!include Segal completion -- table]]
## References
### Introductions and surveys
Introductions and surveys include
* [[Alain Valette]], _Introduction to the Baum-Connes conjecture_ ([pdf](http://www.univ-orleans.fr/mapmo/membres/chatterji/Valette.pdf))
* [[Nigel Higson]], _The Baum-Connes conjecture_ ([pdf](http://media.cit.utexas.edu/math-grasp/Higson_supplemental_1.pdf))
* [[Paul Baum]], _The Baum-Connes conjecture, localisation of categories and quantum groups_, 2008 ([pdf](http://www.mimuw.edu.pl/~pwit/TOK/sem8/files/Baum_bcclcqg.pdf))
Textbook discussion:
* {#Blackadar} [[Bruce Blackadar]], Section 11.7 and 20.2.7 of: _[[K-Theory for Operator Algebras]]_, Cambridge University Press 1986, second ed. 1999 ([doi:10.1007/978-1-4613-9572-0](https://link.springer.com/book/10.1007/978-1-4613-9572-0), [pdf](http://wolfweb.unr.edu/homepage/bruceb/Book6.pdf))
* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))
See also
* Wikipedia, _[Baum-Connes conjecture](http://en.wikipedia.org/wiki/Baum%E2%80%93Connes_conjecture)_
### Original articles
The original article is
* [[Paul Baum]], [[Alain Connes]], _K-theory for Lie groups and foliations_, Enseign. Math. 46 (2000), 3–42.
{#BaumConnes00}
Proof of the conjecture for hyperbolic groups is in
* [[Vincent Lafforgue]], _La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques_, Journal of Noncommutative Geometry, Volume 6, Issue 1, 2012, pages 1-197 ([arXiv:1201.4653](http://arxiv.org/abs/1201.4653))
Technical subtleties are discussed in
* [[Nigel Higson]], [[Vincent Lafforgue]], [[Georges Skandalis]], _Counterexamples to the Baum-Connes conhecture_ ([pdf](http://www.personal.psu.edu/ndh2/math/Papers_files/Higson,%20Lafforgue,%20Skandalis%20-%202002%20-%20Counterexamples%20to%20the%20Baum-Connes%20conjecture.pdf))
The generalization to [[Lie groupoids]] is due to
* [[Jean-Louis Tu]], _The Baum-Connes conjecture for groupoids_, 1999 ([[JLTBaumConnesForGroupoids.pdf:file]])
{#Tu99}
Proofs for some cases are in
* [[Georges Skandalis]], [[Jean-Louis Tu]], G. Yu, _The coarse Baum-Connes conjecture and groupoids_ ([pdf](http://www.math.univ-metz.fr/~tu/publi/coarse.pdf))
[[KK-theory]] tools and the descent map are introduced in
* [[Gennady Kasparov]], _Equivariant KK-theory and the Novikov conjecture_, Inventiones Mathematicae, vol. 91, p.147, 1988([web](http://adsabs.harvard.edu/abs/1988InMat..91..147K))
{#Kasparov88}
The "Green-Julg theorem" for commutative algebra and finite group is due to [[Michael Atiyah]], for commutative algebra and general group due to
* P. Green, _Equivariant if-theory and crossed product C*- algebras_, pp. 337-338 in Operator algebras and applications (Kingston, Ont., 1980), vol. 1, edited
by R. V. Kadison, Proc. Sympos. Pure Math. 38, Amer. Math. Soc, Providence, 1982.
{#Green82}
and the general case is due to an unpublished result by Green and [[Jonathan Rosenberg]] and independently due to
* P. Julg, _K-theorie equivariante et produits croises_, C. R. Acad. Sci. Paris Ser. I Math. 292:13 1981, 629-632.
{#Julg81}
Further discussion is in
* Siegfried Echterhoff, _The Green-Julg theorem_ [pdf](http://wwwmath.uni-muenster.de/u/paravici/Focused-Semester/lecturenotes/green-julg-Echterhoff.pdf)
{#Echterhoff}
* Walther Paravicini, _A generalised Green-Julg theorem for proper groupoids and Banach algebras_, ([arXiv:0902.4365](http://arxiv.org/abs/0902.4365))
* V. Lafforgue, [pdf](http://www.mmas.univ-metz.fr/~gnc/bibliographie/HarmonicAnalysis/LafforgueICM.pdf)
A modification of the Baum-Connes conjecture with coefficient where many counterexamples (to the conjecture with coefficients) are eliminated is in
* [[Paul Baum]], Erik Guentner, Rufus Willett, _Expanders, exact crossed products, and the Baum-Connes conjecture_, [arxiv/1311.2343](http://arxiv.org/abs/1311.2343)
Discussion in terms of [[localization]]/[[homotopy theory]] is in
* [[Ralf Meyer]], [[Ryszard Nest]], _The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259.
[[!redirects Green-Julg theorem]]
|
Baum-Douglas geometric cycle
|
https://ncatlab.org/nlab/source/Baum-Douglas+geometric+cycle
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _Baum-Douglas geometric cycle_ on a given [[manifold]] $X$ is a representative for [[K-homology]] classes on $X$. It is given by a [[submanifold]] $Q \hookrightarrow X$ equipped with [[spin^c structure]] and with a [[complex vector bundle]]. The [[equivalence relation]] identifying such data that represents the same [[K-homology]] class includes a compatible [[bordism]] relation.
Viewed as a [[correspondence]] of the form
$$
X \stackrel{}{\leftarrow} (Q,E) \to \ast
$$
a Baum-Douglas geometric cycle is a special case of the spans that represent classes in [[KK-theory]] (between manifolds) according to ([Connes-Skandalis 84, section 3](#ConnesSkandalis84)).
In [[string theory]] Baum-Douglas cycles constitute one formalization of the concept of [[D-brane]] carrying a [[Chan-Paton gauge field]] ([Reis-Szabo 05](#ReisSzabo05), [Szabo 08](#Szabo08)): the submanifold $Q$ represents the [[worldvolume]] of the D-brane and the [[complex vector bundle]] it carries the [[Chan-Paton gauge field]].
## Related concepts
* [[motivic function]]
## References
The original articles:
* [[Paul Baum]], R. Douglas, *K-homology and index theory*, in: R. Kadison (ed.), *Operator Algebras and Applications, Proceedings of Symposia in Pure Math. **38** AMS (1982) 117-173 [[ams:pspum-38-1](https://bookstore.ams.org/pspum-38-1)]
* [[Paul Baum]], R. Douglas. _Index theory, bordism, and K-homology_, Contemp. Math. **10** (1982) 1-31
A generalization to [[twisted K-theory|twisted]] homology is discussed in section of
* [[Bai-Ling Wang]], _Geometric cycles, index theory and twisted K-homology_, Journal of Noncommutative Geometry ([arXiv:0710.1625](http://arxiv.org/abs/0710.1625))
A generalization to geometric (co)-cocycles for KK-theory is in section 3 of
* {#ConnesSkandalis84} [[Alain Connes]], [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984) ([pdf](http://www.alainconnes.org/docs/longitudinal.pdf))
More generally, a construction of general [[homology theories]] in a similar fashion is discussed in
* S. Buoncristiano, C. P. Rourke and B. J. Sanderson, _A geometric approach to homology theory_, Cambridge Univ. Press, Cambridge, Mass. (1976)
and a construction of [[bivariant cohomology theories]] in this spirit is in
* Martin Jakob, _Bivariant theories for smooth manifolds_, Applied Categorical Structures 10 no. 3 (2002)
The interpretation as a formalization of [[D-branes]] in [[string theory]] is highlighted in
* {#ReisSzabo05} [[Rui Reis]], [[Richard Szabo]], _Geometric K-Homology of Flat D-Branes_ ,Commun.Math.Phys. 266 (2006) 71-122 ([arXiv:hep-th/0507043](https://arxiv.org/abs/hep-th/0507043))
* {#Szabo08} [[Richard Szabo]], _D-branes and bivariant K-theory_, Noncommutative Geometry and Physics 3 1 (2013): 131. ([arXiv:0809.3029](http://arxiv.org/abs/0809.3029))
[[!redirects Baum-Douglas geometric cycles]]
|
Baum-Douglas geometric cycles
|
https://ncatlab.org/nlab/source/Baum-Douglas+geometric+cycles
|
#Contents#
* table of contents
{:toc}
## Idea
For a given [[manifold]] $X$ a _Baum-Douglas geoemtric cycle_ on $X$ is data consisting of a [[submanifold]] $Q \hookrightarrow X$ carrying a [[vector bundle]] $E\to X$ such that this represents a class in [[K-homology]] under a suitable [[equivalence relation]].
Viewed as a [[correspondence]] of the form
$$
X \stackrel{}{\leftarrow} (Q,E) \to \ast
$$
a Baum-Douglas geometric cycle is a special case of the spans that represent classes in [[KK-theory]] (between manifolds) according to ([Connes-Skandalis 84, section 3](#ConnesSkandalis84)).
## References
The general idea of geometric cycles for [[homology theories]] goes back to
* [[William Fulton]], [[Robert MacPherson]], _Categorical framework for the study of singular spaces_, Memoirs of the AMS, 243, 1981
The original articles by Baum and Douglas are
* [[Paul Baum]], R. Douglas, _K-homology and index theory: Operator Algebras and Applications (R. Kadison editor), volume 38 of Proceedings of Symposia in Pure Math., 117-173,
Providence RI, 1982. AMS.
* [[Paul Baum]], R. Douglas. _Index theory, bordism, and K-homology_, Contemp. Math. 10: 1-31 1982.
The proof that these geometric cycles indeed model all of [[K-homology]] is due to
* [[Paul Baum]], [[Nigel Higson]], [[Thomas Schick]], _On the Equivalence of Geometric and Analytic K-Homology_ ([arXiv:math/0701484](http://arxiv.org/abs/math/0701484))
* Jeff Raven, _An equivariant bivariant Chern character_, PhD Thesis
A generalization to geometric (co)-cocycles for [[KK-theory]] is in section 3 of
* [[Alain Connes]], [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984) ([pdf](http://www.alainconnes.org/docs/longitudinal.pdf))
{#ConnesSkandalis84}
and for [[equivariant KK-theory]] in
* [[Heath Emerson]], [[Ralf Meyer]], _Bivariant K-theory via correspondences_, Adv. Math. 225 (2010), 2883-2919 ([arXiv:0812.4949](http://arxiv.org/abs/0812.4949))
For further such generalizations see at [[bivariant cohomology theory]].
Generalization to [[twisted K-homology]] is in
* [[Bai-Ling Wang]], _Geometric cycles, index theory and twisted K-homology_ ([arXiv:0710.1625](http://arxiv.org/abs/0710.1625))
[[!redirects Baum-Douglas geometric cycles]]
|
Bayes rule
|
https://ncatlab.org/nlab/source/Bayes+rule
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
#Bayes rule#
* table of contents
{:toc}
## Statement
In [[probability theory]], the __Bayes Rule__ (or _Bayes\'s Law_, _Bayes\' Theorem_, or another permutation) is the statement that the [[conditional probabilities]] $P(H\vert E)$ for an event $H$, assuming an event $E$ is related to the condition probability $P(E\vert H)$ of $E$ assuming $H$, and the plain probabilities $P(H)$ and $P(E)$ for $H$ and $E$ separately, by
$$
P(H|E) \;=\; \frac{P(E|H) P(H)} {P(E)}
\,.
$$
This follows directly from the defining formula $P(A|B) = P(A \wedge B)/P(B)$ for [[conditional probability]]. The rule may also be written in the expanded form
$$ P(H|E) = \frac{P(E|H) P(H)} {P(E|H) P(H) - P(E|\neg{H}) P(H) + P(E|\neg{H})} ,$$
which additionally uses some of the axioms of probability, or somewhere in between these two forms.
As a theorem, it is quite trivial; the point is in its application as a rule for updating the probability of some hypothesis ($H$) on the basis of some evidence ($E$) (which is key to [[Bayesianism]]), using the _prior_ probability of the hypothesis before the evidence is obtained ($P(H)$) and (in the expanded form) the conditional probabilities of obtaining that evidence in the situation where the hypothesis is true ($P(E|H)$) and in the situation where the hypothesis is false ($P(E|\neg{H})$). (Ideally, the last two can be determined on a purely theoretical basis, but since the probability of $E$ usually depends on other hypotheses of unknown veracity, the application is not always so simple.)
## Applications
### In quantum physics
In [[quantum mechanics]], the [[collapse of the wavefunction]] may be seen as a generalization of Bayes\'s Rule to [[quantum probability theory]]. This is key to the [[Bayesian interpretation of quantum mechanics]].
[[!redirects Bayes Rule]]
[[!redirects Bayes rule]]
[[!redirects Bayes' Rule]]
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[[!redirects Bayes Law]]
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[[!redirects Bayes Theorem]]
[[!redirects Bayes theorem]]
[[!redirects Bayes' Theorem]]
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[[!redirects Bayes's Theorem]]
[[!redirects Bayes's theorem]]
[[!redirects Bayes' formula]]
[[!redirects Bayes' formulas]]
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[[!redirects Bayes formulas]]
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[[!redirects Bayes's formulas]]
|
Bayesian interpretation of quantum mechanics
|
https://ncatlab.org/nlab/source/Bayesian+interpretation+of+quantum+mechanics
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
# The Bayesian interpretation of quantum physics
* table of contents
{: toc}
## Idea
[[mathematics|Mathematically]], [[quantum mechanics]], and in particular [[quantum statistical mechanics]], can be viewed as a generalization of [[probability theory]], that is as _[[quantum probability theory]]_. The [[Bayesianism|Bayesian]] interpretation of probability can then be generalized to a Bayesian [[interpretation of quantum mechanics]], and thus (in principle) of all [[physics]].
The Bayesian interpretation is founded on these principles:
1. Quantum states (pure or mixed) are analogous to (indeed generalizations of) probability distributions, which are to be interpreted in a Bayesian way, as indicating knowledge, belief, etc.
2. Time evolution of pure quantum states by the time-dependent [[Schroedinger equation|Schrödinger's equation]] is analogous to evolution of classical statistical systems by [[Liouville's equation]] (and von Neumann has an equation for the evolution of [[density matrices]] that generalizes both).
3. [[wave function collapse|Collapse of the wave function]] (see also [[propositions as projections]]) is analogous to (indeed a generalization of) updating a probability distribution; [[Born rule|Born's Rule]] and [[Bayes rule|Bayes' Rule]] are analogues.
One should perhaps speak of *a* Bayesian interpretation of quantum mechanics, since there are different forms of Bayesianism. (This article has been written by an objective Bayesian who naturally thinks of Bayesian probabilities as reflecting information rather than belief, betting commitments, etc. For other flavours of Bayesianism, substitute 'knowledge' below with a more appropriate term.)
## Formalism
There are various ways to formulate [[quantum mechanics]] mathematically, but the following elements are fairly common:
* There is an algebra $A$ of [[observables]]; some or all of the elements of this algebra correspond to physically observable quantities.
* There is a space $S$ of [[states]], by which we mean [[mixed states]] in general.
* Given a state $\psi$ and an observable $O$, we obtain a [[probability distribution]] (a [[probability measure]] on the [[real line]] or something like it).
Roughly speaking, this probability distribution tells us the probability of observing the value of $O$ to take particular values given that the system has state $\psi$. Let us write $O_*\psi(E)$ for the probability that the value of $O$ will be observed to belong to the event $E$ (which is a [[measurable set]] of real numbers when $O_*\psi$ is a probability measure on the real line) for a system in state $\psi$.
So for example, we could actually be talking about a [[classical physics|classical]] system, in which we have a [[state space]] or [[phase space]] $X$, an observable is a function on $X$, a [[pure state]] is a point in $X$, and a general state is a probability measure on $X$.
Or we could be talking about a quantum system given by a [[Hilbert space]] $H$, in which an observable is a [[self-adjoint operator]] on $H$, a pure state is a $1$-dimensional subspace of $H$, and a general state is a [[density matrix]] on $H$.
Or we could use a more abstract formulation, such as those of [[AQFT]], [[FQFT]], and so forth; but all of these still have observables and states.
See also [[JBW-algebraic quantum mechanics]] for a mathematical formalism that may be motivated by treating the probability distributions $O_*\psi$ as fundamental (among other considerations).
## Interpretation
The point of the Bayesian interpretation is that the probabilities $O_*\psi(E)$ are to be interpreted as Bayesian probabilities. That is, they are not objective features of reality (the territory) but rather an expression of what a rational observer might know about the world (a map).
Thus, a state $\psi$ represents a _state of knowledge_ about the world. The algebra $A$ of observables may be objective (or fully known), but the particular state $\psi$ that the system is in depends on what is known about that particular system. There is (except in the classical case) *no* possible complete specification of the actual values of all observables, only a probabilistic specification of what one might know about them.
In the [[Schroedinger picture]] (assuming a notion of [[time]]), states [[time evolution|evolve]] deterministically, unitarily, and with conservation of [[entropy]]. (Or in the [[Heisenberg picture]], they don\'t evolve at all, with the observables evolving instead.) But a state may change otherwise, if one\'s knowledge changes. If this is an increase in knowledge as a result of a [[measurement]], then this change in the state may be called the 'collapse of the wavefunction'. But this collapse takes place in the map, not the territory; unlike time evolution, it is not a physical process.
Given a specification of $S$ and $A$, it may be that there exist certain states $\psi$ in $S$ such that $O_*\psi$ is a [[delta measure]] (giving a probability of $1$ for one value and $0$ for all others) for every $O$ in $A$. Then the system is classical. Depending on the mathematical formalism used, such states may not actually belong to $S$ (which might, for example, own only the [[absolutely continuous measure|absolutely continuous]] probability measures in some sense, as is natural in the [[JBW-algebraic quantum mechanics|W*-algebraic approach]]); so in general, we may say that the system is __classical__ if there exists a [[net]] $(\psi_n)_n$ of states such that, for each observable $O$, the net $(O_*\psi_n)_n$ of measures on the real line [[convergence in measure|converges in measure]] to a delta measure. (Or so I imagine; this definition might not really be general enough.)
## History
People have implied (for example at the beginning of [Caves, Fuchs, & Schack, 2001](#CFS2001)) that this is what [[Niels Bohr]] meant all along when he put forth the [[Copenhagen interpretation]] (for more on this suggestion see also at _[[Bohr topos]]_), and people have implied (in [Fuchs, 2002](#Fuchs2002)) that it is what [[Albert Einstein]] was groping towards when he attacked Bohr, and still others ([[Ray Streater]] as cited in [Wikipedia](#Wikipedia)) have implied that it is what [[John von Neumann]] was doing when he eschewed interpretation for mathematical axioms. Any time that somebody has described a quantum state as containing information, or being given by an experimenter\'s knowledge, or being different for one observer than for another, the Bayesian interpretation is implicit. Arguably, it is implicit in any statement that the wavefunction describes probabilities, if [[probability]] is treated as Bayesian. However, the *explicit* exposition of this interpretation seems to have come rather late.
The earliest linking of Bayesianism to quantum states as states of knowledge, as far as I have seen, is Usenet discussion in 1994 ([Baez et al, 2003](#Baez2003)). [[John Baez]] was promoting similar ideas the previous year ([Baez, 1993](#Baez1993)) (and this is not the origin of these ideas either), but this still allows other interpretations. The idea of a 'Bayesian interpretation' came to prominence in 2001, drawing out of work on [[quantum information theory]] ([Caves, Fuchs, & Schack, 2001](#CFS2001)). Further work has been done principally by [[Christopher Fuchs]].
## Related interpretations
The Bayesian interpretation fits into a broader family of [[interpretations of quantum mechanics]] known as _epistemic_ (or _$\psi$-epistemic_ to be more precise). Although 'epistemic' literally refers to knowledge (suggesting objective Bayesianism in particular), the term may be used more broadly to distinguish from _ontic_ interpretations, in which the state $\psi$ is an objective feature of reality. In contrast, an epistemic interpretation is one in which the state is subjective in some way (whether relative to one\'s knowledge or otherwise). Much of what is written above applies more broadly to any epistemic interpretation, although I\'m not sure how much; at least some epistemic interpretations go on to posit a more fundamental reality (involving [[hidden-variable theory|hidden variables]]), while the Bayesian interpretation does not require this.
In the other direction, 'Quantum Bayesianism' or 'QBism' is used specifically to refer to the position advanced over the course of the 21st century principally by [[Christopher Fuchs]]. Although Fuchs\'s first papers on the subject (starting with [Caves, Fuchs, & Schack, 2001](#CFS2001)) referred explicitly to 'states of knowledge', Fuchs has since adopted an increasingly subjective approach, drawing many philosophical conclusions that go beyond mere Bayesianism. These ideas should be attributed to QBism specifically rather than to the Bayesian interpretation generally. For a historical view of how QBism came to be distinguished from earlier Bayesian interpretations of quantum probability, See [Stacey (2019)](#Stacey2019). The technical side of this work has generally not yet taken a categorical flavor, though see [van de Wetering (2018)](#vandeWetering2018).
## Related entries
* [[interpretation of quantum mechanics]]
* [[quantum probability]]
* [[collapse of the wave function]]
## References
* {#Baez1993} [[John Baez]] (1993). This Week's Finds in Mathematical Physics (Week 27). [Web](http://math.ucr.edu/home/baez/week27.html).
* {#Baez2003} John Baez et al (2003). Bayesian Probability Theory and Quantum Mechanics. Collection of Usenet posts from 1994, with commentary. [Web](http://math.ucr.edu/home/baez/bayes.html).
* {#CFS2001} Carlton Caves, [[Christopher Fuchs]], Ruediger Schack (2001). Unknown Quantum States: The Quantum de Finetti Representation. [arXiv:quant-ph/0104088](http://arxiv.org/abs/quant-ph/0104088).
* {#Fuchs2002} [[Christopher Fuchs]] (2002). Quantum Mechanics as Quantum Information (and only a little more). [arXiv:quant-ph/0205039](https://arxiv.org/abs/quant-ph/0205039).
* {#FMS2014} [[Christopher Fuchs]], [[David Mermin]], Ruediger Schack (2014). An Introduction to QBism with an Application to the Locality of Quantum Mechanics. American Journal of Physics 82:8, 749--754. [arXiv:1311.5253](https://arxiv.org/abs/1311.5253).
* {#FSS2014} [[Christopher Fuchs]], [[Maximilian Schlosshauer]] (forward), [[Blake Stacey]] (editor) (2014). My Struggles with the Block Universe. [arXiv:1405.2390](https://arxiv.org/abs/1405.2390).
* {#Mermin2018} [[David Mermin]] (2018). Making better sense of quantum mechanics. [arXiv:1809.01639](https://arxiv.org/abs/1809.01639).
* {#vandeWetering2018} John van de Wetering (2018). Quantum Theory is a Quasi-stochastic Process Theory. EPTCS 266, 179--196. [arXiv:1704.08525](https://arxiv.org/abs/1704.08525).
* {#Stacey2019} [[Blake Stacey]] (2019). Ideas abandoned en route to QBism. [arXiv:1911.07386](https://arxiv.org/abs/1911.07386).
See also
* {#Wikipedia} Wikipedia. Quantum Bayesianism. [Web](https://en.wikipedia.org/wiki/Quantum_Bayesianism).
[[!redirects Bayesian interpretation of quantum mechanics]]
[[!redirects Bayesian interpretations of quantum mechanics]]
[[!redirects Bayesian interpretation of quantum physics]]
[[!redirects Bayesian interpretations of quantum physics]]
[[!redirects Bayesian interpretation of physics]]
[[!redirects Bayesian interpretations of physics]]
[[!redirects quantum Bayesianism]]
[[!redirects Quantum Bayesianism]]
[[!redirects QBism]]
|
Bayesian inversion
|
https://ncatlab.org/nlab/source/Bayesian+inversion
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In elementary [[probability theory]], [[Bayes' formula]] refers to a version of the formula
$$
P(A)\,P(B|A) \,=\, P(B)\,P(A|B)
\,,
$$
and relates the [[conditional probability]] of $B$ given $A$ to the one of $A$ given $B$.
From the [[category-theoretic probability|category-theoretic point of view]], this formula expresses a duality, sometimes called **Bayesian inversion**, which often gives rise to a [[dagger category|dagger structure]].
## Intuition
In logical reasoning, implications in general cannot be reversed: $A \to B$, alone, does not imply $B\to A$.
In [[probability theory]], instead, conditional statements exhibit a [[duality]] which is absent in pure logical reasoning.
Consider for example a city in which *all taxis are yellow*. (The implication is $taxi\to yellow$.)
If we see a yellow car, of course we can't be sure it's a taxi. However, it's *more likely* that it's a taxi compared to a randomly colored car. This increase in likelihood is larger if the fraction of yellow cars is small. Indeed, [[Bayes' rule]] says that
$$
P(taxi|yellow) = \frac{P(yellow\,taxi)}{P(yellow)} = \frac{P(taxi)}{P(yellow)} .
$$
More generally, in a city where *most* taxis are yellow, there is a high conditional probability that a given taxi is yellow,
$$
P(yellow|taxi) .
$$
The higher this probability is, the higher is the probability that a given yellow car is a taxi, again according to [[Bayes' rule]]:
$$
P(taxi|yellow) = \frac{P(yellow\,taxi)}{P(yellow)} = \frac{P(taxi)\,P(yellow|taxi)}{P(yellow)}
$$
In [[categorical probability]], this phenomenon can be modeled by saying that to each "conditional" morphism in the form $\{taxi, not\,taxi\}\to\{colors\,of\,cars\}$ there corresponds a canonical morphism $\{colors\,of\,cars\}\to\{taxi, not\,taxi\}$, called the *Bayesian inverse*.
In some cases, this symmetry gives rise to a [[dagger category]].
## In traditional probability theory
In traditional [[probability theory]], Bayesian inversions are a special case of [[conditional probability]].
Some care must be taken to avoid dividing by zero or incurring into paradoxes via limiting procedures.
### Discrete case
In the discrete case, a [[probability distribution]] on a [[set]] $X$ is simply a function $p:X\to[0,1]$ such that
$$
\sum_{x\in X} p(x) = 1 .
$$
A [[stochastic map]] $k:X\to Y$ is a function
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
row sep=0,%
]
X\times Y \ar{r} & {[0,1]} \\
(x,y) \ar[mapsto]{r} & k(y|x)
\end{tikzcd}
such that for all $x\in X$,
$$
\sum_{y\in Y} k(y|x) = 1 .
$$
If we now equip $X$ and $Y$ with discrete probability distributions $p$ and $q$, obtaining discrete [[probability spaces]], we say that $k$ is a [[stochastic map#measurepreserving_stochastic_maps|measure-preserving stochastic map]] $(X,p)\to(Y,q)$ if for every $y\in Y$,
$$
\sum_{x} k(y|x) \, p(x) = q(y) .
$$
A **Bayesian inverse** of $k:(X,p)\to(Y,q)$ is then defined to be a measure-preserving stochastic map $k^\dagger:(Y,q)\to(X,p)$ such that for every $x\in X$ and $y\in Y$, the following [[Bayes formula]] holds.
$$
p(x) \,k(y|x) = q(y)\,k^\dagger(x|y)
$$
In the discrete case, Bayesian inverses always exist, and can be obtained by taking
$$
k^\dagger(x|y) \coloneqq \frac{p(x)\,k(y|x)}{q(y)}
$$
for those $y\in Y$ with $q(y)\ne 0$, and an arbitrary number on the $y\in Y$ with $q(y)=0$.
(To ensure the normalization condition, on such $y$ one can for example take $k^\dagger(x|y)=\delta_{x_0,x}$, where $x_0$ is a fixed element of $X$. Note that $X$ is nonempty since it admits a probability measure.)
Bayesian inverses are not unique, but they are uniquely defined on the support of $q$. That is, they are unique up to [[almost surely|almost sure]] [[equality]].
### Measure-theoretic case
The situation is more delicate outside the discrete case.
Given [[probability spaces]] $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$, and a [[Markov kernel#measurepreserving_kernels|measure-preserving Markov kernel]] $k:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$, a **[[Markov kernel#bayesian_inversion|Bayesian inverse]]** of $k$ is a Markov kernel $k^\dagger:(Y,\mathcal{B},q)\to(X,\mathcal{A},p)$ such that for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$, the following Bayes-type formula holds.
$$
\int_A k(B|x)\,p(dx) = \int_B k^\dagger(A|y)\,q(dy)
$$
As one can see from [[Markov kernel#almost_sure_equality|Markov kernel - Almost sure equality]], this formula specifies a kernel only up to [[almost surely|almost sure]] [[equality]], just as in the discrete case.
Existence, on the other hand, is more tricky. In general, a kernel $k^\dagger$ as above may fail to exist. The problem is that in order for $k^\dagger$ to be a well-defined [[Markov kernel]] $(Y,\mathcal{B})\to(X,\mathcal{A})$, we need the following two conditions:
* the map $y\mapsto k^\dagger(A|y)$ is [[measurable function|measurable]] in $y$ for all $A\in\mathcal{A}$;
* the assignment $A\mapsto k^\dagger(A|y)$ is a [[probability measure]] in $A$ for all $y\in Y$.
The first condition can be taken care of using [[conditional expectation]].
That however does not assure the second condition. It can be shown, however, that if $(X,\mathcal{A})$ is [[standard Borel]] and either
* $(Y,\mathcal{B})$ is standard Borel too
or
* the kernel $k:X\to Y$ is in the form $\delta_f$ for a measurable $f:X\to Y$,
then a Bayesian inverse always exists.
(See also [[Markov kernel#bayesian_inversion|Markov kernel - Bayesian inversion]] and [[Markov kernel#conditionals|Markov kernel - Conditionals]].)
## In categorical probability
In [[categorical probability]], Bayesian inverses are axiomatized in a way which reflects the measure-theoretic version of the concepts.
One then can choose to work in categories where such axioms are satisfied.
### In Markov categories
In [[Markov categories]], Bayesian inverses are defined in a way that parallels the construction for Markov kernels.
The abstraction of a [[probability space]] is given by an [[object]] $X$ in a [[Markov category]], together with a state $p:I\to X$. As usual, the abstraction of a kernel is a morphism $f:X\to Y$.
A **Bayesian inverse** of $f$ with respect to $p$ is a morphism $f^\dagger:Y\to X$ such that the following equation holds, where $q=f\circ p$.
\begin{tikzpicture}[%
thick,%
scale=0.8,%
every node/.style={scale=1.25},%
none/.style={fill=none,draw=none},%
morphism/.style={fill=white, draw=black, shape=rectangle},%
bn/.style={fill=black, draw=black, shape=circle, inner sep=1.8pt},%
over arrow/.style={-, black, preaction={draw=white, line width=2mm}},%
]
\node [style=bn] (0) at (-3, -1) {};
\node [style=bn] (1) at (3, -1) {};
\node [style=none] (2) at (-3, -2) {};
\node [style=none] (3) at (3, -2) {};
\node [style=none] (4) at (-4, 0) {};
\node [style=none] (5) at (-2, 0) {};
\node [style=none] (6) at (2, 0) {};
\node [style=none] (7) at (4, 0) {};
\node [style=none] (8) at (-4, 1.5) {};
\node [style=none] (9) at (-2, 1.5) {};
\node [style=none] (10) at (2, 1.5) {};
\node [style=none] (11) at (4, 1.5) {};
\node [style=none] (16) at (-4, 2) {$X$};
\node [style=none] (17) at (-2, 2) {$Y$};
\node [style=none] (18) at (2, 2) {$X$};
\node [style=none] (19) at (4, 2) {$Y$};
\node [style=none] (20) at (0, -0.25) {$=$};
\draw [in=165, out=-90] (4.center) to (0);
\draw [in=15, out=-90] (5.center) to (0);
\draw (0) to (2.center);
\draw [in=165, out=-90] (6.center) to (1);
\draw [in=15, out=-90] (7.center) to (1);
\draw (1) to (3.center);
\draw (8.center) to (4.center);
\draw (9.center) to (5.center);
\draw (10.center) to (6.center);
\draw (11.center) to (7.center);
\node [style=morphism] (12) at (-4, 0.25) {$f^\dagger_p$};
\node [style=morphism] (13) at (4, 0.25) {$f$};
\node [style=morphism] (14) at (-3, -2) {$q$};
\node [style=morphism] (15) at (3, -2) {$p$};
\end{tikzpicture}
This recovers the classical probability definitions when instantiated in [[Stoch]] and its subcategories.
Just as in traditional probability, Bayesian inverses are unique only up to [[almost sure equality]].
Also, just as in traditional probability, they may fail to exist. Being an instance of [[Markov category#conditionals|conditionals]], however, they always exists when conditionals exist, such as in the category [[BorelStoch]].
(See also [[Markov category#conditionals|Markov category - conditionals]].)
### In dagger categories
In the [[category of couplings]], the idea of Bayesian inversion is made explicit from the start by means of the [[dagger category|dagger structure]].
Given [[probability spaces]] $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$, a [[coupling (probability theory)|coupling]] between them can be seen equivalently as going from $X$ to $Y$ or from $Y$ to $X$. This duality, when the couplings are expressed via [[Markov kernels]], reflects exactly Bayesian inversion.
Therefore, at the level of joint distributions, one can consider the duality given by Bayesian inversions to be already part of the symmetry of the category.
Categorical abstractions of the category of coupling via [[dagger categories]] have therefore the concept of Bayesian inversion already built in.
## Quantum versions
(For now, see the references.)
## See also
* [[categorical probability]], [[Markov category]]
* [[transport plan]], [[category of couplings]]
* [[Markov kernel]], [[stochastic map]], [[Stoch]]
* [[conditional probability]], [[conditional expectation]]
* [[probability theory]], [[statistics]], [[Bayesian statistics]]
* [[dagger category]]
## References
For categorical probability:
* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))
* {#cd_categories} Kenta Cho and [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))
* Dario Stein and [[Sam Staton]], _Probabilistic Programming with Exact Conditions_, JACM, 2023. ([arXiv](https://arxiv.org/abs/2312.17141))
* Noé Ensarguet and [[Paolo Perrone]], _Categorical probability spaces, ergodic decompositions_, and transitions to equilibrium, 2023. ([arXiv:2310.04267](https://arxiv.org/abs/2310.04267))
For the quantum case:
* {#coecke_spekkens12} [[Bob Coecke]] and Robert W. Spekkens, _Picturing classical and quantum Bayesian inference_, Synthese, 186(3), 2012. ([arXiv](https://arxiv.org/abs/1102.2368))
* {#quantum_markov} [[Arthur J. Parzygnat]], _Inverses, disintegrations, and Bayesian inversion in quantum Markov categories_, 2020. ([arXiv](https://arxiv.org/abs/2001.08375))
* {#noncommutative_bayes} [[Arthur J. Parzygnat]] and Benjamin P. Russo, _A noncommutative Bayes theorem_, Linear Algebra Applications 644, 2022. ([arXiv](https://arxiv.org/abs/2005.03886))
* {#conditional_quantum} [[Arthur J. Parzygnat]], _Conditional distributions for quantum systems_, EPTCS 343, 2021. ([arXiv](https://arxiv.org/abs/2102.01529))
* {#retrodiction} [[Arthur J. Parzygnat]], Francesco Buscemi, _Axioms for retrodiction: achieving time-reversal symmetry with a prior_, Quantum 7(1013), 2023. [arXiv](https://arxiv.org/abs/2210.13531)
* {#quantum_bayes} James Fullwood and [[Arthur J. Parzygnat]], _From time-reversal symmetry to quantum Bayes rules_, PRX Quantum 4, 2023. ([arXiv](https://arxiv.org/abs/2212.08088))
* {#noncomm_disintegrations} [[Arthur J. Parzygnat]], Benjamin P. Russo, _Non-commutative disintegrations: existence and uniqueness in finite dimensions_, Journal of Noncommutative Geometry 17(3), 2023. ([arXiv](https://arxiv.org/abs/1907.09689))
category: probability
[[!redirects Bayesian inversions]]
[[!redirects Bayesian inverse]]
[[!redirects Bayesian inverses]]
[[!redirects bayesian inversion]]
[[!redirects bayesian inversions]]
[[!redirects bayesian inverse]]
[[!redirects bayesian inverses]]
|
Bayesian reasoning
|
https://ncatlab.org/nlab/source/Bayesian+reasoning
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
=--
=--
# Bayesian reasoning
* table of contents
{: toc}
## Idea
Bayesian [[reasoning]] is an application of [[probability theory]] to [[inductive reasoning]] (and [[abductive reasoning]]). It relies on an interpretation of probabilities as expressions of an agent's uncertainty about the world, rather than as concerning some notion of objective chance in the world. The perspective here is that, when done correctly, inductive reasoning is simply a generalisation of [[deductive reasoning]], where knowledge of the truth or falsity of a proposition corresponds to adopting the extreme probabilities $1$ and $0$.
Several advantages to this interpretation have been proposed. For one thing, it is possible to have (rational) degrees of belief about a wide range of propositions, including matters which are settled, yet unknown. For example, it is possible to speak of the probability of the value of a constant of nature falling within a given interval on the basis of various pieces of evidence, or of the outcome of a coin that has already been tossed. This interpretation of probability beyond limiting frequencies leads, so advocates such as [[Edwin Jaynes]] claim, to a more integrated treatment of probability theory and statistics.
## Dutch Book justification
It can be shown by so-called "Dutch Book" arguments, that a rational agent must set their degrees of belief in the outcomes of events in such a way that they satisfy the axioms of probability theory. The idea here is that to believe a proposition to degree $p$ is equivalent to being prepared to accept a wager at the corresponding odds. For instance, if I believe there is a 0.75 chance of rain today, then I should be prepared to accept a wager so that I receive $S$ units of currency if it does not rain, and pay out $S/3$ units if it does rain. Note that $S$ may be chosen to be negative by the bettor.
It can be shown then that such betting odds must satisfy the probability axioms, otherwise it will be possible for someone to place multiple bets which will cause the bookmaker to suffer a certain loss whatever the outcome. For example, in the case above, my degree of belief that it will _not_ rain today should be 0.25. Were I to offer, say, 0.5, someone could stake $(-3)$ units on the first bet and $(-2)$ units on the second bet, and will gain $1$ unit whether or not it rains. Of course, real bookmakers have odds which sum to more than 1, but they suffer no guaranteed loss since clients are only allowed positive stakes.
##Cox's axioms
Some consider the reliance on the idea of the undesirability of certain financial loss to be unbefitting for a justification of what is supposed to be an extension of ordinary deductive logic ([Jaynes 2003](#Jaynes)). Axiomatisations in terms of the properties one should expect of degrees of plausibility have been given, and it can be shown from such axioms that these degrees satisfy the axioms of probability. Richard Cox is responsible for one such axiomatisation (for the moment see [Wikipedia: Cox's theorem](http://en.wikipedia.org/wiki/Cox's_theorem)).
##Conditionalizing
Using [[Bayes' Rule]], degrees of belief can be updated on receipt of new evidence.
$$
P(h|e) = P(e|h) \cdot \frac{P(h)}{P(e)},
$$
where $h$ is a hypothesis and $e$ is evidence.
The idea here is that when $e$ is observed, your degree of belief in $h$ should be changed from $P(h)$ to $P(h|e)$. This is known as **conditionalizing**. If $P(h|e) \gt P(h)$, we say that $e$ has provided **confirmation** for $h$.
Typically, situations will involve a range of possible hypotheses, $h_1, h_2, \ldots$, and applying Bayes' Rule will allow us to compare how these fare as new observations are made. For example, comparing the fate of two hypotheses,
$$
\frac{P(h_1|e)}{P(h_2|e)} = \frac{P(e|h_1)}{P(e|h_2)}\cdot \frac{P(h_1)}{P(h_2)}.
$$
How to assign prior probabilities to hypotheses when you don't think you have an exhaustive set of rivals is not obvious. When astronomers in the nineteenth century tried to account for the anomalies in the position of Mercury's perihelion, they tried out all manner of explanations: maybe there was a planet inside Mercury's orbit, maybe there was a cloud of dust surrounding the sun, maybe the power in the inverse square law ought to be (2 - $\epsilon$),... Assigning priors and changing these as evidence comes in is one thing, but it would have been wise to have reserved some of the prior for 'none of the above'.
Interestingly, one of the first people to give a qualitative sketch of how such an approach would work was [[George Polya]] in 'Mathematics and Plausible Reasoning' ([Polya](#Polya)), where examples from mathematics are widely used. The idea of a Bayesian account of plausible reasoning in mathematics surprises many, it being assumed that mathematicians rely solely on [[deductive reasoning|deduction]]. (See also Chap. 4 of [Corfield03](#Corfield).)
##Objective Bayesianism
For some Bayesians, degrees of belief must satisfy further restrictions. One extreme form of this view holds that given a particular state of knowledge, there is a single best set of degrees of belief that should be adopted for any proposition.
Some such restrictions are generally accepted. If, for example, all I know of an event is that it has $n$ possible outcomes, the objective Bayesian will apply the principle of indifference to set their degrees of belief to $1/n$ for each outcome. On the other hand, if there is background knowledge concerning differences between the outcomes, indifference need not hold. This principle of indifference can be generalized to other kinds of invariance, such as the Jeffreys prior ([wiki](http://en.wikipedia.org/wiki/Jeffreys_prior)).
Other objective Bayesian principles include maximum entropy (see [Jaynes 2003](#Jaynes)). For instance, Jaynes argues that if all that is known of a die is that the mean value of throws is equal to, say, 4, then a prior distribution over $\{1, 2, 3, 4, 5, 6\}$ should be chosen which maximizes [[entropy]], subject to the constraint that the mean is 4. Many familiar distributions are maximum entropy distributions, subject to moment constraints. For instance, the Normal distribution, $N(\mu, \sigma^2)$, is the distribution over the reals which maximises entropy subject to having mean $\mu$ and variance $\sigma^2$.
##Exchangeability
Frequentist statistics makes much use of independent and identically distributed (iid) random variables, for example in sampling situations. If, say, we were to toss a coin repeatedly and record the outcomes, the frequentist would typically understand this as sampling from a Bernoulli distribution for some fixed value $p$ of the coin showing heads. From the sample one could then calculate an estimate and confidence interval for the true value of $p$.
Many Bayesians, in particular [[Bruno de Finetti]], argue that this makes no sense since probability is not in the world, but rather it represents the strengths of our beliefs in different outcomes. Their formulation in such repeated sampling cases is to say that if our degrees of belief are such that, for all $n$, the probability we assign to any sequence of $n$ tosses is invariant under any permutation of $n$ elements, then we can represent our degrees of belief for all sequences as arising from a mixture of Bernoulli distributions for some prior distribution over the value of $p$.
More formally, given a sequence of random variables $\{X_i\}^\infty_{i = 1}$ each taking the values $0$ and $1$, de Finetti's Representation Theorem says that the sequence is exchangeable if and only if there is a random variable $\Theta: \Omega \to [0, 1]$, with distribution $\mu_{\Theta}$, such that
$$
P\{X_1=x_1,\ldots,X_n=x_n\}=\int_{[0,1]} \theta^s (1 - \theta)^{n - s} d \mu_{\Theta}(\theta),
$$
in which $s = \sum^n_{i=1} x_i$.
Often, for ease of calculation, Beta distributions are chosen as priors on $p$, which can be taken as representing one's confidence as though one had already seen a certain number of heads and tails. Bayesians are sometimes criticized for the subjectivity inherent in the choice of a prior, but in many cases, such as this one, prior distributions will eventually be 'washed out' by the weight of the evidence. The de Finetti theorem has a generalization for multivariate distributions ([BBF](#BBF)).
Of course, exchangeability may not represent the strength of one's prior beliefs accurately. For example, in the case of coin tossing, I may have a suspicion of there being something in the tossing mechanism which would make the result of one toss depend on its predecessor. Then it would be quite reasonable for me, say, to have accorded a higher prior probability to the sequence of one hundred heads followed by one hundred tails than to some of its permutations. There are, however, generalizations of de Finetti's representation theorem to Markov chain situations ([Diaconis and Freedman](#DF80)).
## Related entries
* [[conditional expectation]]
* [[Bayesian interpretation of quantum mechanics]]
## References
### General
* {#Jaynes} [[Edwin Jaynes]], _Probability Theory: The Logic of Science_, Cambridge University Press, 2003.
* {#Polya} [[George Polya]], _Mathematics and Plausible Reasoning: Vol. II: Patterns of Plausible Inference_, Princeton University Press, 1954.
* {#Corfield} [[David Corfield]], Chap. 4 of: _Towards a Philosophy of Real Mathematics_, Cambridge University Press (2003) [[doi:10.1017/CBO9780511487576](https://doi.org/10.1017/CBO9780511487576)]
* [[Sumio Watanabe]], _Mathematical theory of Bayesian statistics_, Cambridge University Press (2018) [[ISBN:9780367734817](https://www.routledge.com/Mathematical-Theory-of-Bayesian-Statistics/Watanabe/p/book/9780367734817), [pdf](http://196.189.45.87/bitstream/123456789/36917/1/Sumio%20Watanabe_2018.pdf)]
See also
* Wikipedia, _[Bayesian inference](https://en.wikipedia.org/wiki/Bayesian_inference)_
### Bayesian reasoning and neuroscience
* David C. Knill, Alexandre Pouget: *The Bayesian Brain: The Role of Uncertainty in Neural Coding and Computation*, in Trends in Neurosciences **27** 12 (2004) 712–719 [[doi:10.1016/j.tins.2004.10.007](https://doi.org/10.1016/j.tins.2004.10.007)]
### Category-theoretic treatment
Discussion of Bayesian inference with methods from [[category theory]]:
* Kirk Sturtz, _Bayesian Inference using the Symmetric Monoidal Closed Category Structure_, ([ arXiv:1601.02593](https://arxiv.org/abs/1601.02593))
* Jared Culbertson, Kirk Sturtz, _Bayesian machine learning via category theory_, ([arXiv:1312.1445](https://arxiv.org/abs/1312.1445))
* Kotaro Kamiya, John Welliaveetil, _A category theory framework for Bayesian learning_ ([arXiv:2111.14293](https://arxiv.org/abs/2111.14293))
* [[Toby St Clere Smithe]], _Mathematical Foundations for a Compositional Account of the Bayesian Brain_ [[arXiv:2212.12538](https://arxiv.org/abs/2212.12538)]
### Exchangeability
* {#DF80} [[Persi Diaconis]] and David Freedman, "De Finetti's theorem for Markov chains." Annals of Probability, 8(1), 115-130, 1980.
* {#BBF} A. Bach, H. Blank, H. Francke, _Bose-Einstein statistics derived from the statistics of classical particles_, Lettere Al Nuovo Cimento Series 2, Volume 43, Issue 4, pp 195-198.
* {#FGP21} [[Tobias Fritz]], Tomáš Gonda, [[Paolo Perrone]], _De Finetti's Theorem in Categorical Probability_, ([arXiv:2105.02639](https://arxiv.org/abs/2105.02639))
### Bayesian inference in physics
Discussion of applications in [[astronomy]], [[cosmology]] and [[particle physics]] includes
* [[Jörg Rachen]], _Bayesian Classification of Astronomical Objects -- and what is behind it_ ([arXiv:1302.2429](http://arxiv.org/abs/1302.2429))
* [[Roberto Trotta]], _Bayesian Methods in Cosmology_, ([arXiv:1701.01467](https://arxiv.org/abs/1701.01467))
* {#Lyons13} Louis Lyons, _Bayes and Frequentism: a Particle Physicist's perspective_, ([arXiv:1301.1273](https://arxiv.org/abs/1301.1273))
* Joshua Landon, Frank X. Lee, Nozer D. Singpurwalla, _A Problem in Particle Physics and Its Bayesian Analysis_, ([arXiv:1201.1141](https://arxiv.org/abs/1201.1141))
[[!redirects Bayesian reasoning]]
[[!redirects Bayesian induction]]
[[!redirects Bayesian probability]]
[[!redirects Bayesianism]]
[[!redirects Bayesian inference]]
[[!redirects Bayesian inferences]]
[[!redirects Bayesian statistics]]
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BBDG decomposition theorem
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https://ncatlab.org/nlab/source/BBDG+decomposition+theorem
|
+-- {: .rightHandSide}
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###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
There are various "decomposition theorems" in various fields of mathematics. This entry will be about the **[[Beilinson]]--Bernstein--Deligne--Gabber decomposition theorem** which is probably the strongest single result of wide applicability in modern [[geometric representation theory]].
## References
* A. A. Beilinson, J. Bernstein, P. Deligne _Faisceaux pervers_, Astérisque __100__(1980).
* Mark Andrea A. de Cataldo, Luca Migliorini, _The decomposition theorem, perverse sheaves and the topology of algebraic maps_, Bull. Amer. Math. Soc. __46__ (2009), 535-633, [html abs, refs](http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/home.html), [pdf article](http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf)
[[!redirects decomposition theorem]]
[[!redirects the decomposition theorem]]
[[!redirects Beilinson--Bernstein--Deligne--Gabber decomposition theorem]]
[[!redirects Beilinson–Bernstein–Deligne–Gabber decomposition theorem]]
[[!redirects Beilinson-Bernstein-Deligne-Gabber decomposition theorem]]
|
BCJ relations
|
https://ncatlab.org/nlab/source/BCJ+relations
|
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
(...)
## Related concepts
* [[effective QFT]] incarnations of [[open/closed string duality]],
relating ([[supergravity|super]]-)[[gravity]] to ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]:
* [[KLT relations]]
* [[BCJ relations]]
* [[classical double copy]]
## References
Review:
* Mathias Tolotti, _BCJ-relations in Quantum Field Theory_ ([pdf](https://openscience.ub.uni-mainz.de/bitstream/20.500.12030/1524/1/3941.pdf))
[[!redirects BCJ-relations]]
[[!redirects BCJ relation]]
[[!redirects BCJ-relation]]
|
BD operad
|
https://ncatlab.org/nlab/source/BD+operad
|
#Contents#
* table of contents
{:toc}
## Idea
The _Beilinson-Drinfeld operad_ is the [[operad]] whose [[algebras over an operad]] are [[BD algebras]].
See at _[[relation between BV and BD]]_.
## Definition
The underlying graded vector spaces are
$$
BD \coloneqq P_0 \otimes \mathbb{R}[ [\hbar] ]
$$
and the [[differential]] is
$$
d ( (-)\cdots (-))
=
\hbar \{-,-\}
\,.
$$
(...)
## Related concepts
[[!include deformation quantization - table]]
## References
The BD operad was introduced in
* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_
A review in the context of [[factorization algebras of observables]] is in section 2.4 of
* [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in perturbative quantum field theory : $P_0$-operad_ ([wiki](http://math.northwestern.edu/~costello/factorization_public.html#[[P_0%20operad]]), [pdf](http://math.northwestern.edu/~gwilliam/factorization.pdf))
[[!redirects BD-operad]]
|
BDR 2-vector bundle
|
https://ncatlab.org/nlab/source/BDR+2-vector+bundle
|
> under construction
> for the time being this here are nothing but [references](#References) and rough notes taken in a talk long ago...
_BDR 2-vector bundles_ are a notion of [[categorified]] [[vector bundle]] motivated by the concept of [Kapranov-Voevodsy's 2-vector spaces](http://ncatlab.org/nlab/show/2-vector+space#KV2VectorSpace).
See also at _[[iterated algebraic K-theory]]_.
#Contents#
* table of contents
{:toc}
## 2-Vector bundles following Baas-Dundas-Rognes
We are looking for a generalization of the notion of [[vector bundle]] in
[[higher category theory]].
Let $X$ be a [[topological space]] and $\{U_\alpha \to X\}_{\alpha \in A}$ an [[open cover]],
where the index set $A$ is assumed to be a [[poset]].
Defition. A **charted 2-vector bundle** (i.e. a [[cocycle]] for a BDR 2-vector bundle)
of rank $n$ is
* for $\alpha \lt \beta \in A$ on $U_\alpha \cap U_\beta =: U_{\alpha\beta}$
a matrix $(E^{\alpha\beta}_{i j})_{i,j = 1}^{n}$ of [[vector bundle]]s
$E^{\alpha \beta}_{i j} \to U_{\alpha \beta}$
such that the determinant of the underlying matrix of dimensions
is $det(dim(E^{\alpha \beta}_{i j})) = \pm 1$.
* on triple overlaps $U_{\alpha \beta \gamma}$
for $\alpha \lt \beta \lt \gamma \in A$ we have
[[isomorphism]]s
$$
\phi^{\alpha \beta \gamma}
:
\oplus_{j} E^{\alpha \beta}_{i j} \otimes E^{\beta \gamma}_{j k}
\stackrel{\simeq}{\to}
E^{\alpha \gamma}_{i k}
$$
* such that the $\phi$ satisfy on quadruple overlaps the evident cocycle condition
(as described at [[gerbe]] and [[principal 2-bundle]]).
Next we need to define morphisms of such charted 2-vector bundles. These involve the
evident refinements of covers and fiberwise transformations.
Write $2Vect(X)$ for the equivalence classes of charted 2-vector bundles under these morphisms.
**Remark** If we restrict attention to $n = 1$ then this gives the same as
$U(1)$-gerbes/[[bundle gerbe]]s.
**Theorem** (Baas-Dundas-Rognes)
There exists a [[classifying space]] $\mathcal{K}(V)$
such that for $X$ a finite [[CW-complex]] there is an [[isomorphism]]
$$
[X, \mathcal{K}(V)] =
{\lim_\to}_{a : Y \to X}
Gr(2Vect(Y))
$$
between [[homotopy]] classes of continuous maps $X \to \mathcal{K}(V)$
and equivalence classes of the group completed [[infinity-stackification|stackification]] of
2-vector bundles,
where the [[colimit]] is over acyclic [[Serre fibration]]s (Note: these are not acyclic fibrations in the usual sense, rather their fibres have trivial integral [[homology]]) and $Gr(-)$ indicates
the [[Grothendieck group]] completion using the monoid structure arising from the direct sum of 2-vector bundles.
**Proof** In BDR, Segal Birthday Proceedings
**Note** $2Vect_n(X) = [X, |B Gl_n (V)| ]$.
BDR called $\mathcal{K}(V)$ the **2-K-theory** of the [[bimonoidal category]]
of [[Kapranov-Voevodsky 2-vector space]]s.
## The homotopy type of the classifying space
**Theorem** (Baas-Dundas-Rognes-Richter)
The K-theory of BDR 2-vector bundles is the [[algebraic K-theory]] of [[ku]] (see at _[[iterated algebraic K-theory]]_)
$\mathcal{K}(V) \simeq K(ku)$
Here:
* $ku$ is the connected version of the [[spectrum]] of complex [[topological K-theory]];
* $K(-)$ denotes forming the [[algebraic K-theory]] spectrum of [[ring spectrum|ring spectra]].
So by the general formula for [[algebraic K-theory]] for [[ring spectrum|ring spectra]], this is
$$
K(ku) \simeq \mathbb{Z} \times B Gl(ku)^+
\,
$$
Some flavor of $\mathcal{K}(V)$.
* $V$ is the (a [[skeleton]] of the [[core]] of) of [[category]] of complex [[vector space]]s.
* [[object]]s are [[natural number]]s, $n$ corresponding to $\mathbb{C}^n$;
* [[morphism]]s are $Hom(k,k) = GL(k)$ and there are no morphisms between different
$k,l$.
This category $V$ is naturally a [[bimonoidal category]] under [[coproduct]] and
[[tensor product]] of [[vector space]]s.
K-theory is about understanding linear algebra on a ring, so we want to understand the
linear algebra of this [[monoidal category]].
We write $Mat_n(V) $ for the $n \times n$ matrices of linear isomorphisms between
finite dimensional vector spaces.
Such matrices can be multiplied using the usual formula for matrix products
on the [[tensor product]] and [[direct sum]] of vector space and linear maps.
Write $Gl_n(V)$ for the subcollection of those matrices for which the
determinant of their matrix of dimensions is $\pm 1$.
Now define
$$
\mathcal{K}(V) = \Omega B (\coprod_{n \geq 0} B Gl_n (V))
\,.
$$
Notice that this is a direct generalization of the corresponding formula
for the algebraic K-theory of a ring $R$,
$$
K(R) = \Omega B (\coprod_{n \geq 0} B Gl_n(R))
\,.
$$
## $K(ku)$ as a form of elliptic cohomology
Ausoni and Rognes compute the [[homology]] groups
(for a certain sense of homology) of $K(ku)$.
take rational homotopy
* $H^*(-, \mathbb{Q})$
for $p$ a [[prime]], multiplying by $p$ gives an isomorphism on this.
p = $\nu_0$
* Let $KU^*(-)$ be complex oriented [[topological K-theory]], then
$$
KU_* = \mathbb{Z}[u^{\pm 1}]
$$
for $|u| = 2$ (the [[Bott class]]) we have that multiplying by $u$ is an
isomorphism and $u^{p-1} = \nu_1$
The $\nu_i$ come from the [[Brown-Peterson spectrum]] $B P$ and
$\pi_* BP = \mathbb{Z}_{(p)}[\nu_1, \nu_2, \cdots]$
motto: the higher the $\nu_i$ the more you detect.
Christian Ausoni figured out something that implies that
"$K(ku)$ detects as much in the [[stable homotopy category]] as
any other form of [[elliptic cohomology]]."
## From gerbes to 2-vector bundles
It is hard to directly construct charted 2-vector bundles. We have more examples
of [[gerbe]]s. So we want to get one from the other.
**Example**
We have
$$
\mathbb{S}^1 = \mathbb{C}P^1 \subset \mathbb{C}P^\infty = B U(1) = K(\mathbb{Z},2)
$$
(see [[Eilenberg-MacLane space]] and [[classifying space]])
$$
\mathbb{S}^3 = \Sigma \mathbb{S}^2 \to \Sigma B U(1) \subset
\Sigma B U \to B B U_{\otimes}
\subset units(ku)
\to B GL(ku)
$$
using $\Sigma B U(1) \to B BU(1) \to B B U_{\otimes}$
we can take a $U(1)$-[[gerbe]] classified by maps into $B^2 U(1)$ and
induce from it the associated 2-vector bundle.
the canonical map
$$
\mathbb{S}^3 \to K(\mathbb{Z},3)
$$
may be thought of as classifying the gerbe called the [[magnetic monopole]]-gerbe
Postcomposing with $\mu : K(\mathbb{Z},3) \to K(ku)$
we have
Fact: $\mu$ gives a generator in $\pi_3 K(\mathbb{Z},3) = H^3(\mathbb{S}^3)$
**Theorem** (Ausoni-Dundas-Rognes)
$$
j(\mu) = 2 \zeta - \nu
$$
in $\pi_3(K(ku))$
so regarded as a 2-vector bundle $\mu$ is not a generator.
ADR: $\zeta$ is "half a monopole".
$$
\pi_3(K(ku)) = \mathbb{Z} \oplus \mathbb{Z}/24 \mathbb{Z}
$$
(the first summand is $\zeta$, the second $\nu$).
Thomas Krogh has an [[orientation]] theory for 2-vector bundles which says that $j(\nu)$
is not orientable.
## 2K-theory of bimonoidal categories {#2K-theory}
Let $(R, \oplus, \otimes, 0,1, c_{\oplus})$
be a [[bimonoidal category]], i.e. a [[categorification|categorified]] [[rig]].
This can be broken down as
1. $(R, \oplus, 0 , c_{\oplus})$ a permutative category, a categorified abelian [[monoid]];
1. $(R , \otimes, 1)$ is a [[monoidal category]], assumed to be _strict_ monidal in the following;
1. a distributivity law.
**Examples**
1. $E = Core$[[FinSet]],
the [[core]] of the
category of finite sets and morphisms only between sets of the
same cardinality.
In the [[skeleton]], objects are [[natural number]]s $n \in \mathb{N}$,
$\oplus$ and $\otimes$ is addition and multiplication on $\mathbb{N}$,
respectively. Here $c_{\oplus}$ is the evident natural [[isomorphism]]
between direct sums of finite sets.
1. $V = Core$[[Vect]] the [[core]]
of the category of finite dim vector spaces, with morphisms only between
those of the same dimension.
1. ...
**Definition** For $R$ a bimonoidal category, write $Mat_n(R)$
for the $n \times n$ matrices with entries morphisms in $R$. Then matrix multiplication is defined using the bimonoidal structure on $R$. This gives a weak [[monoid]] structure.
Let $Gl_n(R)$ be the category of weakly invertible such matrices. This is the [[full subcategory]] of $Mat_n(R)$. We get a diagram of [[pullback]] squares
$$
\array{
Gl_n(R) &\hookrightarrow& Mat_n(R)
\\
\downarrow && \downarrow
\\
Gl_n(\pi_0(R))&\to& Mat_n(\pi_0(R))
\\
\downarrow && \downarrow
\\
Gl_n(Gr(\pi_0(R)))&\to& Mat_n(Gr(\pi_0(R)))
}
\,,
$$
where $Gr(-)$ denotes [[Grothendieck group]]-completion.
**Definition** (Baas-Dundas-Rognes, 2004)
For $R$ a [[bimonoidal category]], the **2K-theory** of $R$ is
$$
\mathcal{K}(R) := \Omega B \coprod_{n \geq 0} | B Gl_n(R) |
$$
where the $\Omega$ is forming [[loop space]], the leftmost $B$ is forming classifying space of a category and the inner $B$ is a flabby version of classifying space of a category.
This can also be written
$$
\cdots \simeq \mathbb{Z} \times |B Gl_n(R)|^+
\,.
$$
Here $B_q Gl_n(R)$ is a [[simplicial category]]
...
**Theorem** (Baas-Dundas-Rognes)
Let $R$ be a [[small category|small]] [[Top]]-[[enriched category|enriched]] [[bimonoidal category]]
such that
1. $R$ is a [[groupoid]];
1. for all $X \in R$ we have that $X \oplus (-)$ is [[faithful functor|faithful]].
Then $\mathcal{K}(R) \simeq K(H R)$ is the ordinary [[algebraic K-theory]] of the [[ring spectrum]] $H R$.
Notice that for $H R$ to be a spectrum we only need the additive structure $(R, \oplus, 0, c_{\oplus})$. The point is that the other monoidal structure $\otimes$ indeed makes this a [[ring spectrum]]. This is a not completely trivial statement due to a bunch of people, involving [[Peter May]] and Elmendorf-Mandell (2006).
**Examples**
1. For the category $R := E = Core(FinSet)$
of finite sets as above we have that $H E$ is
the [[sphere spectrum]].
1. For $R := V = Core(FinDimVect)$ the [[core]] of [[complex vector spaces|complex]] [[finite dimensional vector spaces]] we have $H V$ is the complex [[K-theory spectrum]].
1. For $V_{\mathbb{R}}$ analogously we get the real K-theory spectrum.
So by the above theorem
1. $\mathcal{K}(E) \simeq K(S) \simeq A(*)$
1. $\mathcal{K}(V) \simeq K(ku)$;
1. etc.
**Remarks**
1. A. Osono: The equivalence $\mathcal{K}(R) \simeq K(H R)$ of [[topological space]]s is even an equivalence of [[infinity loop space]]s;
1. Application of that:
a) for $E$ a [[ring spectrum]]: find a model
$$
E \simeq H R(R)
$$
and use the equivalence $\mathbb{K}(R) \simeq K(H R E)$ to understand
_arithmetic_ properties of $E$.
b) Often one knows $K(H(R))$ via calculations. here $\mathcal{K}(R)$ might
help to get some deeper understanding.
c) Theorem ([[Birgit Richter]]): for $R$ a bimonoidal category with
anti-involution, then you get an involution of $\mathcal{K}(R)$.
## Related concepts
* [[red-shift conjecture]]
## References
{#References}
The original articles on BDR 2-vector bundles are
* [[Nils Baas]], [[Ian Dundas]], [[John Rognes]], _Two-vector bundles and forms of elliptic cohomology_, in: _Topology, geometry and quantum field theory_, volume 308 of London Math. Soc. Lecture Note Ser., pages 18–45.
Cambridge Univ. Press, Cambridge, (2004).
Their [[classifying spaces]] are discussed in
* [[Nils Baas]], [[Ian Dundas]], [[Birgit Richter]], [[John Rognes]], _Ring completion of rig categories_ ([arXiv:0706.0531](http://arxiv.org/abs/0706.0531))
(a previous version of this carried the title _Two-vector bundles define a form of elliptic cohomology_)
* [[Nils Baas]], M. Böckstedt, [[Tore Kro]], _Two-Categorical Bundles and Their Classifying Spaces_ (2008) ([arXiv:math/0612549](http://arxiv.org/abs/math/0612549)), .
Divisibility of the gerbe on the 3-sphere, seen as a 2-vector bundle is in
* [[Christian Ausoni]], [[Bjørn Ian Dundas]] and [[John Rognes]], _Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere_, Doc. Math. 13, 795-801 (2008). ([dml:45387](https://eudml.org/doc/45387), [pdf](http://www.math.univ-paris13.fr/~ausoni/papers/ADR08.pdf))
Orientation of BDR 2-vector bundles is discussed in
* Thomas Kragh, _Orientations and Connective Structures on 2-vector Bundles_ Mathematica Scandinavica, 113 (2013) no 1, ([journal](http://www.mscand.dk/article/view/15482)), ([arXiv:0910.0131](http://arxiv.org/abs/0910.0131))
[[!redirects BDR 2-vector bundles]]
|
beable
|
https://ncatlab.org/nlab/source/beable
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The term _beable_ was proposed in ([Bell 75](#Bell75)) as a replacement in [[quantum physics]] of the traditional term _[[observable]]_. While the latter is typically given a precise mathematical meaning which does express the way in which the [[physical system]] may be, the word "observable" alludes to the complicated and subtle issue of something (a [[quantum measurement]] device) or even somebody (a concious experimentor) "observing" these ways of the system to be. The point of the term "beable" is to help conceptually cleanly separate the [[being]] of quantum systems from whatever it means to observe them.
In ([Bell 75](#Bell75)), to further justify this, recourse is made to [[Niels Bohr]]'s expressed view that whatever [[quantum physics]] really is, it must be possible to communicate statements about it in terms of classical [[logic]]. Hence beables mostly refer to sets of commuting operators, [[classical contexts]]. This same argument was much later used to motivate [[Bohr toposes]]. In this context, the [[Kochen-Specker theorem]] proves the absence of certain global sections (Proposition 1). Halvorson and Clifton define a beable subalgebra (p9) with respect to a given state $\rho$. We first consider the important special case of the _definite algebra_ $D_\rho$ for a state $\rho$. This is the subalgebra on which $\rho$ is dispersion free. This gives, for each commutative subalgebra of $C\subset D_\rho$, a choice of an element of its spectrum. These choices are compatible. Hence, we obtain not a global section, but only local section. The general notion of beable subalgebra only requires $\rho$ to be a _mixture_ of dispersion-free states.
In practice of theoretical physics most everytime one writes "observable" it should, by this logic, rather be "beable". While this might be reasonable, as a convention of conversation it has not been picked up.
## Related concepts
[[!include states and observables -- content]]
## References
The original discussion of beables in the context of [[causal locality]] in [[AQFT]] is in
* [[John Bell]], _The theory of local beables_, [Epistemological Letters](http://en.wikipedia.org/wiki/Epistemological_Letters) (1975) ([pdf](http://cds.cern.ch/record/980036/files/197508125.pdf))
{#Bell75}
and further in
* [[John Bell]], _Beables for quantum field theory_ (1984) ([pdf](http://prac.us.edu.pl/~ztpce/QM/Bell_beables.pdf))
A survey is in
* Scholarpedia, _[Bell's theorem - beables](http://www.scholarpedia.org/article/Bell's_theorem#beable)_
A more technical development can be found here:
* [[Hans Halvorson]], Rob Clifton, _Maximal Beable Subalgebras of Quantum-Mechanical Observables_ ([arXiv:quant-ph/9905042](http://arxiv.org/abs/quant-ph/9905042))
Discussion in the context of [[interpretation of quantum mechanics]] includes
* Yuichiro Kitajima, _Interpretations of Quantum Mechanics in Terms of Beable Algebras_, ([pdf](http://link.springer.com/article/10.1007/s10773-005-4051-0))
[[!redirects beables]]
|
beauty > history
|
https://ncatlab.org/nlab/source/beauty+%3E+history
| |
Beck module
|
https://ncatlab.org/nlab/source/Beck+module
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
One usually defines [[cohomology]] with respect to some [[coefficient]] objects:
* For [[group cohomology]] of a [[group]] $G$, the coefficients come from a (left) $G$-[[module]].
* For [[Lie algebra cohomology]] of a [[Lie algebra]] $\mathfrak{g}$, the coefficients come from a (left) $\mathfrak{g}$-module.
* For [[Hochschild cohomology]] of an [[associative algebra]] $A$, the coefficients come from an $A$-[[bimodule]].
**Beck modules** are a simultaneous generalisation of all three types of module.
## Definition
Let $\mathcal{C}$ be a [[category]] with [[pullback]]s and let $A$ be an object in $\mathcal{C}$. A **Beck module** over $A$ is an [[abelian group object]] in the slice category $\mathcal{C}_{/ A}$. In particular, if $A$ is the terminal object, this reduces to the notion of an abelian group object in $\mathcal{C}$. We write $\mathbf{Ab}(\mathcal{C}_{/ A})$ for the category of Beck modules over $A$.
([Beck 67, def. 5](#Beck67))
## Properties
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be an [[effective regular category]] (resp. [[locally presentable category]]) and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is an [[abelian category]] (resp. locally presentable category).
=--
+-- {: .proof}
###### Proof
If $\mathcal{C}$ is a effective regular category (resp. locally presentable category), then so is $\mathcal{C}_{/ A}$. Thus, the claim reduces to the fact that the category of abelian group objects in an effective regular category (resp. locally presentable category) is an abelian category (resp. locally presentable category).
=--
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be an effective regular category with [[filtered colimits]] and let $A$ be an object in $\mathcal{C}$. If filtered colimits in $\mathcal{C}$ preserve finite [[limits]], then $\mathbf{Ab}(\mathcal{C}_{/ A})$ (is an abelian category and) satisfies axiom AB5.
=--
+-- {: .proof}
###### Proof
The forgetful functor $\mathcal{C}_{/ A} \to \mathcal{C}$ [[created limit|creates]] pullbacks and filtered colimits, so filtered colimits in $\mathcal{C}_{/ A}$ also preserve finite limits. The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and filtered colimits, so filtered colimits in $\mathbf{Ab}(\mathcal{C}_{/ A})$ preserve kernels. In view of the earlier proposition, it follows that $\mathbf{Ab}(\mathcal{C}_{/ A})$ satisfies axiom AB5.
=--
+-- {: .num_cor}
###### Corollary
Let $\mathcal{C}$ be a locally finitely presentable effective regular category and let $A$ be an object in $\mathcal{C}$. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is a [[Grothendieck category]].
=--
+-- {: .proof}
###### Proof
Combine the two propositions above.
=--
## Derivations
Let $\mathcal{C}$ be a category with pullbacks, let $A$ be an object in $\mathcal{C}$, and let $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ be the forgetful functor. Given a Beck module $M$ over $A$, an $M$-valued **derivation** of $A$ is a morphism $1_A \to U M$ in $\mathcal{C}_{/ A}$, where $1_A$ is the terminal object in $\mathcal{C}_{/ A}$, and we write
$$Der (A, M) = \mathcal{C}_{/ A} (1_A, U M)$$
for the set of $M$-valued derivations of $A$. The **Beck module of differentials** over $A$ is an object $\Omega_A$ in $\mathbf{Ab}(\mathcal{C}_{/ A})$ [[representable functor|representing]] the functor $Der (A, -) : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathbf{Set}$.
The Beck module $\Omega_A$ is not guaranteed to exist in general. When the functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint, $\Omega_A$ is simply the value of the left adjoint at $1_A$.
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be a locally presentable category and let $A$ be an object in $\mathcal{C}$. Then the forgetful functor $U : \mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ has a left adjoint.
=--
+-- {: .proof}
###### Proof
The forgetful functor $\mathbf{Ab}(\mathcal{C}_{/ A}) \to \mathcal{C}_{/ A}$ creates limits and $\kappa$-filtered colimits (for some $\kappa$ large enough), so we may apply the accessible [[adjoint functor theorem]].
=--
## Examples
### Beck modules over associative algebras
{#OverAssociativeAlgebras}
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be the category of (not necessarily commutative) [[rings]] and let $A$ be a ring. Then $\mathbf{Ab}(\mathcal{C}_{/ A})$ is equivalent to the category of $A$-bimodules.
=--
+-- {: .proof}
###### Proof
Let $\epsilon : B \to A$ be ring homomorphism. To give it the structure of a Beck module over $A$, we must give ring homomorphisms $\eta : A \to B$ and $\mu : B \times_A B \to B$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (b_0, b_1)) = \epsilon (b_0) = \epsilon (b_1)$, as well as various other equations. Given elements $b_0, b_1, b_2, b_3$ of $B$ such that $\epsilon (b_0) = \epsilon (b_2)$ and $\epsilon (b_1) = \epsilon (b_3)$, we have the following interchange law:
$$\mu (b_0 + b_1, b_2 + b_3) = \mu (b_0, b_2) + \mu (b_1, b_3)$$
Hence, if $a = \epsilon (b_1) = \epsilon (b_2)$,
$$\mu (b_1, b_2) = \mu (\eta (a) + b_1 - \eta (a), \eta (a) + b_2 - \eta (a)) = \mu (\eta (a), \eta (a)) + \mu (b_1 - \eta(a), b_2 - \eta (a)) = \eta (a) + \mu (b_1 - \eta(a), b_2 - \eta (a))$$
but $\epsilon (b_1 - \eta (a)) = \epsilon (b_2 - \eta (a)) = 0$, so
$$\mu (b_1 - \eta(a), b_2 - \eta(a)) = \mu (0, b_2 - \eta (a)) + \mu (b_1 - \eta (a), 0) = b_2 - \eta (a) + b_1 - \eta (a)$$
and we conclude that
$$\mu (b_1, b_2) = b_1 - \eta (\epsilon (b_1)) + b_2 = b_1 + b_2 - \eta (\epsilon (b_2))$$
and in particular, $\mu$ is entirely determined by $\eta$ and $\epsilon$. We also have the following interchange law,
$$\mu (b_0 b_1, b_2 b_3) = \mu (b_0, b_2) \mu (b_1, b_3)$$
and in particular,
$$\mu (\eta (\epsilon (b_2)) b_1, b_2 \eta (\epsilon (b_1))) = \mu (\eta (\epsilon (b_2)), b_2) \mu (b_1, \eta (\epsilon (b_1))) = b_2 b_1$$
hence,
$$b_2 b_1 = \eta (\epsilon (b_2)) b_1 - \eta (\epsilon (b_2 b_1)) + b_2 \eta (\epsilon (b_1))$$
so if $\epsilon (b_1) = \epsilon (b_2) = 0$, then $b_2 b_1 = 0$.
Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing abelian group structure on $\ker \epsilon$. In addition, the homomorphism $\eta : A \to B$ gives $M$ the structure of an $A$-bimodule, and we see that $B$ is naturally isomorphic to the [[square-0 extension]] ring $A \oplus M$, with componentwise addition and the multiplication given below,
$$(a_0, m_0) \cdot (a_1, m_1) = (a_0 a_1, a_0 m_1 + m_0 a_1)$$
regarded as a Beck module over $A$ by defining $\epsilon : A \oplus M \to A$, $\eta : A \to A \oplus M$, and $\mu : (A \oplus M) \times_A (A \oplus M) \to A \oplus M$ as follows:
$$\epsilon (a, m) = a$$
$$\eta (a) = (a, 0)$$
$$\mu ((a, m_0), (a, m_1)) = (a, m_0 + m_1)$$
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ A})$ and the category of $A$-bimodules, as claimed.
=--
+-- {: .num_prop}
###### Proposition
Let $A$ be a ring. Then the Beck module $\Omega_A$ is isomorphic to the $A$-bimodule of [[Kähler differentials]] (relative to $\mathbb{Z}$).
=--
+-- {: .proof}
###### Proof
Let $M$ be an $A$-bimodule, regard $A \oplus M$ as a ring as above, and let $\epsilon : A \oplus M \to A$ be the obvious projection. A ring homomorphism $\phi : A \to A \oplus M$ satisfying $\epsilon \circ \phi = id_A$ is the same thing as an additive homomorphism $\delta : A \to M$ satisfying the following equations,
$$\delta (a_0 a_1) = \delta (a_0) a_1 + a_0 \delta (a_1)$$
i.e. a derivation $A \to M$ (over $\mathbb{Z}$). Thus, the Beck module $\Omega_A$ has the same universal property as the $A$-bimodule of Kähler differentials.
=--
### Beck modules over groups
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be the category of (not necessarily abelian) [[groups]] and let $G$ be a group. Then $\mathbf{Ab}(\mathcal{C}_{/ G})$ is equivalent to the category of left $G$-modules.
=--
+-- {: .proof}
###### Proof
Let $\epsilon : H \to G$ be group homomorphism. To give it the structure of a Beck module over $G$, we must give group homomorphisms $\eta : G \to H$ and $\mu : H \times_G H \to H$ such that $\epsilon \circ \eta = id_A$, $\epsilon (\mu (h_0, h_1)) = \epsilon (h_0) = \epsilon (h_1)$, as well as various other equations. Given elements $h_0, h_1, h_2, h_3$ of $H$ such that $\epsilon (h_0) = \epsilon (h_2)$ and $\epsilon (h_1) = \epsilon (h_3)$, we have the following interchange law:
$$\mu (h_0 h_1, h_2 h_3) = \mu (h_0, h_2) \mu (h_1, h_3)$$
and in particular,
$$\mu (\eta (\epsilon (h_2)) h_1, h_2 \eta (\epsilon (h_1))) = \mu (\eta (\epsilon (h_2)), h_2) \mu (h_1, \eta (\epsilon (h_1))) = h_2 h_1$$
but on the other hand, if $g = \epsilon (h_1) = \epsilon (h_2)$, then
$$\mu (h_1, h_2) = \mu (\eta (g) \eta (g)^{-1} h_1, \eta (g) \eta (g)^{-1} h_2) = \mu (\eta (g), \eta (g)) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \eta (g) \mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2)$$
and writing $e$ for the unit of $G$ and $H$, we have $\epsilon (\eta (g)^{-1} h_1) = \epsilon (\eta (g)^{-1} h_2) = e$, hence
$$\mu (\eta (g)^{-1} h_1, \eta (g)^{-1} h_2) = \mu (e, \eta (g)^{-1} h_2) \mu (\eta (g)^{-1} h_1, e) = \eta (g)^{-1} h_2 \eta (g)^{-1} h_1$$
so we conclude that
$$\mu (h_1, h_2) = h_2 \eta (g)^{-1} h_1$$
and in particular, $\mu$ is entirely determined by $\eta$.
Let $M = \ker \epsilon$. The above shows that the internal abelian group structure on $B$ restricts to the pre-existing group structure on $\ker \epsilon$. (In particular, $M$ is an abelian group!) We make $M$ into a left $G$-module as follows:
$$g \cdot m = \eta (g) m \eta (g)^{-1}$$
We can then construct the [[semi-direct product]] $M \rtimes G$, which has the following multplication:
$$(m_0, g_0) \cdot (m_1, g_1) = (m_0 \eta (g_0) m_1 \eta (g_0)^{-1}, g_0 g_1)$$
There is a group homomorphism $M \rtimes G \to H$ defined by $(m, g) \mapsto m \eta (g)$, and it is bijective: surjectivity is clear, and injectivity is a consequence of the fact that $M \cap \operatorname{im} \eta = \{ e \}$. We may regard $M \rtimes G$ as a Beck module over $G$ by defining $\epsilon : M \rtimes G \to G$, $\eta : G \to M \rtimes G$, and $\mu : (M \rtimes G) \times_G (M \rtimes G) \to M \rtimes G$ as follows:
$$\epsilon (m, g) = g$$
$$\eta (g) = (0, g)$$
$$\mu ((m_0, g), (m_1, g)) = (m_1 m_0, g)$$
Thus, we have an equivalence between $\mathbf{Ab}(\mathcal{C}_{/ G})$ and the category of left $G$-modules, as claimed.
=--
+-- {: .num_prop}
###### Proposition
Let $G$ be a group and let $M$ be a left $G$-module. Under the above identification of Beck modules over $G$ with left $G$-modules, $M$-valued derivations of $G$ are precisely crossed homomorphisms $G \to M$, i.e. maps $\delta : G \to M$ satisfying the following equation:
$$\delta (g_0 g_1) = \delta (g_0) + g_0 \cdot \delta (g_1)$$
=--
+-- {: .proof}
###### Proof
Let $\epsilon : M \rtimes G \to G$ be the evident projection. A group homomorphism $\phi : G \to M \rtimes G$ such that $\epsilon \circ \phi = id_G$ is the same thing as a map $\delta : G \to M$ satisfying the equation below,
$$(\delta (g_0), g_0) \cdot (\delta (g_1), g_1) = (\delta (g_0 g_1), g_0 g_1)$$
which is equivalent to the defining equation for crossed homomorphisms.
=--
## The tangent category
One may assemble the individual categories of Beck modules over the objects of $\mathcal{C}$ into a category fibred over $\mathcal{C}$, called the [[tangent category]].
## References
The concept is due to
* {#Beck67} [[Jon Beck]], _Triples, algebras and cohomology_, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html))
and was popularized in
* {#Quillen70} [[Daniel G. Quillen]], _On the (co-)homology of commutative rings_, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc., 1970.
See also
* [[Michael Barr]], _Acyclic models_, Chapter 6, §1.
An application to [[knot theory]] is given in
* [[Markus Szymik]], _Alexander-Beck modules detect the unknot_, Fund. Math. 246 (2019) 89-108.
[[!redirects Beck modules]]
|
Beck-Chevalley condition
|
https://ncatlab.org/nlab/source/Beck-Chevalley+condition
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **[[Jon Beck|Beck]]--[[Claude Chevalley|Chevalley]] condition**, also sometimes called just the *Beck condition* or the *Chevalley condition*, is a "commutation of [[adjoint functor|adjoint]]s" property that holds in many "[[base change|change of base]]" situations.
## Motivation
The Beck-Chevalley condition may be understood as a natural compatibility condition for
**[(1)](#MotivationFromIntegralTransforms)** [[integral transforms]] in [[geometry]]
**[(2)](#MotivationFromLogicAndTypeTheory)** [[quantifiers]] in [[formal logic]]/[[type theory]]
### From integral transforms
{#MotivationFromIntegralTransforms}
From the point of view of [[geometry]], in contexts such as of [[integral transforms]] one considers [[correspondences]] between [[spaces]] $A$, $B$ given by [[spans]] of [[maps]] between them:
\begin{tikzcd}[sep=20]
&
X
\ar[dl, "f"{swap}]
\ar[dr, "g"]
\\
A && B
\end{tikzcd}
Via [[composition]] of such correspondences by [[fiber product]] of adjacent legs, they form the [[1-morphisms]] in a [[2-category]] *[[Span]]*.
Assuming some [[base change]] [[adjoint pair]] $f^\ast \vdash f_\ast$, thought of as
* *push-forward* $f_\ast \,\colon\, \mathcal{D}(X) \longrightarrow \mathcal{D}(A)$
and
* *pull-back* $f^\ast \,\colon\, \mathcal{D}(A) \longrightarrow \mathcal{D}(X)$
is [[functor|functorially]] associated with [[maps]] $f$ (i.e. such that there are [[natural isomorphisms]] $(f_2 f_1)_\ast \,\simeq\, (f_2)_\ast (f_1)_\ast$ and dually), the [[integral transform]] encoded by any span as above is the "pull-push" operation given by $g_* f^*$.
Now the Beck-Chevalley condition (on such assignment of [[base change]] adjoints to maps) essentially says that this pull-push construction is ([[2-functor|2-]])[[functor|functorial]] under [[composition]] of [[spans]]:
Concretely, given a pair of composable spans:
\begin{tikzcd}[sep=20]
&
X
\ar[dl, "f"{swap}]
\ar[dr, "g"]
&&
Y
\ar[dl, "h"{swap}]
\ar[dr, "k"]
\\
A && B && C
\end{tikzcd}
and their composite in [[Span]] formed by the [[fiber product]] of the adjacent legs
\begin{tikzcd}[sep=20]
&&
X \times_B Y
\ar[dl, "q"{swap}]
\ar[dr, "p"]
\ar[dd, phantom, "{\lrcorner}"{rotate=-45, pos=.3}]
\\
&
X
\ar[dl, "f"{swap}]
\ar[dr, "g"]
&&
Y
\ar[dl, "h"{swap}]
\ar[dr, "k"]
\\
A && B && C
\end{tikzcd}
then functoriality of pull-push means that the
*two* different ways to pull-push through the diagram coincide:
Pull-pushing through the spans separately results in
$$
(k_\ast h^\ast)
\circ
(g_\ast f^\ast)
\;=\;
k_\ast
\,
\underbrace{h^\ast \, g_\ast}
\,
f^\ast
\,,
$$
while pull-pushing through the composite span results in
$$
(k p)_* (f q)^*
=
k_*
\,
\underbrace{p_* \, q^*}
\,
f^*
\,.
$$
For these two operations to coincide, up to [[isomorphism]], hence for pull-push to respect composition of spans, we need an isomorphism between the operations shown over braces, hence of the form
\[
\label{BCConditionInMotivationFromIntegralTransform}
h^\ast \, g_\ast
\;\simeq\;
p_* \, q^*
\]
between the two ways of push-pulling along the sides of any [[pullback square]]
\begin{tikzcd}[sep=20]
&
X \times_B Y
\ar[dl, "q"{swap}]
\ar[dr, "p"]
\ar[dd, phantom, "{\lrcorner}"{rotate=-45, pos=.3}]
\\
X
\ar[dr, "g"]
&&
Y
\ar[dl, "h"{swap}]
\\
& B &
\end{tikzcd}
This (eq:BCConditionInMotivationFromIntegralTransform) is the Beck-Chevalley condition: A natural necessary condition to ensure that [[integral transforms]] by pull-push [[base change]] through [[correspondences]] is functorial under [[composition]] of [[correspondences]] by [[fiber product]] of adjacent legs.
More motivation along these lines also be found at *[[dependent linear type theory]]*. As a formulation of propagation in cohomological [[quantum field theory]] this is [Sc14](dependent+linear+type+theory#Schreiber14) there.
### From logic and type theory
{#MotivationFromLogicAndTypeTheory}
From the point of view of [[formal logic]] and [[dependent type theory]], the three items $f_! \dashv f^\ast \vdash f_\ast$ in a [[base change]] [[adjoint triple]] constitute the [[categorical semantics]] of [[quantification]] and [[context extension]] ([[existential quantification]]/[[dependent pair type]] $\dashv$ [[context extension]] $\dashv$ [[universal quantification]]/[[dependent product type]]), and so the Beck-Chevalley condition says that these are compatible with each other: concretely that [[substitution]] of [[free variables]] commutes with [[quantification]] --- a condition which [[syntax|syntactically]] is "self-evident", at least it is evidently desirable.
Original articles with this logical perspective on the BC condition: [Lawvere (1970), p. 8](#Lawvere70); [Seely (1983), p. 511](#Seely83); [Pavlović (1991), §1](#Pavlović91); [Pavlović (1996), p. 164](#Pavlović96).
For more on this logical aspect see [below](#InTypeTheory).
## Definition
Suppose given a [[commutative square]] (up to [[isomorphism]]) of [[functors]]:
$$\array{ & \overset{f^*}{\to} & \\
^{g^*}\downarrow && \downarrow^{k^*}\\
& \underset{h^*}{\to} & }$$
in which $f^*$ and $h^*$ have [[left adjoint]]s $f_!$ and $h_!$, respectively. (The classical example is a [[Wirthmüller context]].) Then the [[natural isomorphism]] that makes the square commute
$$
k^* f^* \to h^* g^*
$$
has a [[mate]]
$$
h_! k^* \to g^* f_!
$$
defined as the composite
$$
h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_!
\,.
$$
We say the original square satisfies the **Beck--Chevalley condition** if this mate is an [[isomorphism]].
More generally, it is clear that for this to make sense, we only need a transformation $k^* f^* \to h^* g^*$; it doesn't need to be an isomorphism. We also use the term *Beck--Chevalley condition* in this case,
### Left and right Beck--Chevalley condition
Of course, if $g^*$ and $k^*$ also have [[left adjoints]], there is also a Beck--Chevalley condition stating that the corresponding mate $k_! h^* \to f^* g_!$ is an isomorphism, and this is not equivalent in general. Context is usually sufficient to disambiguate, although some people speak of the "left" and "right" Beck--Chevalley conditions.
Note that if $k^* f^* \to h^* g^*$ is not an isomorphism, then there is only one possible Beck-Chevalley condition.
### Dual Beck--Chevalley condition
If $f^*$ and $h^*$ have *right* adjoints $f_*$ and $h_*$, there is also a dual Beck--Chevalley condition saying that the mate $g^* f_* \to h_* k^*$ is an isomorphism. By adjointness, if $f^*$ and $h^*$ have right adjoints and $g^*$ and $k^*$ have left adjoints, then $g^* f_* \to h_* k^*$ is an isomorphism if and only if $k_! h^* \to f^* g_!$ is.
### For bifibrations
Originally, the Beck-Chevalley condition was introduced in ([Bénabou-Roubaud, 1970](#BenabouRoubaud70)) for [[bifibrations]] over a base category with pullbacks. In *loc.cit.* they call this condition **Chevalley condition** because he introduced it in his 1964 seminar.
A [[bifibration]] $\mathbf{X} \to \mathbf{B}$ where $\mathbf{B}$ has pullbacks satisfies the **Chevalley condition** iff for every commuting square
$$\array{ & \overset{\psi^\prime}{\rightarrow} & \\
\downarrow^{\varphi^\prime} && \downarrow^{\varphi}\\
& \underset{\psi}{\rightarrow} & }$$
in $\mathbf{X}$ over a pullback square in the base $\mathbf{B}$ where $\varphi$ is [[cartesian morphism|cartesian]] and $\psi$ is cocartesian it holds that $\varphi^\prime$ is cartesian
iff $\psi^\prime$ is cocartesian. Actually, it suffices
to postulate one direction because the other one follows.
The nice thing about this formulation is that there is no
mention of "canonical" morphisms and no mention of [[cleavages]].
A fibration $P$ has products satisfying the Chevalley condition iff the opposite fibration $P^{op}$ is
a bifibration satisfying the Chevalley condition in the above sense.
According to the [[Benabou–Roubaud theorem]], the Chevalley condition is crucial for establishing the connection between the descent in the sense of fibered categories and the [[monadic descent]].
### "Local" Beck--Chevalley condition {#Local}
Suppose that $f^*$ and $h^*$ do not have entire left adjoints, but that for a particular object $x$ the left adjoint $f_!(x)$ exists. This means that we have an object "$f_! x$" and a morphism $\eta_x\colon x \to f^* f_! x$ which is initial in the [[comma category]] $(x / f^*)$. Then we have $k^*(\eta) \colon k^* x \to k^* f^* f_! x \to h^* g^* f_! x$, and we say that the square satisfies the *local Beck-Chevalley condition at $x$* if $k^*(\eta)$ is initial in the comma category $(k^* x / h^*)$, and hence exhibits $g^* f_! x$ as "$h_! k^* x$" (although we have not assumed that the entire functor $h_!$ exists).
If the functors $f_!$ and $h_!$ do exist, then the square satisfies the (global) Beck-Chevalley condition as defined above if and only if it satisfies the local one defined here at every object.
## In logic / type theory
{#InTypeTheory}
If the functors in the formulation of the Beck-Chevalley condition are [[base change]] functors in the [[categorical semantics]] of some [[dependent type theory]] (or just of a [[hyperdoctrine]]) then the BC condition is equivalently stated in terms of logic as follows.
A [[commuting diagram]]
$$
\array{
D &\stackrel{h}{\to}& C
\\
{}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{g}}
\\
A &\stackrel{f}{\to}& B
}
$$
is interpreted as a morphism of [[contexts]]. The [[pullback]] (of [[slice categories]] or of fibers in a [[hyperdoctrine]]) $h^*$ and $f^*$ is interpreted as the [[substitution]] of [[variables]] in these contexts. And the [[left adjoint]] $\sum_k \coloneqq k_! $ and $\sum_q \coloneqq g_!$, the [[dependent sum]] is interpreted (up to [[truncated|(-1)-truncation]], possibly) as [[existential quantifier|existential quantification]].
In terms of this the Beck-Chevalley condition says that if the above diagram is a [[pullback]], then **substitution commutes with existential quantification**
$$
\sum_k h^* \phi \stackrel{\simeq}{\to} f^* \sum_g \phi
\,.
$$
+-- {: .num_example}
###### Example
Consider the diagram of [[contexts]]
$$
\array{
[\Gamma, x : X] &\stackrel{}{\to}& [\Gamma, x : X, y : Y]
\\
\downarrow && \downarrow
\\
\Gamma &\to& [\Gamma, y : Y]
}
\;\;\;
\simeq
\;\;\;
\array{
\Gamma \times X &\stackrel{(p_1,p_2,t)}{\to}& \Gamma \times X \times Y
\\
{}^{\mathllap{p_1}}\downarrow && \downarrow^{\mathrlap{(p_1,p_3)}}
\\
\Gamma &\stackrel{(id,t)}{\to} & \Gamma \times Y
}
\,,
$$
with the horizontal morphism coming from a [[term]] $t : \Gamma \to Y$ in [[context]] $\Gamma$ and the vertical morphisms being the evident [[projection]], then the condition says that we may in a [[proposition]] $\phi$ substitute $t$ for $y$ before or after quantifying over $x$:
$$
\sum_{x : X} \phi(x,t) \simeq (\sum_{x : X} \phi(x,y))[t/y]
\,.
$$
=--
## Examples
### Various
* The [[codomain fibration]] of any [[category]] with [[pullbacks]] is a bifibration, and satisfies the Beck--Chevalley condition at every pullback square. In particular, [[locally cartesian closed categories]] always satisfy the Beck--Chevalley condition.
* If $C$ is a [[regular category]] (such as a [[topos]]), the bifibration $Sub(C) \to C$ of [[subobjects]] satisfies the Beck--Chevalley condition at every pullback square.
* The [[family fibration]] $Fam(C)\to Set$ of any category $C$ with small sums satisfies the Beck--Chevalley condition at every pullback square in $Set$.
* [Mackey's restriction formula](double+coset#MackeyFormula) for group representations.
### Images and pre-images
\begin{example}
\label{BeckChevalleyForPreImageBetweenPowerSets}
**(Beck-Chevalley for images and pre-images of sets)**
\linebreak
Given a [[set]] $S \,\in\, Set$, write $\mathcal{P}(S)$ for its [[power set]] regarded as a [[poset]], meaning that
1. $\mathcal{P}(S)$ is the set of [[subsets]] $A \subset S$,
1. regarded as a [[category]] by declaring that there is a [[morphisms]] $A \to B$ precisely if $A \subset B$ as subsets of $S$.
Then for $f \,\colon\, S \longrightarrow T$ a [[function]] between sets, we obtain the followin three [[functors]] between their [[power sets]]:
1. $f_! \,\coloneqq\, f(-) \,\colon\, \mathcal{P}(S) \longrightarrow \mathcal{P}(T)$
forms [[images]] under $f$
1. $f^\ast \,\coloneqq\, f^{-1} \,\colon\, \mathcal{P}(T) \longrightarrow \mathcal{P}(S)$
forms [[preimages]] under $f$,
1. $f_ast \,\coloneqq\, T \setminus f\big(S \setminus (-)\big) \,\colon\, \mathcal{P}(S) \longrightarrow \mathcal{P}(T)$
forms the [[complement]] of [[images]] of [[complements]].
These three functors constitute an [[adjoint triple]]
$$
f(-) \;\dashv\; f^{-1} \;\dashv\; f_\ast
$$
as is readily verified and also discussed at some length in the entry *[[interactions of images and pre-images with unions and intersections]]*.
In particular, the familiar/evident fact that forming [[images]] and [[preimages]] [[preserved colimit|preserves]] [[unions]] may be understood as a special case of the fact that [[left adjoints preserve colimits]].
Now given a [[pullback diagram]] in [[Set]]
\[
\label{PullbackOfSets}
\array{
D'
&\overset{\beta}{\longrightarrow}&
D
\\
\mathllap{{}^{\psi}}\big\downarrow
&{}^{{}_{(pb)}}&
\big\downarrow\mathrlap{{}^{\phi}}
\\
C'
&\underset{\alpha}{\longrightarrow}&
C
}
\]
then these operations satisfy the left Beck-Chevalley condition in the sense that the following [[diagram]] of [[poset]]-[[homomorphisms]] ([[functors]]) [[commuting diagram|commutes]]:
\[
\label{RightBeckChevalleyDiagramForPreImages}
\array{
\mathcal{P}(D')
&\overset{\beta_!}{\longrightarrow}&
\mathcal{P}(D)
\\
\mathllap{{}^{\psi^{-1}}}\big\uparrow
&&
\big\uparrow\mathrlap{{}^{\phi^{-1}}}
\\
\mathcal{P}(C')
&\underset{\alpha_!}{\longrightarrow}&
\mathcal{P}(C)
}
\]
To see this it is sufficient to check commutativity on [[singleton]]-[[subsets]] (because every other subset is a [[union]] of singletons and, as just remarked, [[interactions of images and pre-images with unions and intersections|images and preimages preserve unions]])
and given a [[singleton]] set $\{c'\} \subset C'$ on an element $c' \,\in\, C'$, we directly compute as follows:
\[
\label{PreimagesInAPullbackDiagram}
\begin{array}{l}
\beta \circ \psi^{-1}\big(\{c'\}\big)
\\
\;\simeq\;
\beta\Big(
\big\{
(c',d)
\,\big\vert\,
d \,\in\, D
\,,
\phi(d) = \alpha(c')
\big\}
\Big)
\\
\;=\;
\big\{
d \,\in\, D
\,\big\vert\,
\phi(d) = \alpha(c')
\big\}
\\
\;=\;
\phi^{-1}\big(
\{\alpha(c')\}
\big)
\\
\;=\;
\phi^{-1} \circ \alpha\big(\{c'\}\big)
\,.
\end{array}
\]
Here it is the first two steps which make use of the assumption that (eq:PullbackOfSets) is a [[pullback square]], namely which means that $D'$ is compatibly [[bijection|bijective]] to the [[fiber product]] of $C'$ with $D$ over $C$:
$$
\array{
D'
&\simeq&
\big\{
(c', d) \in C' \times D
\,\big\vert\,
\phi(d)
\,=\,
\alpha(c')
\big\}
\\
\mathllap{{}^{\beta}}
\big\downarrow
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\big\downarrow\mathrlap{{}^{ (c',d) \mapsto d }}
\\
D
&=&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
D
}
$$
\end{example}
This elementary example \ref{BeckChevalleyForPreImageBetweenPowerSets} actually constitutes half of the proof of the archetypical example of Prop. \ref{BCForPresheavesOnPullbacksOfOpfibrations} below.
### Systems of functor categories
{#PullbacksOfOpfibrations}
\begin{proposition}
\label{BCForPresheavesOnPullbacksOfOpfibrations}
Given
1. $\phi \colon D \to C$ an [[opfibration]] of [[small categories|small]] [[strict categories]],
1. a [[pullback]] [[diagram]] in the _[[1-category]]_ [[Cat]] of [[small category|small]] [[strict categories]] and [[functors]], of the form
\[
\label{StrictPullbackOfOpfibration}
\array{
D'
&\overset{\beta}{\longrightarrow}&
D
\\
\mathllap{{}^{\psi}}\big\downarrow
&{}^{{}_{(pb)}}&
\big\downarrow\mathrlap{{}^{\phi}}
\\
C'
&\underset{\alpha}{\longrightarrow}&
C
}
\]
(hence presenting a [[2-pullback]]/[[homotopy pullback]], see Rem. \ref{PullbackOfOpfibrationModelsTwoPullback} below),
1. $\mathcal{C}$ any [[category]] with all [[small colimits]]
then the induced diagram of [[functor categories]]
$$
\array{
[D', \mathcal{C}]
&\overset{\beta^*}{\longleftarrow}&
[D, \mathcal{C}]
\\
\Big\downarrow\mathrlap{{}^{\psi_!}}
&&
\Big\downarrow\mathrlap{{}^{\phi_!}}
\\
[C', \mathcal{C}]
&\underset{\alpha^*}{\longleftarrow}&
[C,\mathcal{C}]
}
$$
(for $(-)^\ast$ denoting [[precomposition]] and $(-)_!$ denoting [[left Kan extension]])
satisfies the Beck-Chevalley condition, in that there is a [[natural isomorphism]]
$$
\psi_! \circ \beta^*
\;\simeq\; \alpha^* \circ \phi_!
\,.
$$
\end{proposition}
For this statement in the more general context of [[quasicategories]] see [Joyal (2008), prop. 11.6](#Joyal08).
\begin{proof}
The proof relies on two observations:
{#FirstObservationInProofForSystesmOfFunctorCategories} (1.) Since $\phi$ is [[opfibration|opfibered]], for every object $c \in C$ the inclusion functor
\begin{tikzcd}[sep=20pt]
&
\phi^{-1}(c)
\ar[
rrrr,
"{ iota }"
]
& && &
\phi/c
\\
x
\ar[
rr,
"{ f }"
]
&&
y
&&
x
\ar[
rr,
"{ f }"
]
&&
y
\\[-15pt]
\phi(x)
\ar[rr, equals]
\ar[dr, equals]
&&
\phi(y)
\ar[dl, equals]
&&
\phi(x)
\ar[rr, "{ \phi(f) }"]
\ar[dr, "{ p_x }"{swap}]
&&
\phi(y)
\ar[dl, "{ p_y }"]
\\
& c
& &&
& c
\end{tikzcd}
(of the (strict) [[fiber]] $\phi^{-1}(c)$ into the [[comma category]] $\phi/c$) is a [[final functor]].
(This has a simple argument in the case that the categories $D$, $C$ are in fact [[groupoids]]: In this case, $\phi$ being an opfibration equivalently means that it is an [[isofibration]], by [this Exp.](isofibration#GrothIsIsoFibBetweenGroupoids), which evidently implies that the above inclusion $\iota$ is [[essentially surjective functor|essentially surjective]]. But since $\iota$ is also manifestly a [[full functor]], it follows that it is final, by [this Prop.](final+functor#FinalFunctorBetweenGroupoids).)
Therefore the [pointwise formula](Kan+extension#PointwiseByConicalLimits) for the [[left Kan extension]] $\phi_!$ is equivalently given by taking the [[colimit]] over the [[fiber]], instead of over the [[comma category]]:
\[
\label{PointwiseLeftKanExtensionInOpfiberedCase}
\phi_!(X)_c
\;\simeq\;
\underset
{\underset{\phi^{-1}\big(\{c\}\big)}{\longrightarrow}}
{\lim}
\,
X
\,.
\]
(2.) Since (eq:StrictPullbackOfOpfibration) is a (strict) pullback of (strict) categories (i.e. an ordinary pullback of [[sets]] of [[objects]] and of [[sets]] of [[morphisms]]), the operations of taking [[images]] and [[preimages]] $(-)^{-1}$ ([[fibers]]) of the [[object]]-[[functions]] of the given functors satisfy the [[poset|posetal]] Beck-Chevalley condition from Exp. \ref{BeckChevalleyForPreImageBetweenPowerSets}:
\[
\label{PreimagesInAPullbackDiagram}
\begin{array}{l}
\beta \circ \psi^{-1}\big(\{c'\}\big)
\\
\;\simeq\;
\beta\Big(
\big\{
(c',d)
\,\big\vert\,
d \,\in\, D
\,,
\phi(d) = \alpha(c')
\big\}
\Big)
\\
\;=\;
\big\{
d \,\in\, D
\,\big\vert\,
\phi(d) = \alpha(c')
\big\}
\\
\;=\;
\phi^{-1}\big(
\{\alpha(c')\}
\big)
\\
\;=\;
\phi^{-1} \circ \alpha\big(\{c'\}\big)
\,.
\end{array}
\]
for all [[objects]] $c' \,\in\, Obj(C')$.
Now using first (eq:PointwiseLeftKanExtensionInOpfiberedCase) and then (eq:PreimagesInAPullbackDiagram) and then again (eq:PointwiseLeftKanExtensionInOpfiberedCase) yields the following sequence of [[isomorphisms]]
$$
\begin{aligned}
\big(\psi_! \beta^* (X)\big)(c')
&
\;\simeq\;
\underset
{\underset{
\beta \circ \psi^{-1}\big(\{c'\}\big)
}{\longrightarrow}}
{\lim}
\,
X
\\
&
\;\simeq\;
\underset
{\underset{
\phi^{-1} \circ \alpha\big(\{c'\}\big)
}{\longrightarrow}}
{\lim}
\,
X
\\
&
\;\simeq\;
\big(\alpha^* \phi_! (X)\big)(c')
\,,
\end{aligned}
$$
all of them [[natural isomorphism|natural]] in $c' \,\in\, Obj(C')$.
\end{proof}
\begin{remark}
As a [[counter-example]] showing that in Prop. \ref{BCForPresheavesOnPullbacksOfOpfibrations} the condition for $\phi \colon D \to C$ to be an [[opfibration]] is necessary for the statement to hold, consider
* $C \coloneqq \{0\to 1\}$ the [[interval category]],
* $D \coloneqq \ast,\;\;C' \coloneqq \ast$ the [[terminal category]],
* $\phi\coloneqq const_0,\;\;\alpha \coloneqq const_1$.
Then the [[pullback]] $D' = \varnothing$ is the [[empty category]]. Therefore
* $[D',\mathcal{C}] = [\varnothing, \mathcal{C}]$ is the [[terminal category]], so that $\psi_!\circ \beta^\ast$ is a [[constant functor]]
(concretely, $\psi_! \colon \ast \simeq [\varnothing, \mathcal{C}] \to [C',\mathcal{C}]$ is [[left adjoint]] to the projection functor to the [[terminal category]] and [hence](initial+object#AdjointsToConstantFunctors) picks the [[initial object]] of $[C',\mathcal{C}]$),
* while $\alpha^* \circ \phi_!$ is the [[identity functor]]
(to see this observe first that $[C,\mathcal{C}]$ now may be identified with the [[arrow category]] of $\mathcal{C}$, and second that, under this identification, $\phi_! \colon X \mapsto id_X$ sends any [[object]] of $\mathcal{C}$ to its [[identity morphism]], as is verified by observing that this is clearly [[left adjoint]] to the [[domain]]-restriction $\phi^\ast$).
\end{remark}
\begin{remark}
\label{PullbackOfOpfibrationModelsTwoPullback}
In Prop. \ref{BCForPresheavesOnPullbacksOfOpfibrations},
the [[pullback]] (eq:StrictPullbackOfOpfibration) of an [[opfibration]] of [[strict categories]] formed in the [[1-category]] of [[strict categories]] may be understood as modelling the [[2-pullback]] (cf. [there](2-pullback#StrictPullback)) of the corresponding [[cospan]] in the actual [[2-category]] [[Cat]].
In particular, if the [[strict categories]] involved in the pullback diagram (eq:StrictPullbackOfOpfibration) are all [[groupoids]], then a functor such as $\phi$ being a [[Kan fibration]] is equivalent to its [[nerve]] being a [[Kan fibration]] (see at *[[right Kan fibration]]*, [this and](right%2Fleft+Kan+fibration#FibrationInGroupoidsAsRightKanFibrations) [this Prop](right%2Fleft+Kan+fibration#OverKanComplex)), in which case the pullback diagram (eq:StrictPullbackOfOpfibration) models the [[homotopy pullback]] (by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback)).
\end{remark}
### Enriched functor categories
Consider
* $(\mathbf{V}, \otimes, \mathbb{1})$ a [[cocomplete category|cocomplete]] [[symmetric monoidal category]], regarded as a [[cosmos for enrichment]],
* $\mathbf{C}$ a [[cocomplete category|cocomplete]] $\mathbf{V}$-[[enriched category]] which is also [[tensoring|tensored]] over $\mathbf{V}$, denoted:
$$
(-)\cdot(-)
\;\colon\;
\mathbf{V} \times \mathbf{C}
\longrightarrow
\mathbf{C}
$$
and write
* for any [[small category|small]] $\mathbf{V}$-[[enriched category]] $\mathcal{X} \,\in\, \mathbf{V}Cat^{small}$:
$$
\mathbf{V}Func(\mathcal{X},\,\mathbf{C})
\;\in\;
Cat
$$
for the $\mathbf{V}$-[[enriched-functor category]] from $\mathcal{X}$ to $\mathbf{C}$,
* for any $\mathbf{V}$-enriched functor $f \,\colon\, \mathcal{X} \longrightarrow \mathcal{Y}$:
$$
\mathbf{V}Func(\mathcal{X},\,\mathbf{C})
\underoverset
{\underset{f^\ast}{\longleftarrow}}
{\overset{f_!}{\longrightarrow}}
{\;\;\; \bot \;\;\;}
\mathbf{V}Func(\mathcal{Y},\,\mathbf{C})
$$
for the [[pair]] of [[adjoint functors]] given by [[precomposition]] and by [[left Kan extension]], the latter being expressible by the following [[coend]]-formula (see [there](Kan+extension#eq:LeftKanExtensionViaCoendFormula)):
\[
\label{CoendFormulaForLeftKanExtension}
\mathscr{V}_{(-)}
\;\in\; \mathbf{V}Func(\mathcal{X},\,\mathbf{C})
\;\;\;\;\;\;\;\;
\vdash
\;\;\;\;\;\;\;\;
(f_! \mathscr{V})_y
\;\simeq\;
\int^{x \in \mathcal{X}}
\mathcal{X}\big(f(x),\,y\big)
\cdot
\mathscr{V}_{x}
\,.
\]
The following is a rather simplistic but maybe instructive example:
\begin{example}
\label{ForPushPullofEnrichedPresheavesThroughProjectionDiagram}
In the case of a *[[cartesian closed category|cartesian closed]]* [[cosmos]] $(\mathbf{V}, \times, \ast)$
consider for $\mathcal{X}, \mathcal{Y}, \mathcal{X}' \,\in\, \mathbf{V}Cat^{small}$
a commuting diagram of $\mathbf{V}$-[[enriched functors]] of the following form:
$$
\array{
\mathcal{X} \times \mathcal{X}'
&\overset{f \times id}{\longrightarrow}&
\mathcal{Y} \times \mathcal{X}'
\\
\mathllap{{}^{pr_{\mathcal{X}}}}
\big\downarrow
&{}^{{}_{(pb)}}&
\big\downarrow
\mathrlap{{}^{pr_{\mathcal{Y}}}}
\\
\mathcal{X}
&\underset{\;\;\; f \;\;\;}{\longrightarrow}&
\mathcal{Y}
}
$$
(which is a [[pullback]] square in the [[1-category]] of [[strict category|strict]] $\mathbf{V}$-[[enriched categories]]).
Then the Beck-Chevalley condition on $\mathbf{C}$-valued $\mathbf{V}$-functors holds, in that the following [[commuting diagram|diagram commutes]]:
$$
\array{
\mathbf{V}Func(
\mathcal{X} \times \mathcal{X}'
,\,
\mathbf{C}
\big)
&\overset{(f \times id)_!}{\longrightarrow}&
\mathbf{V}Func(
\mathcal{Y} \times \mathcal{X}'
,\,
\mathbf{C}
)
\\
\mathllap{{}^{(pr_{\mathcal{X}})^\ast}}
\big\uparrow
&&
\big\uparrow
\mathrlap{{}^{(pr_{\mathcal{Y}})^\ast}}
\\
\mathbf{V}Func(\mathcal{X},\,\mathbf{C})
&\underset{\;\;\; f_! \;\;\;}{\longrightarrow}&
\mathbf{V}Func(\mathcal{Y},\,\mathbf{C})
}
$$
To see this, we may check object-wise as follows:
$$
\begin{array}{l}
\big(
(f \times id)_!
(pr_{\mathcal{X}})^\ast
\mathscr{V}
\big)_{(y,x')}
\\
\;=\;
\int^{(x,x'_0)}
\mathcal{Y}\big(f(x),y\big)
\times
\mathcal{X}'\big(x'_0,x'\big)
\cdot
\big(
(pr_{\mathcal{X}})^\ast\mathscr{V}_{x,x'_0}
\big)
\\
\;=\;
\int^{(x,x'_0)}
\mathcal{Y}\big(f(x),y\big)
\times
\mathcal{X}'\big(x'_0,x'\big)
\cdot
\mathscr{V}_{x}
\\
\;=\;
\int^{x'_0}
\int^{x}
\mathcal{Y}\big(f(x),y\big)
\times
\mathcal{X}'\big(x'_0,x'\big)
\cdot
\mathscr{V}_{x}
\\
\;\simeq\;
\int^{x'_0}
\mathcal{X}'\big(x'_0,x'\big)
\cdot
\int^{x}
\mathcal{Y}\big(f(x),y\big)
\cdot
\mathscr{V}_{x}
\\
\;\simeq\;
\int^{x}
\mathcal{Y}\big(f(x), y\big)
\cdot
\mathscr{V}_{x}
\\
\;\simeq\;
\big(
f_!
\mathscr{V}
\big)_y
\\
\;\simeq\;
\big(
(pr_{\mathcal{Y}})^\ast
(
f_!
\mathscr{V}
)
\big)_{y,x'}
\,,
\end{array}
$$
where we used (apart from the definition of [[precomposition]]), in order of appearance:
1. the [[coend]]-expression (eq:CoendFormulaForLeftKanExtension) for the [[left Kan extension]],
1. the [[Fubini theorem]] for coends ([here](end#Fubini)),
1. the fact that the [[closed monoidal category|closed tensor]] $(-) \times (-)$ [[preserves colimits]] and hence [[coends]] in each variable,
1. the [[co-Yoneda lemma]] in the coend-form [here](co-Yoneda+lemma#coYonedaLemma)
and finally the expression (eq:CoendFormulaForLeftKanExtension) again.
\end{example}
### Proper base change in étale cohomology
For [[coefficients]] of [[torsion group]], [[étale cohomology]]
satisfies Beck-Chevalley along [[proper morphisms]]. This is the statement of the _[[proper base change theorem]]_. See there for more details.
### Grothendieck's yoga of six operations
A Beck-Chevalley condition is part of the [[axioms]] of [[Grothendieck's yoga of six operations]], e.g. [Cisinski & Déglise (2009), §A.5](#CisinskiDéglise09), [Gallauer (2021), p. 16](#Gallauer21), [Scholze (2022), p. 11](#Scholze22).
## Related pages
* [[exact square]]
* [[Benabou-Roubaud theorem]]
## References
{#References}
> The Beck-Chevalley condition has arisen in the theory of [[monadic descent|descent]] - as developed from [[Grothendieck]] 1959. [[Jon Beck]] and [[Claude Chevalley]] studied it independently from each another. [...] It is conspicuous that neither of them ever published anything on it. [[Pavlović 1991, §14](#Pavlović91)]
Original articles:
* {#BenabouRoubaud70} [[Jean Bénabou]], [[Jacques Roubaud]], _Monades et descente_, C. R. Acad. Sc. Paris, Ser. A **270** (1970) 96-98 [[gallica:12148/bpt6k480298g/f100](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f100), [[BenabouRoubaud-MonadesEtDescente.pdf:file]], English translation (by [[Peter Heinig]]): [MO:q/279152](https://mathoverflow.net/q/279152)]
> (proving the [[Bénabou-Roubaud theorem]])
* {#Lawvere70} [[William Lawvere]], p. 8 of: *[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]*, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970) 1-14 [[pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf)]
> (in terms of [[base change]] as the [[categorical semantics]] for [[universal quantification]]/[[dependent product]] and [[existential quantification]]/[[dependent sum]])
expanded on in
* {#Seely83} [[Robert A. G. Seely]], *Hyperdoctrines, Natural Deduction and the Beck Condition*, Zeitschr. f. math. Logik und Grundlagen d. Math. **29** (1983) 505-542 [[doi:10.1002/malq.19830291005]( https://doi.org/10.1002/malq.19830291005), [pdf](https://www.math.mcgill.ca/seely/ZML/ZML.PDF)]
Review:
* {#Pavlovic1990} [[Duško Pavlović]], pp. 105 in: *Predicates and Fibrations*, PhD thesis, Utrecht (1990) [[pdf](https://dusko.org/wp-content/uploads/2020/03/1990-proefschrift-dusko.pdf), [[Pavlovic-PredicatesAndFibrations.pdf:file]]]
* {#Pavlović91} [[Duško Pavlović]], Section 1 of: *Categorical interpolation: Descent and the Beck-Chevalley condition without direct images*, in: *Category Theory*, Lecture Notes in Mathematics **1488** (1991) [[doi:10.1007/BFb0084229](https://doi.org/10.1007/BFb0084229), [pdf](http://dusko.org/wp-content/uploads/2020/03/1990-Como-BCDE.pdf), [[Pavlovic-CategoricalInterpolation.pdf:file]]]
* [[Simon John Ambler]], §5.5.1 in: *First Order Linear Logic in Symmetric Monoidal Closed Categories*, PhD thesis, Edinburgh (1991) [[ECS-LFCS-92-194](https://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-194/ECS-LFCS-92-194.pdf), [[Ambler-FOLL.pdf:file]]]
> (in view of [[linear type theory]])
* {#Pavlović96} [[Duško Pavlović]], *Maps II: Chasing Diagrams in Categorical Proof Theory*, Logic Journal of the IGPL, **4** 2 (1996) 159–194 [[doi:10.1093/jigpal/4.2.159](https://doi.org/10.1093/jigpal/4.2.159), [pdf](http://dusko.org/wp-content/uploads/2020/03/00-95-IGPL-mapsII.pdf)]
* [[Paul Balmer]], around §7.5 of: *Stacks of group representations*, J. European Math. Soc. **17** 1 (2015) 189-228 [[arXiv:1302.6290](https://arxiv.org/abs/1302.6290), [doi:10.4171/jems/501](https://doi.org/10.4171/jems/501)]
Discussion for [[subobject lattices]]:
* [[Saunders MacLane]], [[Ieke Moerdijk]], chapter IV.9 (around p. 205) of: _[[Sheaves in Geometry and Logic]]_, Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)]
Review in the context of [[Grothendieck's yoga of six operations]]:
* {#CisinskiDéglise09} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], section A.5 of: *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer (2019) [[arXiv:0912.2110](http://arxiv.org/abs/0912.2110), [doi:10.1007/978-3-030-33242-6](https://doi.org/10.1007/978-3-030-33242-6)]
* {#Gallauer21} [[Martin Gallauer]], pp. 16 of: *An introduction to six-functor formalism*, lecture at *[The Six-Functor Formalism and Motivic Homotopy Theory](https://sites.google.com/view/summer-school-motivic/home)*, Università degli Studi di Milano (Sept. 2021) [[arXiv:2112.10456](https://arxiv.org/abs/2112.10456), [pdf](https://homepages.warwick.ac.uk/staff/Martin.Gallauer/docs/m6ff.pdf)]
* {#Scholze22} [[Peter Scholze]], p. 11 of: *Six Functor Formalisms*, [lecture notes ](https://people.mpim-bonn.mpg.de/scholze/Course%20Winter%202223.html) (2022) [[pdf](https://people.mpim-bonn.mpg.de/scholze/SixFunctors.pdf), [[Scholze-SixOperations.pdf:file]]]
Discussion for [[derived functors]] with focus on the example of [[base change]] of [[retractive spaces]] and [[homotopy categories]] of [[parameterized spectra]]:
* [[Michael Shulman]], *Comparing composites of left and right derived functors*, New York Journal of Mathematics **17** (2011) 75-125 [[arXiv:0706.2868](https://arxiv.org/abs/0706.2868), [eudml:229181](https://eudml.org/doc/229181)]
* [[Mike Shulman]], *Framed bicategories and monoidal fibrations*, Theory and Applications of Categories, **20** 18 (2008) 650–738 [[tac:2018](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html), [arXiv:0706.1286](https://arxiv.org/abs/0706.1286)]
Discussion in the generality of [[enriched category theory]]:
* [[Michael Shulman]], e.g. Thm. 9.3 in: *Enriched indexed categories*, Theory Appl. Categ. **28** (2013) 616-695 [[doi:1212.3914](https://arxiv.org/abs/1212.3914), [tac:28-21](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html)]
Discussion in the generality of [[(infinity,1)-categories|$\infty$-categories]]:
* {#HopkinsLurie13} [[Michael Hopkins]], [[Jacob Lurie]], Prop. 4.3.3 in: *[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]* (2013) [[pdf](http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)]
* [[Urs Schreiber]], around Def. 5.5 of: *[[schreiber:Quantization via Linear homotopy types]]* [[arXiv:1402.7041](https://arxiv.org/abs/1402.7041)]
* {#GaitsgoryLurie} [[Dennis Gaitsgory]], [[Jacob Lurie]], Section 2.4.1 of: _Weil's conjecture for function fields_ (2014-2017) [["first volume of expanded account" pdf](https://www.math.ias.edu/~lurie/papers/tamagawa-abridged.pdf)]
for [[infinity-cosmos|$\infty$-cosmoi]]:
* [[Emily Riehl]], [[Dominic Verity]], Prop. 5.3.9 in: *Kan extensions and the calculus of modules for ∞-categories*, Algebr. Geom. Topol. **17** (2017) 189-271 [[doi:10.2140/agt.2017.17.189](https://doi.org/10.2140/agt.2017.17.189), [arXiv:1507.01460](https://arxiv.org/abs/1507.01460)]
Discussion for presheaf categories in the context of [[quasicategories]] ([[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]]):
* {#Joyal08} [[André Joyal]], [p. 175](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=27) of: *[[The Theory of Quasi-Categories and its Applications]]*, lectures at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM (2008) [[pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]]]
Discussion for [[Goursat categories]]:
* [[Marino Gran]], [[Diana Rodelo]], *Beck-Chevalley condition and Goursat categories*, Journal of Pure and Applied Algebra **221** 10 (2017) 2445-2457 [[arXiv:1512.04066](https://arxiv.org/abs/1512.04066), [doi:10.1016/j.jpaa.2016.12.031](https://doi.org/10.1016/j.jpaa.2016.12.031)]
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|
Becker-Gottlieb transfer
|
https://ncatlab.org/nlab/source/Becker-Gottlieb+transfer
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The Becker-Gottlieb transfer is a variant of [[push-forward in generalized cohomology]] of [[cohomology theories]] along [[proper map|proper]] [[submersions]] of [[smooth manifolds]].
The Becker-Gottlieb transfer operation has been refined to [[differential cohomology]] in ([Bunke-Gepner 13](#BunkeGepner13)).
Its compatibility in [[differential algebraic K-theory]] with the differential refinement of the [[Borel regulator]] is the content of the _transfer index conjecture_ ([Bunke-Tamme 12, conjecture 1.1](#BunkeTamme12), [Bunke-Gepner 13, conjecture 5.3](#BunkeGepner13)).
For the moment see at _[regulator -- Becker-Gottlieb transfer](Beilinson+regulator#GottliebTransfer)_ for more.
## Definition
See e.g. ([Haugseng 13, def. 3.9](#Haugseng13)).
## Related concepts
* [[sheaf with transfer]]
## References
The original articles
* {#BeckerGottlieb75} [[James Becker]], [[Daniel Gottlieb]], _The transfer map and fiber bundles_, Topology , 14 (1975) ([pdf](https://www.math.purdue.edu/~gottlieb/Bibliography/Transfb.pdf), <a href="https://doi.org/10.1016/0040-9383(75)90029-4">doi:10.1016/0040-9383(75)90029-4</a>)
(which also gives a proof of the [[Adams conjecture]]).
* [[James Becker]], [[Daniel Gottlieb]], _Vector fields and transfers_ Manuscr. Math. , 72 (1991) pp. 111–130 ([pdf](https://www.math.purdue.edu/~gottlieb/Papers/manuscripta.pdf), [doi:10.1007/BF02568269](https://link.springer.com/article/10.1007/BF02568269))
Interpretation in terms of [[dualizable objects]]:
* [[Albrecht Dold]], [[Dieter Puppe]], *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics, **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)]
* {#BeckerGottlieb} [[James Becker]], [[Daniel Gottlieb]]: *A History of Duality in Algebraic Topology* [[pdf](http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf), [[BeckerGottlieb-DualityHistory.pdf:file]]]
Review:
* [[Dai Tamaki]], [[Akira Kono]], Section 4.5 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))
* [[eom]], _[Becker-Gottlieb transfer](http://www.encyclopediaofmath.org/index.php/Becker-Gottlieb_transfer)_
Discussion in the context of [[differential algebraic K-theory]] is in
* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], section 2.1 of _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))
* {#BunkeGepner13} [[Ulrich Bunke]], [[David Gepner]], _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_ ([arXiv:1306.0247](http://arxiv.org/abs/1306.0247))
In prop. 4.14 of
* {#Haugseng13} [[Rune Haugseng]], _The Becker-Gottlieb Transfer Is Functorial_ ([arXiv:1310.6321](http://arxiv.org/abs/1310.6321)) (withdrawn now)
Becker-Gottlieb transfer was identified with the [[Umkehr map]] induced from a [[Wirthmüller context]] in which in addition $f_\ast$ satisfies its [[projection formula]] (a "[[transfer context]]", def.4.9)
The article
* {#KleinMalkiewich16} [[John Klein]], [[Cary Malkiewich]], _The transfer is functorial_, [arXiv:1603.01872](http://arxiv.org/abs/1603.01872),
claimed to establish the functoriality of the Becker-Gottlieb transfer for fibrations with finitely dominated fibers on the level of [[homotopy categories]] (without higher coherences), but contained an unfixable mistake (cf. _[Corrigendum to 'The transfer is functorial'](https://www.sciencedirect.com/science/article/pii/S0001870822005692)_).
[[!redirects transfer index conjecture]]
|
becoming
|
https://ncatlab.org/nlab/source/becoming
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
According to _[[Science of Logic]]_ ([SoL §152](Science+of+Logic#152), [Lawvere 91, p. 11](#Lawvere91)) "becoming" is the [[unity of opposites]] of [[nothing]] and [[being]]. The reverse is _[[ceasing]]_. The [[Aufhebung]] of both is [[Dasein]]:
| | | [[Dasein]] | | |
|--|--|--|--|--|
| [[becoming]] : | [[nothing]] | $\;\;\;\dashv$ | [[being]] | : [[ceasing]] |
The universal factorization via $\emptyset$-[[unit of a monad|unit]] and $\ast$-[[counit of a comonad]]:
$$
\array{
\emptyset &\longrightarrow& X &\longrightarrow& \ast
\\
\\
nothing && becoming && being
}
$$
of the unique [[function]] from the [[empty type]] to the [[unit type]] through any other [[type]] $X$.
Indeed, this is the statement of [SoL §174](Science+of+Logic#174): _there is nothing which is not an intermediate state between being and nothing._
## References
* [[Hegel]], _[[Science of Logic]]_, Volume One: _The Objective Logic_, Book One: _The Doctrine of Being_, Chapter 1 _Being_, C _Becoming_
* [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844
* {#Lawvere91} [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_1991
[[!redirects Werden]]
[[!redirects werden]]
|
behavior
|
https://ncatlab.org/nlab/source/behavior
|
# Behaviours
* table of contents
{: toc}
## Idea
Presumably, this is some abstraction of the notion of behaviour?
## Defintions
A __behaviour__ (or __behavior__) is ....
Given two behaviour $B$ and $C$, a __behaviour morphism__ from $B$ to $C$ is ....
Behaviours and behaviour morphisms form a [[category]] $Beh$.
## References
* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)
[[!redirects behavior]]
[[!redirects behaviors]]
[[!redirects behaviour]]
[[!redirects behaviours]]
[[!redirects Beh]]
|
Behrang Noohi
|
https://ncatlab.org/nlab/source/Behrang+Noohi
|
Behrang Noohi is reader in mathematics at Queen Mary University, London.
* [website](https://www.qmul.ac.uk/maths/profiles/noohib.html)
## Selected writings
On [[transgression in group cohomology]] generalized to Real $\mathbb{Z}/2$-equivariant cohomology (as appropriate for twists of [[KR-theory]]):
* [[Behrang Noohi]], [[Matthew B. Young]], *Twisted loop transgression and higher Jandl gerbes over finite groupoids*, Algebr. Geom. Topol. **22** (2022) 1663-1712 [[arXiv:1910.01422](https://arxiv.org/abs/1910.01422), [doi:10.2140/agt.2022.22.1663](https://doi.org/10.2140/agt.2022.22.1663)]
## Related entries
* [[geometric stack]]
* [[butterfly]]
* [[nonabelian group cohomology]]
* [[string topology]]
category: people
|
Bei Zeng
|
https://ncatlab.org/nlab/source/Bei+Zeng
|
* [institute page 1](https://uwaterloo.ca/institute-for-quantum-computing/about/people/bei-zeng)
* [institute page 2](https://physics.ust.hk/eng/people_detail.php?pplcat=1&id=555)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Bei_Zeng)
## Selected writings
On [[quantum information theory]] ([[entanglement entropy]]) applied to [[topological phases of matter]]/[[topological order]]:
* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:
*[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$
category: people
|
Beilinson conjecture
|
https://ncatlab.org/nlab/source/Beilinson+conjecture
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_Beilinson's conjectures_ ([Beilinson 85](#Beilinson85)) [[conjecture]] for [[arithmetic varieties]] over [[number fields]]
1. that the realification of the [[Beilinson regulator]] exhibits an [[isomorphism]] between the relevant [[algebraic K-theory]]/[[motivic cohomology]] groups and [[Deligne cohomology]] ([[ordinary differential cohomology]]) groups;
(recalled e.g. as [Schneider 88, p. 30](#Schneider88), [Brylinski-Zucker 91, conjecture 5.20](#BrylinskiZucker91), [Deninger-Scholl (3.1.1)](#DeningerScholl), [Nekovar (6.1 (1))](#Nekovar)).
1. induced by this that [[special values of L-functions|special values]] of the ([[Hasse-Weil L-function|Hasse-Weil]]-type) [[L-function]] are proportional to the [[Beilinson regulator]], in analogy with the [[class number formula]] and the [[Birch and Swinnerton-Dyer conjecture]]
(recalled e.g. as [Schneider 88, p. 31](#Schneider88) [Brylinski-Zucker 91, conjecture 5.21](#BrylinskiZucker91), [Deninger-Scholl (3.1.2)](#DeningerScholl), [Nekovar (6.1 (2))](#Nekovar))).
The Beilinson conjecture for [[special values of L-functions]] follows the [[Birch and Swinnerton-Dyer conjecture]] and [[Pierre Deligne]]'s conjecture on special value of L-functions.
## Related concepts
* [[L-function]]
* [[Beilinson regulator]]
* [[standard conjectures]]
## References
The original articles are
* {#Beilinson85} [[Alexander Beilinson]] _Higher regulators and values of L-functions_, Journal of Soviet Mathematics 30 (1985), 2036-2070, ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861))
* {#Beilinson80} [[Alexander Beilinson]], _Higher regulators of curves_, Funct. Anal. Appl. 14 (1980), 116-118, [mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=1800&option_lang=eng).
* {#Beilinson87} [[Alexander Beilinson]], _Height pairing between algebraic cycles_, in _K-Theory, Arithmetic and Geometry_, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, [DOI](http://dx.doi.org/10.1007/BFb0078364).
Reviews include
* {#RapoportSchappacherSchneider88}
[[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_ Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)
* {#Soule84} [[Christophe Soulé]], _Régulateurs_ Séminaire Bourbaki, 27 (1984-1985), Exp. No. 644, 17 p. ([Numdam](http://www.numdam.org/item?id=SB_1984-1985__27__237_0))
* {#Schneider88} [[Peter Schneider]], _Introduction to the Beilinson conjectures_, in [Rapoport-Schappacher-Schneider 88](#RapoportSchappacherSchneider88) ([[SchneiderBeilinsonConjectures.pdf:file]])
* {#Nekovar} [[Jan Nekovar]], section 3 of _Beilinson's Conjectures_ ([pdf](http://people.math.jussieu.fr/~nekovar/pu/mot.pdf))
* {#DeningerScholl} [[Christopher Deninger]], [[Anthony Scholl]], _The Beilinson conjectures_ ([pdf](https://www.dpmms.cam.ac.uk/~ajs1005/preprints/d-s.pdf))
* {#BrylinskiZucker91} [[Jean-Luc Brylinski]], Steven Zucker, conjecture 5.20,5.21 in _An overview of recent advances in Hodge theory_, English translation in _Several complex variables VI_, volume 69 of _Encyclopedia of Math. Sciences_, pages 39-142, 1990 (original in Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 1991, Volume 69, Pages 48–165 ([web](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intf&paperid=177&option_lang=eng), [pdf (Russian original)](http://www.mathnet.ru/php/getFT.phtml?jrnid=intf&paperid=177&what=fullt&option_lang=eng)))
A [[noncommutative algebraic geometry|noncommutative]] analogue is considered in
* [[Dmitri Kaledin]], _Beilinson conjectures in the noncommutative setting_, [pdf](http://www.math.nyu.edu/~tschinke/books/finite-fields/final/03_kaledin.pdf).
[[!redirects Beilinson conjectures]]
[[!redirects Beilinson's conjectures]]
|
Beilinson monad
|
https://ncatlab.org/nlab/source/Beilinson+monad
|
#Contents#
* table of contents
{:toc}
## Idea
A _monad_ in the sense of Beilinson is a [[chain complex]] whose [[chain homology]] is concentrated in degree 0. (For instance p. 21 [here](#ES)).
In other words, it is a certain [[Eilenberg-MacLane object]] in the context of [[homological algebra]].
(This is unrelated to [[monad (disambiguation)|other notions of monads]]).
## References
* [[David Eisenbud]], Frank-Olaf Schreyer, _Relative Beilinson Monad and Direct Image for Families of Coherent Sheaves_ ([arXiv:math/0506391](http://arxiv.org/abs/math/0506391))
{#ES}
[[!redirects Beilinson monads]]
|
Beilinson's Conjectures on Special Values of L-Functions
|
https://ncatlab.org/nlab/source/Beilinson%27s+Conjectures+on+Special+Values+of+L-Functions
|
This entry is about the book
* [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.)
\linebreak
**Beilinson's Conjectures on Special Values of L-Functions**
\linebreak
Perspectives in Mathematics, Volume 4,
Academic Press, Inc. 1988,
Harcourt Brace Jovanovich, Publishers
[ISBN:978-0-12-581120-0](https://www.sciencedirect.com/book/9780125811200/beilinsons-conjectures-on-special-values-of-l-functions)
[Electronic version](https://web.archive.org/web/20190618180518/https://ivv5hpp.uni-muenster.de/u/pschnei/publ/beilinson-volume/)
on [[Deligne cohomology]] and [[Beilinson's conjectures]] on [[special values of L-functions]].
(The book is out of print and unavailable from [the publisher](https://www.sciencedirect.com/book/9780125811200/beilinsons-conjectures-on-special-values-of-l-functions), even electronically.)
#Contents#
* table of contents
{:toc}
## Preface
* <a href="https://ncatlab.org/nlab/files/beilinson-preface.pdf">pdf</a>
Every Spring and Fall, in the woods of the Black Forest, mathematicians from
West-Germany and other countries gather at Oberwolfach for the “Arbeitsgemeinschaft Geyer-Harder” to teach themselves in a joint effort new theories or results. At each meeting the topic and organizers of the following Arbeitsgemeinschaft are chosen after an evening of discussions by democratic vote. The organizers then prepare a detailed program for the conference which is posted in the mathematical institutes.
Everyone interested in taking part in the meeting has to volunteer for one of the talks in the program. The organizers then choose the actual speakers.
The Arbeitsgemeinschaft of April 1986 was devoted to the work of Beilinson
and Bloch on regulators. The meeting centered around Beilinson's article “Higher regulators and values of L-functions”.
It was generally felt at the conference that the joint effort made to understand this fundamental paper should not be lost, and should be helpful to others. Hence this volume.
Most of its chapters are based on talks at the meeting; others have been added in order to give a coherent account of the conjectures and some of the known evidence for them. In the name of all participants we want to express our gratitude to the Mathematisches Forschungsinstitut Oberwolfach, which makes the Arbeitsgemeinschaft
possible.
1987
[[Michael Rapoport]]
[[Norbert Schappacher]]
[[Peter Schneider]]
## Program proposal
* <a href="https://ncatlab.org/nlab/files/beilinson-programmvorschlag.pdf">pdf</a>
## Introduction to the Beilinson Conjectures
[[Peter Schneider]].
* <a href="https://ncatlab.org/nlab/files/Schneider.pdf">pdf</a>
### §1 Complex L-functions
### §2 Deligne cohomology
### §3 Absolute cohomology
### §4 Chern classes
### §5 The conjectures
### §6 Further hints
## Deligne’s Conjecture
[[Maria Heep]], [[Uwe Weselmann]].
* <a href="https://ncatlab.org/nlab/files/HeepWeselmann.pdf">pdf</a>
### §1 Motives
### §2 Duality, functional equation, criticial values
### §3 Deligne’s periods
### §4 Beilinson’s periods
## Deligne-Beilinson Cohomology
[[Hélène Esnault]], [[Eckart Viehweg]].
* <a href="https://ncatlab.org/nlab/files/EsnaultViehweg.pdf">pdf</a>
### §1 The Deligne cohomology
### §2 The Deligne-Beilinson complex
### §3 Products
### §4 Relative cohomology
### §5 Extensions and complements
### §6 The cycle map in the de Rham cohomology
### §7 The cycle map in the Deligne cohomology
### §8 Chern classes in the Deligne-Beilinson cohomology
## $\lambda$-Rings and Adams Operations in Algebraic K-Theory.
[[Wolfgang K. Seiler]].
* <a href="https://ncatlab.org/nlab/files/Seiler.pdf">pdf</a>
## The Theorem of Riemann-Roch
[[Günter Tamme]].
* <a href="https://ncatlab.org/nlab/files/Tamme.pdf">pdf</a>
### §1 The λ-ring structure and the Chern character for K-theory with supports
### §2 Riemann-Roch without denominators
### §3 The smooth Riemann-Roch theorem for K-theory with supports
### §4 The singular Riemann-Roch
### §5 Absolute cohomology and homology
## Comparison of the Regulators of Beilinson and of Borel.
[[Michael Rapoport]].
* <a href="https://ncatlab.org/nlab/files/Rapoport.pdf">pdf</a>
## The Beilinson Conjecture for Algebraic Number Fields.
[[Jürgen Neukirch]].
### Part I: Regulators and values of Artin L-series
* <a href="https://ncatlab.org/nlab/files/Neukirch1.pdf">pdf</a>
#### §1 Regulators for algebraic number fields
#### §2 Regulators for Artin motives
#### §3 L-series of Artin motives
#### §4 Dirichlet L-series
#### §5 Regulators of Dirichlet motives
### Part II: The regulator map for cyclotomic fields
* <a href="https://ncatlab.org/nlab/files/Neukirch2.pdf">pdf</a>
#### §1 The main theorem
#### §2 Universal symbols
#### §3 Special symbols
#### §4 Reduction to the main lemma
#### §5 Proof of the main lemma for n = 1
#### §6 Proof of the main lemma for n ≥ 2
## On the Beilinson Conjectures for Elliptic Curves with Complex Multiplication
[[Christopher Deninger]], [[Kay Wingberg]].
* <a href="https://ncatlab.org/nlab/files/DeningerWingberg.pdf">pdf</a>
### §1 A formula for the regulator of curves
### §2 A weakened version of the Beilinson conjecture for elliptic curves
### §3 Calculating $\langle\omega,[\alpha,\beta]_{\cal D}\rangle$ for elliptic curves over $\bf R$
### §4 Relations between the L-function of an elliptic curve over Q with complex multiplication and Eisenstein-Kronecker-Lerch series
### §5 The absolute cohomology
## Beilinson’s Theorem on Modular Curves.
[[Norbert Schappacher]], [[Anthony J. Scholl]].
* <a href="https://ncatlab.org/nlab/files/SchappacherScholl.pdf">pdf</a>
### §1 The Theorem
### §2 Transformation of L-values
### §3 Eisenstein series and modular units
### §4 Whittaker functions and L-factors
### §5 Evaluation of the regulator integral
### §6 Non-vanishing of the regulator integral
### §7 Integrality
## Deligne Homology, Hodge-D-Conjecture, and Motives
[[Uwe Jannsen]].
* <a href="https://ncatlab.org/nlab/files/Jannsen.pdf">pdf</a>
|
Beilinson-Bernstein localization
|
https://ncatlab.org/nlab/source/Beilinson-Bernstein+localization
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Beĭlinson-Bernstein localization theorem
Consider a complex [[reductive group]] $G$ with [[Lie algebra]] $\mathfrak{g}$, [[Borel subgroup]] $B \subset G$, and [[flag variety]] $\mathcal{B} = G/B$. The localization theory of Beilinson-Bernstein identifies [[representations]] of $\mathfrak{g}$ with [[global sections]] of (twisted) [[D-modules]] on $\mathcal{B}$. In particular, highest weight representations are realized by $B$-equivariant $\mathcal{D}$-modules on $\mathcal{B}$, or in other words, by $\mathcal{D}$-modules on the [[quotient stack]] $B\backslash \mathcal{B}$.
Furthermore, given a [[subgroup]] $K \subset G$, it identifies [[modules]]
for the [[Harish-Chandra pair]] $(\mathfrak{g}, K)$ with global sections of $K$-equivariant twisted $\mathcal{D}$-modules on $\mathcal{B}$.
The case $K = G$ gives the [[Borel-Weil theorem|Borel-Weil]] description of irreducible algebraic (equivalently, finite-dimensional) representations of $G$ as sections of equivariant [[line bundles]] on $\mathcal{B}$.
([Ben-Zvi&Nadler 07](#BenZviNadler07))
## References
* A. Beilinson, J. Bernstein, _Localisation de $\mathfrak{g}$-modules_, C.R. Acad. Sci. Paris, 292 (1981), 15-18, [MR82k:14015](http://www.ams.org/mathscinet-getitem?mr=82k:14015), Zbl 0476.14019
* [[A. Beilinson]], _Localization of representations of reductive Lie algebra_, Proc. of ICM 1982, (1983), 699-716. Zbl 0571.20032
* [[Valery Lunts]], [[Alexander Rosenberg]], _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159, [MR2001f:17028](http://www.ams.org/mathscinet-getitem?mr=1694897), [doi](http://dx.doi.org/10.1007/s000290050044); _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3894), _II. D-Calculus in the braided case. The localization of quantized enveloping algebras_, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3916)
* [[Edward Frenkel]], [[Dennis Gaitsgory]], _Localization of $\mathfrak{g}$-modules on the affine Grassmannian_, Ann. of Math. (2) 170 (2009), no. 3, 1339–1381, [MR2600875](http://www.ams.org/mathscinet-getitem?mr=2600875), [doi](http://dx.doi.org/10.4007/annals.2009.170.1339)
* Toshiyuki Tanisaki, _The Beilinson-Bernstein correspondence for quantized enveloping algebras_, Math. Z. __250__ (2005), no. 2, 299–361, [MR2006h:17025](http://www.ams.org/mathscinet-getitem?mr=2178788), [doi](http://dx.doi.org/10.1007/s00209-004-0754-9) [math.QA/0309349](http://front.math.ucdavis.edu/0309.5349);
* H. Hecht, D. Miličić, W. Schmid, J. A. Wolf, _Localization and standard modules for real semisimple Lie groups, I: The Duality Theorem_
Zbl 0699.22022, [MR910203](http://www.ams.org/mathscinet-getitem?mr=910203), _II: Applications_,
* Hendrik Orem, Lecture notes: The Beilinson-Bernstein Localization
Theorem, [pdf](http://www.ma.utexas.edu/users/horem/bb-notes.pdf)
* [[David Ben-Zvi]], [[David Nadler]], _Loop Spaces and Langlands Parameters_ ([arXiv:0706.0322](http://arxiv.org/abs/0706.0322)))
{#BenZviNadler07}
[[!redirects Beĭlinson-Bernstein localization]]
[[!redirects Beĭlinson-Bernstein localization theorem]]
[[!redirects Beilinson-Bernstein localization theorem]]
|
Beilinson-Drinfeld algebra
|
https://ncatlab.org/nlab/source/Beilinson-Drinfeld+algebra
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _Beilinson-Drinfeld algebra_ (or _BD-algebra_ for short) is like a [[BV-algebra]] with nilpotent [[BV-operator]], but over [[formal power series]] in one formal parameter $\hbar$, and such that the [[Gerstenhaber algebra|Gerstenhaber bracket]] is proportional to that parameter.
This means that one may think of a BD-algebra as an $\hbar$-parameterized formal family of algebras which for $\hbar = 0$ are [[Poisson 0-algebras]] and for $\hbar \neq 0$ they look superficially like [[BV-algebras]]. But see remark \ref{RoleOfTheDifferential} below.
Such BD-algbras are used to formalize [[formal deformation quantization]] in the context of the [[BV-BRST formalism]] (in [Costello-Gwilliam](#CostelloGwilliam)).
## Definition
+-- {: .num_defn #BeilinsonDrinfelAlgebra}
###### Definition
A **quantum BV complex** or **Beilinson-Drinfeld algebra** is
1. a [[differential graded-commutative algebra]] $A$ (whose [[differential]] we denote by $\Delta$) over the ring $\mathbb{R} [ [ \hbar ] ]$ of [[formal power series]] over the [[real numbers]] in a formal constant $\hbar$,
1. equipped with a [[Poisson 0-algebra|Poisson bracket]] $\{-,-\}$ of the same degree as the differential
such that
* the following equation holds for all elements $a,b \in A$ of homogeneous degree ${\vert a\vert}, {\vert b\vert} \in \mathbb{Z}$
$$
\Delta( a \cdot b) = (\Delta a) \cdot b + (-1)^{\vert a\vert} a \Delta b + \hbar \{a,b\}$ for all homogenous elements $a, b \in A
\,.
$$
=--
(e.g. [Costello-Gwilliam, def. 1.4.0.1](#CostelloGwilliam), [Gwilliam 13, def. 2.2.5](#Gwilliam))
+-- {: .num_remark #RoleOfTheDifferential}
###### Remark
If in the equation in def. \ref{BeilinsonDrinfelAlgebra} one replaces $\hbar$ by 1, then it takes the form characteristic of a [[BV-algebra]]. However the differential $\Delta$ here is the differential in the underlying [[chain complex]] for an [[algebra over an operad]] in chain complexes, while in a BV-algebra $\Delta$ is the unary operation encoded in the [[BV-operad]], hence present already for algebras in plain modules/chain complexes over that operad. Accordingly, in the context of BD-algebra then for $\hbar \neq 0$ the bracket is actually trivial up to homotopy.
For more see at _[[relation between BV and BD]]_.
=--
## Related concepts
[[!include deformation quantization - table]]
## References
The notion was introduced in
* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_
A discussion is in section 2.4 of
* {#CostelloGwilliam} [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in quantum field theory Volume 2_ ([pdf](http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf))
See also
* {#Gwilliam13} [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis (2013) ([[GwilliamThesis.pdf:file]])
* Martin Doubek, [[Branislav Jurčo]], Lada Peksová, Ján Pulmann, _Connected sum for modular operads and Beilinson-Drinfeld algebras_, [arXiv:2210.06517](https://arxiv.org/abs/2210.06517)
* Constantin-Cosmin Todea, _BD algebras and group cohomology_, Comptes Rendus. Mathématique __359__ (2021) no. 8, 925--937 [doi](https://doi.org/10.5802/crmath.246)
[[!redirects Beilinson-Drinfeld algebras]]
[[!redirects BD algebra]]
[[!redirects BD algebras]]
[[!redirects BD-algebra]]
[[!redirects BD-algebras]]
|
Beilinson–Deligne cup product
|
https://ncatlab.org/nlab/source/Beilinson%E2%80%93Deligne+cup+product
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential cohomology
+-- {: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The _Beilinson-Drinfeld cup product_ is an explicit presentation of the [[cup product]] in [[ordinary differential cohomology]] (see also at _[[cup product in differential cohomology]]_) for the case that the latter is modeled by the [[Cech cohomology|Cech]]-[[Deligne cohomology]]. It sends (see [[cup product in abelian Cech cohomology]])
$$
\cup: A[p]^\infty_D\otimes B[q]^\infty_D\to (A\otimes_{\mathbb{Z}} B)[p+q]^\infty_D,
$$
where $A$ and $B$ are lattices in $\mathbb{R}^n$, and $\mathbb{R}^m$ for some $n$ and $m$, respectively.
It is a morphism of complexes, so it induces a cup product in [[Deligne cohomology]].
For $A=B=\mathbb{Z}$, the Beilinson-Deligne cup product is associative and commutative up to homotopy, so it induces an associative and commutatvive cup product on [[ordinary differential cohomology|differential cohomology]]
## Definition
Let the [[Deligne complex]] $\mathbf{B}^n(\mathbb{R}//\mathbb{Z})_{conn}$ be given by
$$
\array{
&
\mathbb{Z}
&\hookrightarrow&
C^\infty(-,\mathbb{R})
&\stackrel{d_{dR}}{\to}&
\cdots
&\stackrel{d_{dR}}{\to}&
\Omega^{n}(-)
\\
\\
degree: & 0 && 1 && \cdots && (n+1)
}
$$
where we refer to degrees as indicated in the bottom row.
+-- {: .num_defn}
###### Definition
The _Beilinson-Deligne product_ is the morphism of [[chain complexes]]
of [[sheaves]]
$$
\cup
:
\mathbf{B}^p (\mathbb{R}//\mathbb{Z})_{conn}
\otimes
\mathbf{B}^q (\mathbb{R}//\mathbb{Z})_{conn}
\to
\mathbf{B}^{p+q+1} (\mathbb{R}//\mathbb{Z})_{conn}
$$
given on homogeneous elements $\alpha$, $\beta$ as follows:
$$
\alpha \cup \beta :=
\left\{
\array{
\alpha \wedge \beta = \alpha \beta & if\,deg(\alpha) = 0
\\
\alpha \wedge d_{dR}\beta & if\,deg(\alpha) \gt 0\,and\,deg(\beta) = q+1
\\
0 & otherwise
}
\right.
\,.
$$
=--
## Applications
### In higher Chern-Simons theory
The [[action functional]] of abelian [[higher dimensional Chern-Simons theory]] is given by the [[fiber integration in ordinary differential cohomology]] over the BD cup product of differential cocycles
$$
S_{CS} : H^{2k+2}(\Sigma)_diff \to U(1)
$$
$$
\hat C \mapsto \int_\Sigma \hat C \cup \hat C
\,.
$$
For more on this see [[higher dimensional Chern-Simons theory]].
## Related concepts
* [[Deligne line bundle]]
* [[intersection pairing]]
* [[self-dual higher gauge theory]]
## References
The original references are
* [[Pierre Deligne]], _Théorie de Hodge II_ , IHES Pub. Math. (1971), no. 40, 5–57.
* [[Alexander Beilinson]], _Higher regulators and values of L-functions_ , J. Soviet Math. 30 (1985), 2036—2070
* [[Alexander Beilinson]], _Notes on absolute Hodge cohomology_ , Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986.
A survey is for instance around prop. 1.5.8 of
* [[Jean-Luc Brylinski]], _Loop spaces, characteristic classes and geometric quantization_ Birkhäuser (1993)
and in section 3 of
* [[Helene Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_ in Rapoport, Schappacher, Schneider (eds.) _Beilinson's Conjectures on Special Values of L-Functions_ . Perspectives in Math. 4, Academic Press (1988) 43 - 91 ([pdf](http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf))
For the cup product of [[Cheeger-Simons differential character]]s see also
* [[Mike Hopkins]] and [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_ ([arXiv](http://arxiv.org/abs/math/0211216))
[[!redirects Deligne-Beilinson cup product]]
[[!redirects Deligne-Beilinson cup products]]
[[!redirects Deligne–Beilinson cup product]]
[[!redirects Deligne–Beilinson cup products]]
[[!redirects Deligne--Beilinson cup product]]
[[!redirects Deligne--Beilinson cup products]]
[[!redirects Beilinson-Deligne cup product]]
[[!redirects Beilinson-Deligne cup products]]
[[!redirects Beilinson–Deligne cup product]]
[[!redirects Beilinson–Deligne cup products]]
[[!redirects Beilinson--Deligne cup product]]
[[!redirects Beilinson--Deligne cup products]]
[[!redirects Beilinson-Deligne cup-product]]
[[!redirects differential cup-product]]
[[!redirects differential cup product]]
[[!redirects differential cup products]]
[[!redirects cup product in ordinary differential cohomology]]
[[!redirects cup-product in ordinary differential cohomology]]
[[!redirects cup product on differential cohomology]]
[[!redirects cup product in generalized differential cohomology]]
[[!redirects cup product on generalized differential cohomology]]
[[!redirects cup product in differential generalized cohomology]]
[[!redirects cup product on differential generalized cohomology]]
|
being
|
https://ncatlab.org/nlab/source/being
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
According to [[道德經]] (English translation following [[Xiao-Gang Wen]] [here](http://physics.stackexchange.com/a/94612/5603)):
> The nameless nonbeing is the origin of universe;
> The named being is the mother of all observed things.
> Within nonbeing, we enjoy the mystery of the universe.
> Among being, we observe the richness of the world.
> Nonbeing and being are two aspects of the same mystery.
> From nonbeing to being and from being to nonbeing is the gateway to all understanding.
Rather similarly, according to ([Hegel 12](#Hegel12)) _pure being_ is the [[unity of opposites|opposite]] of [[nothing]] whose unity is pure [[becoming]].
| | | [[Dasein]] | | |
|--|--|--|--|--|
| [[becoming]] : | [[nothing]] | $\;\;\;\dashv$ | [[being]] | : [[ceasing]] |
According to the formalization of this proposed by ([Lawvere 91](#Lawvere91)), this is described by the [[adjoint modality]]
$$
(\emptyset \dashv \ast)
$$
of the [[idempotent monad]] constant on a [[terminal object]] $\ast$ and its [[left adjoint]] $\emptyset$.
In this interpretation any other [[adjoint modality]] of the form $(\Box \dashv \bigcirc)$ characterizes a more "determinate" form of being ( _Dasein_ in the terminology of ([Hegel 12](#Hegel12))).
A [[category]] equipped with such a notion of being is, naturally, called a _[[category of being]]_.
## Related concepts
* [[ontology]], [[metaphysics]]
* [[category of being]]
* [[Aufhebung]]
* [[modality]]
* [[beable]]
* [[existential quantifier]]
## References
* {#Hegel12} [[Hegel]], _[[Science of Logic]]_, Volume One: _The Objective Logic_, Book One, _The Doctrine of Being_, Chapter 1 _Being_
* {#Lawvere91} [[Bill Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ (1991)
* [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844
* German Wikipedia, _[Sein](http://de.wikipedia.org/wiki/Sein)_
[[!redirects Sein]]
[[!redirects sein]]
|
Beiträge zur Begründung der transfiniten Mengenlehre
|
https://ncatlab.org/nlab/source/Beitr%C3%A4ge+zur+Begr%C3%BCndung+der+transfiniten+Mengenlehre
|
This page collects material related to the article
* {#Cantor1895} [[Georg Cantor]],
_Beiträge zur Begründung der transfiniten Mengenlehre_,
Math. Ann. 46 (1895) pp.481-512,
reprinted from p. 282 on in
[[Ernst Zermelo]] (ed.),
_Georg Cantor -- Gesammelte Abhandlungen mathematischen und philosophischen Inhalts_,
Springer Berlin 1932
([online English translation](https://archive.org/details/contributionstof00cant))
which is the origin of the modern concept of [[cardinality]] of [[sets]]. (But see also the commentary in [[William Lawvere]], _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_).
#Contents#
* table of contents
{:toc}
## Section 1. The conception of Power or Cardinal Number**
By an "aggregate" (Menge) we are to understand any collection into a whole (Zusammenfassung zu
einem Ganzen) $M$ of definite and separate objects of our intuition or our thought. These objects are called the "elements" of $M$.
In signs we express this thus :
(i) $M = \{m\}$.
We denote the uniting of many aggregates $M$, $N$,
$P$, $\cdots$, which have no common elements, into a
single aggregate by
(2) $(M, N, P, \cdots)$.
The elements of this aggregate are, therefore, the elements of $M$, of $N$, of $P$, . . ., taken together. We will call by the name "part" or "partial aggregate " of an aggregate M any other aggregate $M_1$ whose elements are also elements of $M$. If $M_2$ is a part of $M_1$ and $M_1$ is a part of $M$, then $M_2$ is a part of $M$.
Every aggregate $M$ has a definite "power", which
we will also call its "cardinal number".
We will call by the name "power" or "cardinal number" of $M$ the general concept which, by means
of our active faculty of thought, arises from the aggregate $M$ when we make abstraction of the
nature of its various elements $m$ and of the order
in which they are given.
[482] We denote the result of this double act of abstraction, the cardinal -number or power of M, by
(3) $\overline{\overline{M}}$
Since every single element $m$, if we abstract from its nature, becomes a "unit," the cardinal number $M$ is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate $M$.
We say that two aggregates $M$ and $N$ are "equivalent,"
in signs
(4) $M \sim N$ or $N \sim M$
if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element
of the other. To every part $M_1$ of $M$ there corresponds, then, a definite equivalent part $N_1$ of $N$, and inversely.
$[$ ...$]$
Every aggregate is equivalent to itself
(5) $M \sim M$
If two aggregates are equivalent to a third, they are
equivalent to one another ; that is to say :
(6) from $M \sim P$ and $N \sim P$ follows $M \sim N$.
Of fundamental importance is the theorem that two aggregates $M$ and $N$ have the same cardinal number if, and only if, they are equivalent : thus,
(7) from $M \sim N$ we get $\overline{\overline{M}} = \overline{\overline{N}}$,
and
(8) from $\overline{\overline{M}} = \overline{\overline{N}}$ we get $M \sim N$.
Thus the equivalence of aggregates forms the necessary and sufficient condition for the equality of their cardinal numbers.
[483] In fact, according to the above definition of power, the cardinal number $M$ remains unaltered if in the place of each of one or many or even all
elements $m$ of $M$ other things are substituted. If, now, $M \sim N$, there is a law of co-ordination by means of which $M$ and $N$ are uniquely and reciprocally
referred to one another; and by it to the
element $m$ of $M$ corresponds the element $n$ of $N$.
Then we can imagine, in the place of every element
$m$ of $M$, the corresponding element $n$ of $N$ substituted,
and, in this way, $M$ transforms into $N$ without
alteration of cardinal number. Consequently
$\overline{\overline{M}} = \overline{\overline{N}}$.
The converse of the theorem results from the remark that between the elements of $M$ and the
different units of its cardinal number $M$ a reciprocally univocal (or bi-univocal) relation of correspondence subsists. For, as we saw, $\overline{\overline{M}}$ grows, so to
speak, out of $M$ in such a way that from every element $w$ of $M$ a special unit of $M$ arises. Thus we can say that
(9) $M \sim \overline{\overline{M}}$
In the same way $N \sim \overline{\overline{N}}$. If then $\overline{\overline{M}} = \overline{\overline{N}}$, we have,
by (6), $M \sim N$.
We will mention the following theorem, which
results immediately from the conception of equival ence. If $M$, $N$, $P$, . . . are aggregates which have no common elements, $M'$, $N'$, $P'$, . . . are also aggregates
with the same property, and if
$M \sim M'$, $N \sim N'$, $P \sim P'$, ...,
then we always have
$(M, N, P, . . .) \sim (M', N', P', \cdots)$.
category: reference
|
Bekenstein-Hawking entropy
|
https://ncatlab.org/nlab/source/Bekenstein-Hawking+entropy
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[gravity]], __Bekenstein-Hawking entropy__ is an [[entropy]] assigned to [[black hole]], on the basis of laws of thermodynamics and observers outside black hole. It striking property is that it is proportional to the surface area of the balck hole's [[horizon]].
## Properties
### Interpretation in AdS/CFT correspondence
In the context of [[string theory]] BH entropy is explained by a version of the [[AdS/CFT correspondence]]. Here every [[black brane]] solution in [[supergravity]] is the strong-coupling limit of a [[D-brane]] [[worldvolume]] [[QFT]]. After [[Kaluza-Klein mechanism|KK-reduction]] these black brane configurations become ordinary [[black hole]]s. The [[entropy]] of the [[D-brane]] worldvolume theories on the [[event horizon]] turns out to coincide with the BH entropy of the corresponding black hole.
Detailed computations exist in particular for [[D1-brane]]/[[D5-brane]] systems. This is parts of the [[AdS/CFT correspondence]]. See ([AGMOO, chapter 5](#AGMOO)).
See also
* [[holographic entanglement entropy]]
* _[[string theory results applied elsewhere]]_.
### Interpretation by strong coupling limit of D-branes
Another way to derive Bekenstein-Hawking entropy in [[string theory]] is by computing the entropy of weakly coupled open strings on D-brane configurations in flat [[Minkowski space]] which transmute as the coupling constant is increased to given (supersymmetric) black hole configurations. More on this is at _[[black holes in string theory]]_.
## Related concepts
* gravitational entropy
* [[black hole radiation]]
* **Bekenstein-Hawking entropy**
* [[generalized second law of thermodynamics]]
* [[black holes in string theory]]
* [[holographic entanglement entropy]]
* [[thermal quantum field theory]]
## References
{#References}
### General
{#ReferencesGeneral}
Textbook account:
* {#FrolovZelnikov11} [[Valeri Frolov]], Andrei Zelnikov, _Introduction to black hole physics_, Oxford 2011
Review:
* Dmitri V. Fursaev, *Black Hole Thermodynamics and Perturbative Quantum Gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.06081](https://arxiv.org/abs/2210.06081)]
Review form the point of view of [[thermal field theory]]:
* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), <a href="https://doi.org/10.1016/0370-1573(87)90136-0">doi:10.1016/0370-1573(87)90136-0</a>)
Basic introductory accounts:
* [[Robert Wald]], _The Thermodynamics of Black Holes_ ([arXiv:gr-qc/9912119](http://arxiv.org/abs/gr-qc/9912119))
* [[Jacob Bekenstein]], _[Bekenstein-Hawking entropy](http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy)_, (2008), Scholarpedia, 3(10):7375
and further review:
* S. P. de Alwis, *Comments on Entropy Calculations in Gravitational Systems* [[arXiv:2304.07885](https://arxiv.org/abs/2304.07885)]
A more general discussion which identifies thermodynamic properties of all [[horizons]] appearing on gravity (not just [[black hole]] horizons) was given in
* {#Jacobson95} [[Ted Jacobson]], _Thermodynamics of Spacetime: The Einstein Equation of State_, Phys.Rev.Lett.75:1260-1263, 1995 ([arXiv:gr-qc/9504004](http://arxiv.org/abs/gr-qc/9504004))
This article showed that under some assumptions the [[Einstein equations]] can even be _derived_ from identifying gravitational horizon area with [[entropy]] and them imposing laws of [[thermodynamics]].
For more comments and more references on this observation see
* [[Thanu Padmanabhan]], _Thermodynamical Aspects of Gravity: New insights_, Rep. Prog. Phys. 73 (2010) 046901 ([arXiv:0911.5004](http://arxiv.org/abs/0911.5004))
(Later authors tried to argue that derivations like this show that gravity is not a fundamental force of nature such as [[electromagnetism]] or the [[strong nuclear force]], but rather an [[entropic force]] that arises only from more fundamental forces in a [[thermodynamic limit]]. This however remains at best unclear.)
Discussion of black hole entropy from entropy of [[conformal field theory]] associated with the horizon has been given in
* [[Steve Carlip]], _Entropy from Conformal Field Theory at Killing Horizons_, Class.Quant.Grav.16:3327-3348,1999 ([arXiv:gr-qc/9906126](http://xxx.lanl.gov/abs/gr-qc/9906126))
* [[Steve Carlip]], _Horizon Constraints and Black Hole Entropy_, Class.Quant.Grav.22:1303-1312, 2005 ([arXiv:hep-th/0408123](http://arxiv.org/abs/hep-th/0408123))
and reviewed in
* {#Carlip05} [[Steve Carlip]], _Horizon constraints and black hole entropy_ ([arXiv:gr-qc/0508071](http://arxiv.org/abs/gr-qc/0508071))
* {#Carlip07} [[Steve Carlip]], _Symmetries, Horizons, and Black Hole Entropy_, Gen.Rel.Grav.39:1519-1523,2007; Int.J.Mod.Phys.D17:659-664,2008 ([arXiv:0705.3024](http://arxiv.org/abs/0705.3024))
Further developments on black hole entropy are in
* [[Ashoke Sen]], _Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions_, JHEP04(2013)156 ([arXiv:arXiv:1205.0971](http://arxiv.org/abs/arXiv:1205.0971))
* Aitor Lewkowycz, [[Juan Maldacena]], _Generalized gravitational entropy_ ([arXiv:1304.4926](http://arxiv.org/abs/1304.4926))
A discussion of the "black hole firewall problem":
* [[Ahmed Almheiri]], [[Donald Marolf]], [[Joseph Polchinski]], James Sully, *Black holes: complementarity or firewalls?*, [[arXiv:1207.3123](http://arxiv.org/abs/arXiv:1207.3123)]
### Via Wick-rotated thermal field theory
Discussion via [[Wick rotation]] to [[Euclidean field theory]] on spacetimes with compact/periodic Euclidean time ([[thermal field theory]] on [[curved spacetimes]]) is in
* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), <a href="https://doi.org/10.1016/0370-1573(87)90136-0">doi:10.1016/0370-1573(87)90136-0</a>)
* {#GibbonsPerry78} [[Gary Gibbons]], Malcolm J. Perry, _Black Holes and Thermal Green Functions_, Vol. 358, No. 1695 (1978) ([jstor:79482](https://www.jstor.org/stable/79482))
### Interpretation in string theory
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]]:
* {#StromingerVafa96} [[Andrew Strominger]], [[Cumrun Vafa]], _Microscopic Origin of the Bekenstein-Hawking Entropy_, Phys. Lett. B379: 99-104, 1996 ([arXiv:hep-th/9601029](http://arxiv.org/abs/hep-th/9601029))
Review of interpretation of [[black holes in string theory]] includes
* {#AGMOO} [[Ofer Aharony]], S. S. Gubser, [[Juan Maldacena]], [[Hirosi Ooguri]], Y. Oz, Chapter 5 of _Large N field theories, string theory and gravity_, [arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111)
* [[Per Kraus]], _Lectures on black holes and the $AdS_3/CFT_2$ correspondence_, Lect. Notes Phys.755:193-247, 2008 ([arXiv:hep-th/0609074](https://arxiv.org/abs/hep-th/0609074))
* {#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))
* [[Dieter Lüst]], Ward Vleeshouwers, sections 21-22 of _Black Hole Information and Thermodynamics_ ([arXiv:1809.01403](https://arxiv.org/abs/1809.01403))
* Sebastian De Haro, Jeroen van Dongen, Manus Visser, [[Jeremy Butterfield]], _Conceptual Analysis of Black Hole Entropy in String Theory_ ([arXiv:1904.03232](https://arxiv.org/abs/1904.03232))
* Jeroen van Dongen, Sebastian De Haro, Manus Visser, [[Jeremy Butterfield]], _Emergence and Correspondence for String Theory Black Holes_ ([arXiv:1904.03234](https://arxiv.org/abs/1904.03234))
Discussion in view of [[higher curvature corrections]] includes
* [[Thomas Mohaupt]], _Strings, higher curvature corrections, and black holes_ ([arXiv:hep-th/0512048](http://arxiv.org/abs/hep-th/0512048))
See also
* MO question, [statistical-physics-of-string-theory](http://mathoverflow.net/questions/31789/statistical-physics-of-string-theory)
### Interpretation as entanglement entropy
Discussions of the interpreation of BH entropy as [[holographic entanglement entropy]] include
* Alejandro Satz, [[Ted Jacobson]], _Black hole entropy and the renormalization group_ ([arXiv:1301.3171](http://arxiv.org/abs/1301.3171))
Computation of [[black hole entropy]] in 4d via [[AdS4-CFT3 duality]] from [[holographic entanglement entropy]] in the [[ABJM theory]] for the [[M2-brane]] is discussed in
* Jun Nian, Xinyu Zhang, _Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole_ ([arXiv:1705.01896](https://arxiv.org/abs/1705.01896))
[[!include entanglement island proposal for black hole paradox -- references]]
### Relation to weight systems on chord diagrams
An argument that [[AdS/CFT duality|dual]] [[observables]] on [[black hole thermodynamics]] are generically given by [[single trace operators]] that evaluate to [[weight systems]] on [[chord diagrams]] (such as observables on [[SYK model]]-like systems)
(for more see at _[[weight systems on chord diagrams in physics]]_):
* [[Micha Berkooz]], [[Prithvi Narayan]], [[Joan Simón]], Section 2.1 of _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))
[[!redirects black hole entropy]]
[[!redirects BH entropy]]
[[!redirects black hole thermodynamics]]
|
Bell state
|
https://ncatlab.org/nlab/source/Bell+state
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[quantum physics]], what are called *Bell states* or *EPR states* or *EPR pairs* are certain [[quantum states]] exhibiting maximal [[quantum entanglement]].
In [[qbit]]-based [[quantum computation]], the elementary Bell state is usually prepared via the following small [[quantum circuit]] consisting of a [[Hadamard gate]] followed by a [[quantum CNOT gate]]:
\begin{imagefromfile}
"file_name": "BellStatePreparationCircuit-221026.jpg",
"width": "680",
"unit": "px",
"margin": {
"top": -20,
"bottom": 20,
"right": 0,
"left": 10
}
\end{imagefromfile}
## Related concepts
* [[quantum entanglement]]
* [[quantum information theory]]
* [[quantum teleportation]]
[[!include states and observables -- content]]
## References
The terminology "EPR state" is in honor of [Einstein, Podoldsky & Rosen 1935](EPR+paradox#EinsteinPodoldskyRosen35) and "Bell state" is in honor of [Bell 1964](Bell's+inequalities#Bell64).
Textbook accounts:
* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §1.3.2 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]
In [[string diagram]]-calculus ([[finite quantum mechanics in terms of dagger-compact categories]]):
* {#Coecke10} [[Bob Coecke]], §4.2 of: *Quantum Picturalism*, Contemporary Physics **51** (2010) 59-83 [[arXiv:0908.1787](https://arxiv.org/abs/0908.1787), [doi:10.1080/00107510903257624](https://doi.org/10.1080/00107510903257624)]
See also:
* Wikipedia, *[Bell state](https://en.wikipedia.org/wiki/Bell_state)*
[[!redirects Bell states]]
|
Bell's inequality
|
https://ncatlab.org/nlab/source/Bell%27s+inequality
|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[quantum physics]]/[[quantum information theory]], What came to be called *Bell's inequality* ([Bell 1964](#Bell64)) is an [[inequality]] satisfied by the three pairwise [[correlation functions]] between three [[random variables]] defined on one and the same classical [[probability space]]. As such, it is an elementary statement about classical probability theory which as been argued ([Pitowsky 1989a](#Pitowsky89a)) to have been known already to [[Boole -- The Laws of Thought|Boole (1854)]].
The point of the argument by [Bell 1964](#Bell64) was to highlight that when taking these three random variables to be the results of [[quantum measurements]] of the [[spin]] of an [[electron]] along three pairwise non-orthogonal axes (as in the [[Stern-Gerlach experiment]]) then [[quantum theory]] predicts that this inequality is *violated* -- implying that there is no single classical [[probability space]] (called a *[[hidden variable theory|hidden variable]]* in the context of [[interpretations of quantum mechanics]]) on which these three [[quantum measurement]]-results are jointly [[random variables]].
A number of [[experiments]] have sought to check Bell's inequalities in [[quantum physics]] ("[[Bell tests]]") and all claim to have verified that it is indeed violated in nature (see [Aspect 2015](#Aspect15)), as predicted by [[quantum theory]].
Bell's inequality has been and is receiving an enormous amount of attention, first in discussions of [[interpretations of quantum mechanics]], but more recently and more concretely also in the context of [[quantum information theory]].
## Statement
{#Statement}
### Original formulation
{#OriginalFormulation}
> The following is fairly verbatim recap of the original argument in [Bell 1964](#Bell64). For a streamlined re-statement see further [below](#CompactReformulation).
Let us denote the result _A_ of a measurement that is determined by a unit vector, $\vec{a}$, and some parameter $\lambda$ as $A(\vec{a},\lambda)=\pm 1$ where we further suppose that the outcome of the measurement is either +1 or -1. Likewise, we may do the same for the result _B_ of a second measurement, i.e. $B(\vec{b},\lambda)$. We further make the vital assumption that the result _B_ does not depend on $\vec{a}$ and likewise _A_ does not depend on $\vec{b}$.
Before proceeding, we should note that $\lambda$ here plays the role of a "hidden" parameter or variable. We say it is "hidden" because its precise nature is not known. However, it is still a very real parameter with a probability distribution $\rho(\lambda)$. The expectation value of the product of the two measurements is
\[
\label{AnExpectationValue}
P(\vec{a},\vec{b})=\int d\lambda\rho(\lambda)A(\vec{a},\lambda)B(\vec{b},\lambda).
\]
Because $\rho$ is a normalized probability distribution,
$$
\int d\lambda \rho(\lambda) = 1
$$
and because $A(\vec{a},\lambda)=\pm 1$ and $B(\vec{b},\lambda)=\pm 1$, P cannot be less than -1. It can be equal to -1 at $\vec{a}=\vec{b}$ only if $A(\vec{a},\lambda)=\pm 1 = -B(\vec{a},\lambda)=\pm 1$ except at a set of points $\lambda$ of zero probability. Thus we can write (eq:AnExpectationValue) as
\[
P(\vec{a},\vec{b})=-\int d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda).
\]
If we introduce a third unit vector $\vec{c}$ we can find the difference between the correlation of $\vec{a}$ to the two other unit vectors,
\[
\label{DifferenceOfCorrelations}
P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int d\lambda\rho(\lambda)[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)].
\]
Rearranging this we may write (eq:DifferenceOfCorrelations) as
\[
P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1].
\]
Given the limitations we have placed on the value of _A_, we may write
\[
|P(\vec{a},\vec{b})-P(\vec{a},\vec{c})| \le \int d\lambda\rho(\lambda)[1-A(\vec{b},\lambda)A(\vec{c},\lambda)].
\]
But the second term on the right is simply $P(\vec{b},\vec{c})$ and thus
\[
1 + P(\vec{b},\vec{c}) \ge |P(\vec{a},\vec{b})-P(\vec{a},\vec{c})|
\]
which is the original form of Bell's inequality. Note that this may be written in terms of correlation coefficients,
\[
1 + C(b,c) \ge |C(a,b)-C(a,c)|
\]
where _a_, _b_, and _c_ are now settings on the measurement apparatus.
#### Quantum mechanical violations
The original derivation of Bell's inequalities involved the use of a [[Stern-Gerlach device]] that measures spin along an axis. Suppose $\sigma_{1}$ and $\sigma_{2}$ are spins. The result, _A_, of measuring $\sigma_{1}\cdot\vec{a}$ is then interpreted as being entirely determined by $\vec{a}$ and $\lambda$. Likewise for _B_ and $\sigma_{2}\cdot\vec{b}$. It is also important to remember that the result _B_ does not depend on $\vec{a}$ and likewise _A_ does not depend on $\vec{b}$.
For a singlet state (that is a state with total spin of zero), the quantum mechanical expectation value of measurements along two different axes (see the Wigner derivation below for a more intuitive explanation of the physical nature of this) is
\[
\langle\sigma_{1}\cdot\vec{a},\sigma_{2}\cdot\vec{b}\rangle = - \vec{a}\cdot\vec{b}.
\]
In theory this ought to equal $P(\vec{a},\vec{b})$ but in practice it does not. It is important to remember that we are using _classical_ reasoning throughout our derivations of the various forms of Bell's inequalities.
The setup envisioned here consists of pairs of spin-1/2 particles produced in singlet states that then each pass through separate Stern-Gerlach (SG) devices. Since they are in singlet states, if we measured the first particle of a pair to be aligned with a given axis, say $\vec{a}$, then the second should be measured to be anti-aligned with that same axis, giving a total spin of zero.
In practice we are dealing with _beams_ of particles and thus we can never be absolutely certain that correlated pairs are measured simultaneously and so we ultimately are making statistical predictions. Nevertheless, in a given sample consisting of a large-enough number of randomly distributed spin-1/2 particles, we can be certain that, for example, a definite number are aligned with an axis $\vec{a}$ while a definite number are aligned with an axis $\vec{b}$.
Now take an individual particle and suppose that, for this particle, if we measured $\sigma\cdot\vec{a}$ we would obtain a +1 with certainty (meaning it is aligned with $\vec{a}$) but if we instead chose to measure $\sigma\cdot\vec{b}$ we would obtain a -1 with certainty (meaning it is anti-aligned with $\vec{b}$). Notationally we refer to such a particle as belonging to type $(\vec{a}+,\vec{b}-)$. Clearly for a given pair of particles in a singlet state, if particle 1 is of type $(\vec{a}+,\vec{b}-)$, then particle 2 must be of type $(\vec{a}-,\vec{b}+)$.
#### Locality
For beams of correlated particles measuring along only two axes, we should expect to get a roughly evenly balanced distribution of types as follows:
$$
\array{
\text{ Particle 1 } & & \text{ Particle 2 } \\
(\vec{a}+,\vec{b}-) & \leftrightarrow & (\vec{a}-,\vec{b}+) \\
(\vec{a}+,\vec{b}+) & \leftrightarrow & (\vec{a}-,\vec{b}-) \\
(\vec{a}-,\vec{b}-) & \leftrightarrow & (\vec{a}+,\vec{b}+) \\
(\vec{a}-,\vec{b}+) & \leftrightarrow & (\vec{a}+,\vec{b}-)
}
$$
There is a very important assumption implied here. Suppose a particular pair belongs to the first grouping, that is if an observer _A_ decides to measure the spin along $\vec{a}$ for particle 1, he or she _necessarily_ obtains a plus sign (corresponding to it being aligned with $\vec{a}$) _regardless_ of any measurement observer _B_ may make on particle 2. This is the principle of locality: _A_'s result is predetermined independently of _B_'s choice of what to measure.
#### Wigner's derivation
Now suppose we introduce a third axis, $\vec{c}$, so that we can have, for example, particles of type $(\vec{a}+,\vec{b}+,\vec{c}-)$ corresponding to being aligned if measured on $\vec{a}$ and $\vec{b}$ and anti-aligned on $\vec{c}$. Further let us "count" the pairs that fall into the various groupings and label the populations as follows:
$$
\array{
\text{ Population } & \text{ Particle 1 } & \text{ Particle 2 } \\
N_{1} & (\vec{a}+,\vec{b}+, \vec{c}+) & (\vec{a}-,\vec{b}-,\vec{c}-) \\
N_{2} & (\vec{a}+,\vec{b}+, \vec{c}-) & (\vec{a}-,\vec{b}-,\vec{c}+) \\
N_{3} & (\vec{a}+,\vec{b}-, \vec{c}+) & (\vec{a}-,\vec{b}+,\vec{c}-) \\
N_{4} & (\vec{a}+,\vec{b}-, \vec{c}-) & (\vec{a}-,\vec{b}+,\vec{c}+) \\
N_{5} & (\vec{a}-,\vec{b}+, \vec{c}+) & (\vec{a}+,\vec{b}-,\vec{c}-) \\
N_{6} & (\vec{a}-,\vec{b}+, \vec{c}-) & (\vec{a}+,\vec{b}-,\vec{c}+) \\
N_{7} & (\vec{a}-,\vec{b}-, \vec{c}+) & (\vec{a}+,\vec{b}+,\vec{c}-) \\
N_{8} & (\vec{a}-,\vec{b}-, \vec{c}-) & (\vec{a}+,\vec{b}+,\vec{c}+)
}
$$
Let's suppose that observer _A_ finds particle 1 is aligned with $\vec{a}$, i.e. $\vec{a}+$, and that observer _B_ finds particle 2 is aligned with $\vec{b}$, i.e. $\vec{b}+$. From the above table it is clear that the pair belong to either population 3 or 4. Note that because $N_{i}$ is positive semi-definite we must be able to construct relations like, for instance,
\[
\label{APositivityCondition}
N_{3} + N_{4} \le (N_{3} + N_{7}) + (N_{4} + N_{2}).
\]
Now let $P(\vec{a}+;\vec{b}+)$ be the probability that, in a random selection, _A_ finds particle 1 to be $\vec{a}+$ and _B_ finds particle 2 to be $\vec{b}+$. In terms of populations, we have
\[
\label{Populations1}
P(\vec{a}+;\vec{b}+) = \frac{(N_{3} + N_{4})}{\sum_{i}^{8}N_{i}}.
\]
Similarly we have
\[
\label{Populations2}
P(\vec{a}+;\vec{c}+) = \frac{(N_{2} + N_{4})}{\sum_{i}^{8}N_{i}}
\]
and
\[
\label{Populations3}
P(\vec{c}+;\vec{b}+) = \frac{(N_{3} + N_{7})}{\sum_{i}^{8}N_{i}}.
\]
The positivity condition (eq:APositivityCondition) then becomes
\[
\label{WignerFormOfBellInequality}
P(\vec{a}+;\vec{b}+) \le P(\vec{a}+;\vec{c}+) + P(\vec{c}+;\vec{b}+).
\]
This is Wigner's form of Bell's inequality.
#### Violations and geometry
As we mentioned before, we have used purely classical reasoning to derive the two forms of Bell's inequality that we have thusfar encountered. Recall that the context within which the above were derived was the Stern-Gerlach experiment are we are measuring along axes of the magnetic field. As such, there are angles between these various axes. Thus the quantum mechanically-derived probabilities corresponding to (eq:Populations1), (eq:Populations2), and (eq:Populations3) are
$$
P(\vec{a}+;\vec{b}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{ab}}{2}\right),
$$
$$
P(\vec{a}+;\vec{c}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{ac}}{2}\right),
$$
and
$$
P(\vec{c}+;\vec{b}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{cb}}{2}\right),
$$
respectively. Bell's inequality, (eq:WignerFormOfBellInequality), then becomes
\[
\label{AnotherFormOfBellInequality}
\frac{1}{2}sin^{2}\left(\frac{\theta_{ab}}{2}\right) \le \frac{1}{2}sin^{2}\left(\frac{\theta_{ac}}{2}\right) + \frac{1}{2}sin^{2}\left(\frac{\theta_{cb}}{2}\right).
\]
From a geometric point of view, this inequality is not always possible. For example, suppose, for simplicity that $\vec{a}$, $\vec{b}$, and $\vec{c}$ lie in a plane and suppose that $\vec{c}$ bisects $\vec{a}$ and $\vec{b}$, i.e.
$$
\array{
\theta_{ab} = 2\theta & \text{ and } & \theta_{ac}=\theta_{cb}=\theta.
}
$$
Then (eq:AnotherFormOfBellInequality) is violated for $0 \lt \theta \lt \frac{\pi}{2}$. For example, if $\theta = \frac{\pi}{4}$, (eq:AnotherFormOfBellInequality) would become $0.500 \le 0.292$ which is absurd!
### Compact reformulation
{#CompactReformulation}
A transparent and compact way to derive the actual [[inequality]] of [Bell 1964](#Bell64) (adjusting the original argument only slightly for mathematical elegance) is reviewed in [Khrennikov 2008, §10.1](#Khrennikov08), which we broadly follow:
\begin{proposition}
Given
1. a [[probability space]] $(\Lambda, d\rho)$ with
1. three [[random variables]] taking values in $\{\pm 1\}$ (regarded inside the [[real numbers]]):
\[
\label{TheRandomVariables}
S_i
\;\colon\;
X \longrightarrow \{\pm 1\}
\hookrightarrow
\mathbb{R}
\,,
\;\;\;\;
i\,\in\, \{1,2,3\}
\]
then the [[correlation functions]]
\[
\label{TheCorrelations}
\langle S_{i} \, S_j\rangle
\;\coloneqq\;
\int_{\Lambda}
\;
S_i(\lambda)
\,
S_j(\lambda)
\;
d\rho(\lambda)
\]
satisfy this [[inequality]]:
\[
\label{TheInequality}
\big\vert
\langle S_1 S_2\rangle
-
\langle S_3 S_2\rangle
\big\vert
\;\leq\;
1 - \langle S_1 S_3\rangle
\,.
\]
(where $\left\vert-\right\vert$ denotes the [[absolute value]])
\end{proposition}
\begin{proof}
Recall that the [[expectation value]] of a random variable $P \,\colon\, \Lambda \longrightarrow \mathbb{R}$ is given by its [[Lebesgue integral]] against the [[probability measure]]:
$$
\langle P \rangle
\;\coloneqq\;
\int_\Lambda P(\lambda) \, d\rho(\lambda)
\,,
$$
and that $d\rho$ being a [[probability measure]] implies the normalization
\[
\label{NormalizationOfProbability}
\langle 1 \rangle
\;\equiv\;
\int_\Lambda 1 \, d\rho(\lambda)
\;=\;
1
\,.
\]
Moreover, the assumption (eq:TheRandomVariables) that the random variables $S_i$ take values in $\{\pm 1\}$
immediately implies for all $i,j \,in\, \{1,2,3\}$ that
\[
\label{IdempotencyOfTheRandomVariables}
\big( S_i \cdot S_i \big) \,=\, 1
\,,
\;\;\;\;
\text{i.e.}
\;\;\;
\underset{\lambda \,\in\, \Lambda}{\forall}
S_i(\lambda) \, S_i(\lambda)
\,=\,
(\pm 1)^2
\,=\,
1
\,.
\]
Together this implies -- by repeatedly using the [[Cauchy-Schwarz inequality]] -- the bounds:
$$
\big\vert
\langle S_i \rangle
\big\vert
\;\leq\;
1
\,,
\;\;\;\;\;\;\;
\big\vert
\langle S_i S_j\rangle
\big\vert
\;\leq\;
1
$$
and thus, in particular:
\[
\label{BoundOnCorrelations}
\big\vert
\langle
P \, S_i \, S_j
\rangle
\big\vert
\;\leq\;
\big\vert
\langle
P
\rangle
\big\vert
\,,
\]
for any [[random variable]] $P \,\colon\, \Lambda \to \mathbb{R}$.
Using these (evident) ingredients, we directly compute as follows
$$
\begin{array}{ll}
\big\vert
\langle S_1 S_2\rangle
-
\langle S_3 S_2\rangle
\big\vert
&
\\
\;=\;
\Big\vert
\int_{\Lambda}
S_1(\lambda) \, S_2(\lambda)
\,
d\rho(\lambda)
-
\int_{\Lambda}
S_3(\lambda) \, S_2(\lambda)
\,
d\rho(\lambda)
\Big\vert
&
\text{by}\;\text{(eq:TheCorrelations)}
\\
\;=\;
\Big\vert
\int_{\Lambda}
\big(
S_1(\lambda)
-
S_3(\lambda)
\big)
\,
S_2(\lambda)
\,
d\rho(\lambda)
\Big\vert
&
\text{by linearity of the integral}
\\
\;=\;
\Big\vert
\int_{\Lambda}
\big(
1
-
S_1(\lambda) \, S_3(\lambda)
\big)
S_1(\lambda) \, S_2(\lambda)
\,
d\rho(\lambda)
\Big\vert
&
\text{by}\;\text{(eq:IdempotencyOfTheRandomVariables)}
\\
\;\leq\;
\Big\vert
\int_{\Lambda}
\big(
1
-
S_1(\lambda) \, S_3(\lambda)
\big)
\,
d\rho(\lambda)
\Big\vert
&
\text{by}\;\text{(eq:BoundOnCorrelations)}
\\
\;=\;
1 - \langle S_1 S_3\rangle
&
\text{by}\;\text{(eq:NormalizationOfProbability)}\;\text{and}\; \text{(eq:TheCorrelations)}
\end{array}
$$
This is the inequality (eq:TheInequality).
\end{proof}
## Related concepts
* [[Einstein-Podolsky-Rosen paradox]]
* [[Grothendieck inequality]]
* [[quantum probability]]
* [[interpretation of quantum mechanics]]
* [[quantum information theory]]
Other theorems about the foundations and [[interpretation of quantum mechanics]] include:
* [[order-theoretic structure in quantum mechanics]]
* [[Kochen-Specker theorem]]
* [[Alfsen-Shultz theorem]]
* [[Harding-Döring-Hamhalter theorem]]
* [[Fell's theorem]]
* [[Gleason's theorem]]
* [[Wigner theorem]]
* [[Bub-Clifton theorem]]
* [[Kadison-Singer problem]]
## References
### General
The original article:
* {#Bell64} [[John Bell]], *On the Einstein Podolsky Rosen paradox*, Physics **1** 195 (1964) [[doi:10.1103/PhysicsPhysiqueFizika.1.195](https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195), [pdf](http://www.drchinese.com/David/Bell_Compact.pdf)]
Review:
* [[John F. Clauser]], [[Abner Shimony]], *Bell's theorem. Experimental tests and implications*, Rep. Prog. Phys. **41** (1978) 1881 [[doi:10.1088/0034-4885/41/12/002](https://iopscience.iop.org/article/10.1088/0034-4885/41/12/002)]
* {#Kuperberg05} [[Greg Kuperberg]], section 1.6.2 of: _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]
* [[Valter Moretti]], Thm. 4.49 of: *Fundamental Mathematical Structures of Quantum Theory*, Springer (2019) [[doi:10.1007/978-3-030-18346-2](https://doi.org/10.1007/978-3-030-18346-2)]
and on a background of [[quantum logic]]:
* Laura Molenaar, *Quantum logic and the EPR paradox*, Delft (2014) [[uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b](http://resolver.tudelft.nl/uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b)]
Further on experimental verification:
* {#Aspect15} [[Alain Aspect]], *[Closing the Door on Einstein and Bohr’s Quantum Debate](https://physics.aps.org/articles/v8/123)*, Physics **8** 123 (2015)
Relation to the [[Kochen-Specker theorem]]:
* [[Leandro Aolita]], Rodrigo Gallego, Antonio Acín, Andrea Chiuri, Giuseppe Vallone, Paolo Mataloni, Adán Cabello, *Fully nonlocal quantum correlations*, Phys. Rev. A **85** 032107 (2012) [[arXiv:1105.3598](https://arxiv.org/abs/1105.3598), [doi:10.1103/PhysRevA.85.032107](https://doi.org/10.1103/PhysRevA.85.032107)]
See also:
* Wikipedia, _[Bell's theorem](http://en.wikipedia.org/wiki/Bell_inequality)_
* Wikipedia, *[Bell test](https://en.wikipedia.org/wiki/Bell_test)*
* Wikipedia, *[Leggett-Garg inequality](https://en.wikipedia.org/wiki/Leggett%E2%80%93Garg_inequality)*
* Stanford Encyclopedia of Philosophy, _Bell's theorem_ ([url](http://plato.stanford.edu/entries/bell-theorem/))
In relation to the [[Grothendieck inequality]]:
* [[Boris S. Tsirelson]], *Quantum analogues of the Bell inequalities. The case of two spatially separated domains*, Journal of Soviet Mathematics **36** (1987) 557–570 [[doi:10.1007/BF01663472](https://doi.org/10.1007/BF01663472)]
* [[Boris S. Tsirelson]], *Some results and problems on quantum Bell-type inequalities* Hadronic Journal Supplement **8** 4 (1993) 329-345 [[pdf](https://www.tau.ac.il/~tsirel/download/hadron.pdf), [[Tsirelson-QuantumBellType.pdf:file]] [web](https://www.tau.ac.il/~tsirel/download/hadron.html)]
> (but see the erratum [here](https://www.tau.ac.il/~tsirel/Research/bellopalg/main.html))
* Wikipedia, *[Tsirelson's bound](https://en.wikipedia.org/wiki/Tsirelson%27s_bound)*
### In quantum field theory
In the generality of [[quantum field theory]]:
On Bell inequalities in [[particle physics]] and possible relation to the [[weak gravity conjecture]]:
* Aninda Sinha, Ahmadullah Zahed, *Bell inequalities in 2-2 scattering* [[arXiv:2212.10213](https://arxiv.org/abs/2212.10213)]
On [[BRST complex|BRST invariant]] Bell inequality in [[gauge field theory]]:
* David Dudal, Philipe De Fabritiis, Marcelo S. Guimaraes, Giovani Peruzzo, Silvio P. Sorella: *BRST invariant formulation of the Bell-CHSH inequality in gauge field theories* [[arXiv:2304.01028](https://arxiv.org/abs/2304.01028)]
### Probabilistic opposition
{#ReferencesProbabilisticOpposition}
Identification of Bell's inequalities with much older inequalities in classical [[probability theory]], due to [[George Boole]]'s *[[Boole -- The Laws of Thought|The Laws of Thought]]*, was pointed out by (among others, called the "probabilistic opposition" in [Khrennikov 2007, p. 3](#Khrennikov07)) by:
* {#Pitowsky89a} [[Itamar Pitowsky]], *From George Boole To John Bell — The Origins of Bell’s Inequality*, in: *Bell’s Theorem, Quantum Theory and Conceptions of the Universe*, Fundamental Theories of Physics **37** Springer (1989) [[doi:10.1007/978-94-017-0849-4_6](https://doi.org/10.1007/978-94-017-0849-4_6)]
* {#Pitowsky89b} [[Itamar Pitowsky]], *Quantum Probability -- Quantum Logic*, Lecture Notes in Physics **321**, Springer (1989) [[doi:10.1007/BFb0021186](https://doi.org/10.1007/BFb0021186)]
* [[Luigi Accardi]], *The Probabilistic Roots of the Quantum Mechanical Paradoxes*, in: *The Wave-Particle Dualism*, Fundamental Theories of Physics **3** Springer (1984) [[doi:10.1007/978-94-009-6286-6_16](https://doi.org/10.1007/978-94-009-6286-6_16)]
reviewed in:
* Elemer E Rosinger, *George Boole and the Bell inequalities* [[arXiv:quant-ph/0406004](https://arxiv.org/abs/quant-ph/0406004)]
* {#Khrennikov07} [[Andrei Khrennikov]], *Bell's inequality: Physics meets Probability* [[arXiv:0709.3909](https://arxiv.org/abs/0709.3909)]
* {#Khrennikov08} [[Andrei Khrennikov]], *Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?*, Entropy **10** 2 (2008) 19-32 [[doi:10.3390/entropy-e10020019](https://doi.org/10.3390/entropy-e10020019)]
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Bell's theorem > history
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Belle experiment
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|
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Experiments
+-- {: .hide}
[[!include experiments -- contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Belle experiment_ is an [[experiment]] with focus on the [[physics]] of [[B-mesons]] (similar to the [[LHCb experiment]], see also the [[BaBar experiment]]).
After the discovery of the [[Higgs field]] at the two major other detectors of [[LHC]], these [[B-meson]] [[experiments]] currently stand out as seeing hints for physics beyond the [[standard model of particle physics]] with considerable [[statistical significance]]: the _[[flavour anomalies]]_ (see [Adamczyk 19](#Adamczyk19))
## Related concepts
* related [[experiments]]
* [[LHCb]]
* [[BaBar experiment]]
* observations
* [[flavour anomaly]]
* [[V_cb puzzle]]
* [[pentaquark]]
* [[XYZ particle]]
* [[precision experiment]]
## References
### General
* [Belle homepage](https://belle.kek.jp/)
* [Belle II homepage](https://www.belle2.org/)
* C. Z. Yuan (for the [[Belle II Collaboration]]), _The Belle II Experiment at the SuperKEKB_ ([arXiv:1208.3813](https://arxiv.org/abs/1208.3813))
* [[Belle collaboration]], _Physics Achievements from the Belle Experiment_ ([arXiv:1212.5342](https://arxiv.org/abs/1212.5342))
* S.I. Eidelman, A.V. Nefediev, P.N. Pakhlov, V.I. Zhukova, _Superfactory of bottomed hadrons Belle II_ ([arXiv:2012.05147](https://arxiv.org/abs/2012.05147))
See also
* Wikipedia, _[Belle experiment](https://en.wikipedia.org/wiki/Belle_experiment)_
* Wikipedia, _[Belle II experiment](https://en.wikipedia.org/wiki/Belle_II_experiment)_
More on [[Belle II]]:
* *Belle II Executive Summary* [[arXiv:2203.10203](https://arxiv.org/abs/2203.10203)]
* *Snowmass White Paper: Belle II physics reach and plans for the next decade and beyond* [[arXiv:2207.06307](https://arxiv.org/abs/2207.06307)]
> Belle II is an experiment operating at the [[intensity frontier]].
### On flavour anomalies
[[flavour anomalies]]:
* [[Belle collaboration]], _Measurement of the $\tau$ lepton polarization and $R(D^\ast)$ in the decay $\bar B \to D^\ast \tau^- \bar \nu_\tau$ with one-prong hadronic τ decays at Belle_ ([arXiv:1709.00129](https://arxiv.org/abs/1709.00129))
* {#Adamczyk19} Karol Adamczyk, _Semitauonic B decays at Belle/Belle II_, Proceedings of the [10th International Workshop on the CKM Unitarity Triangle](https://ckm2018.physi.uni-heidelberg.de/) (CKM 2018), Heidelberg, Germany, September 17-21, 2018 ([arXiv:1901.06380](https://arxiv.org/abs/1901.06380))
* [[Belle collaboration]], _Measurement of the $D^{\ast -}$ polarization in the decay $B^0 \to D^{\ast -} \tau^+ \nu_\tau$_ ([arXiv:1903.03102](https://arxiv.org/abs/1903.03102))
* {#Caria19} Giacomo Caria on behalf of the [[Belle collaboration]], _Measurement of $R(D)$ and $R(D^\ast)$ with a semileptonic tag at Belle_, Moriond, EW 22/03/2019 ([pdf](http://moriond.in2p3.fr/2019/EW/slides/6_Friday/3_YSF/1_gcaria_moriond2019.pdf))
* [[Belle Collaboration]], _Test of lepton flavor universality in $B \to K^\ast \ell^+ \ell^-$ decays at Belle_ ([arXiv:1904.02440](https://arxiv.org/abs/1904.02440))
* {#Belle1910} [[Belle collaboration]], _Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^\ast)$ with a semileptonic tagging method_, Phys. Rev. Lett. 124, 161803 (2020)
([arXiv:1910.05864](https://arxiv.org/abs/1910.05864), [doi:10.1103/PhysRevLett.124.161803](https://doi.org/10.1103/PhysRevLett.124.161803))
Outlook for [[Belle II]] to confirm the [[flavour anomalies]], if they are real:
* Tobias Huber, Tobias Hurth, Jack Jenkins, Enrico Lunghi, Qin Qin, K.Keri Vos, _Phenomenology of inclusive $\bar B \to X_s \ell^+ \ell^-$ for the Belle II era_ ([arXiv:2007.04191](https://arxiv.org/abs/2007.04191))
> With $[$ ... $]$ [[Belle II]], we are at the brink of a new era in [[quark]] [[flavour physics]] $[...]$ we present a phenomenological study of the potential for [[Belle II]] to reveal possible new physics in the inclusive decay channel.
> (p. 15:) We see that [[flavour anomalies|anomalies]] in the exclusive sector can be confirmed at the [[statistical significance|5σ]] level by inclusive measurements if the true values of $C_{9,10}^{\mu NP}$ are at the current best-fit point of the exclusive fits
> (p. 16:) After including $B_s \to \mu^+ \mu_-$, the reach in the $[C_9^{\mu NP}, C_{10}^{\mu NP}]$ plane improves considerably. Exclusive [[flavour anomalies|anomalies]] could be confirmed at the [[statistical significance|6σ]] level.
> (p. 20:) Should the true value of $C_9^{NP}$ and $C_{10}^{NP}$ be at the current best-fit points of the global fits, an analysis of inclusive $\bar B_ \to X_s \ell^+ \ell^-$ at [[Belle II]] with $50 ab^{-1}$ of data will exclude the SM point $C_9^{NP} = C_{10}^{NP} = 0$ at the level of $\sim 5$[[statistical significance|σ]]. This again underlines the necessity of a full angular analysis of $\bar B_ \to X_s \ell^+ \ell^-$ at [[Belle II]].
[[!redirects Belle]]
[[!redirects Belle II]]
[[!redirects Belle collaboration]]
[[!redirects Belle collaborations]]
[[!redirects Belle Collaboration]]
[[!redirects Belle Collaborations]]
[[!redirects Belle II collaboration]]
[[!redirects Belle II collaborations]]
[[!redirects Belle II Collaboration]]
[[!redirects Belle II Collaborations]]
[[!redirects Belle II experiement]]
[[!redirects Belle II experiments]]
|
Ben Allanach
|
https://ncatlab.org/nlab/source/Ben+Allanach
|
* [webpage](http://www.damtp.cam.ac.uk/people/b.allanach/)
## Selected writings
On [[flavour anomalies]]:
* {#Allanach19} [[Ben Allanach]], _Finding Z's responsible for $R_{K^{(\ast)}}$_, talk at [Moriond 2019](http://moriond.in2p3.fr/2019/EW/) ([pdf](http://moriond.in2p3.fr/2019/EW/slides/6_Friday/2_afternoon/5_Allanach.pdf))
* {#AllanachGripaiosYou17} [[Ben Allanach]], Ben Gripaios, Tevong You, _The Case for Future Hadron Colliders From $B \to K^{(\ast)}\mu^+ \mu^-$ Decays_, JHEP03(2018)021 ([arXiv:1710.06363](https://arxiv.org/abs/1710.06363))
* [[Ben Allanach]], _$U(1)_{B_3-L_2}$ Explanation of the Neutral Current $B$−Anomalies_ ([arXiv:2009.02197](https://arxiv.org/abs/2009.02197))
category: people
|
Ben Brown
|
https://ncatlab.org/nlab/source/Ben+Brown
|
PhD student at The University of Edinburgh.
[Personal Homepage](https://bencwbrown.co.uk/)
category: people
|
Ben Goertzel
|
https://ncatlab.org/nlab/source/Ben+Goertzel
|
__Ben Goertzel__ is a prominent researcher in [[machine learning]]/artificial intelligence predominantly interested in achieving AGI (artificial general intelligence, the term he actually coined in print).
* Ben Goertzel, _Reflective metagraph rewriting as a foundation for an AGI "Language of Thought"_, [arXiv:2112.08272](https://arxiv.org/abs/2112.08272)
category: people
|
Ben Greenman
|
https://ncatlab.org/nlab/source/Ben+Greenman
|
[home page](http://ccs.neu.edu/home/types)
|
Ben Gripaios
|
https://ncatlab.org/nlab/source/Ben+Gripaios
|
* [webpage](https://www.hep.phy.cam.ac.uk/~gripaios/)
## Selected writings
On [[particle physics]]:
* James Dodd, [[Ben Gripaios]], _The Ideas of Particle Physics_, Cambridge University Press 2020 ([ISBN:9781108727402](https://www.cambridge.org/gb/academic/subjects/physics/particle-physics-and-nuclear-physics/ideas-particle-physics-4th-edition?format=PB), [doi:10.1017/9781108616270]( https://doi.org/10.1017/9781108616270))
On [[equivariant ordinary differential cohomology]] and [[action functionals]] for [[topological field theories]] (such as [[Chern-Simons theory]]):
* Joe Davighi, [[Ben Gripaios]], [[Oscar Randal-Williams]], _Differential cohomology and topological actions in physics_ ([arXiv:2011.05768](https://arxiv.org/abs/2011.05768))
category: people
|
Ben Heidenreich
|
https://ncatlab.org/nlab/source/Ben+Heidenreich
|
* [webpage](https://www.physics.umass.edu/people/ben-heidenreich)
## Selected writings
On the [[weak gravity conjecture]]:
* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Sharpening the Weak Gravity Conjecture with Dimensional Reduction_, JHEP02(2016)140 ([arXiv:1509.06374](https://arxiv.org/abs/1509.06374))
* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Repulsive Forces and the Weak Gravity Conjecture_ ([arXiv:1906.02206](https://arxiv.org/abs/1906.02206))
* [[Ben Heidenreich]], Matteo Lotito, *Proving the Weak Gravity Conjecture in Perturbative String Theory, Part I: The Bosonic String* [[arXiv:2401.14449](https://arxiv.org/abs/2401.14449)]
On [[swampland conjectures]]:
* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Emergence and the Swampland Conjectures_ ([arXiv:1802.08698](https://arxiv.org/abs/1802.08698))
category: people
|
Ben Knudsen
|
https://ncatlab.org/nlab/source/Ben+Knudsen
|
* [webpage](http://www.math.northwestern.edu/~knudsen/)
## Selected writings
On [[configuration spaces of points]] in [[algebraic topology]]:
* [[Ben Knudsen]], _Configuration spaces in algebraic topology_ ([arXiv:1803.11165](https://arxiv.org/abs/1803.11165))
On the [[Morava E-theory]] of [[configuration spaces of points]]:
* [[Lukas Brantner]], [[Jeremy Hahn]], [[Ben Knudsen]], *The Lubin-Tate Theory of Configuration Spaces: I* ([arXiv:1908.11321](https://arxiv.org/abs/1908.11321))
## Eelated $n$Lab entries
* [[factorization homology]]
category: people
|
Ben Moon
|
https://ncatlab.org/nlab/source/Ben+Moon
|
### About
Name: I publish as 'Benjamin', but typically go by 'Ben'.
* [Homepage](https://www.cs.kent.ac.uk/people/rpg/bgm4/)
* [GitHub](https://github.com/GuiltyDolphin)
## Selected writings
On [[dependent linear types]] and [[graded modalities]]:
* [[Benjamin Moon]], [[Harley Eades III]], [[Dominic Orchard]], *Graded Modal Dependent Type Theory*. In: N. Yoshida (ed.) *Programming Languages and Systems ESOP 2021*, Lecture Notes in Computer Science **12648**, Springer (2021) 462-490 [[doi:10.1007/978-3-030-72019-3_17](https://doi.org/10.1007/978-3-030-72019-3_17), [arxiv:2010.13163](https://arxiv.org/abs/2010.13163)]
## Related entries
* [[type theory]]
category:people
[[!redirects Benjamin Moon]]
|
Ben Walter
|
https://ncatlab.org/nlab/source/Ben+Walter
|
* [webpage](http://www.math.purdue.edu/~walterb/)
## Selected writings
On [[Lie coalgebras]], [[rational homotopy theory]] and [[Hopf invariants]] via [[homotopy Whitehead integrals]] ("[[functional cup products]]"):
* {#SinhaWalter13} [[Dev Sinha]], [[Ben Walter]], _Lie coalgebras and rational homotopy theory II: Hopf invariants_, Trans. Amer. Math. Soc. 365 (2013), 861-883 ([arXiv:0809.5084](https://arxiv.org/abs/0809.5084), [doi:10.1090/S0002-9947-2012-05654-6](https://doi.org/10.1090/S0002-9947-2012-05654-6))
## Related $n$Lab entries
* [[model structure on dg-coalgebras]]
* [[model structure on dg-Lie algebras]]
category: people
|
Ben Webster
|
https://ncatlab.org/nlab/source/Ben+Webster
|
Is a professor at Northeastern University.
* [website](https://uwaterloo.ca/scholar/b2webste/home)
* [Secret Blogging Seminar](http://sbseminar.wordpress.com/).
## Selected writings
On [[Hilbert schemes]] as [[Coulomb branches]]:
* [[Ben Webster]], _Coherent sheaves on Hilbert schemes through the Coulomb lens_, 2018 ([[WebsterHilbertScheme18.pdf:file]])
##Expository posts:##
###Algebraic geometry###
[Interpreting the Hecke Algebra II: the sheafification ](http://sbseminar.wordpress.com/2009/04/09/interpreting-the-hecke-algebra-ii-the-sheafification/)
[Geometry and triply graded knot homology](http://sbseminar.wordpress.com/2009/04/09/geometry-and-triply-graded-knot-homology/)
[The Grothendieck trace formula as categorification, I: the category and the comparison theorem ](http://sbseminar.wordpress.com/2009/02/12/the-grothendieck-trace-formula-as-categorification-the-category-and-the-comparison-theorem/)
[About that field isomorphism...](http://sbseminar.wordpress.com/2009/02/15/about-that-field-isomorphism/)
###Knot theory###
[How to get an algebra from a knot invariant](http://sbseminar.wordpress.com/2009/04/13/how-to-get-an-algebra-from-a-knot-invariant/)
[Hochschild homology](http://sbseminar.wordpress.com/2007/07/22/hochschild-homology/)
[Soergel bimodules](http://sbseminar.wordpress.com/2007/07/23/soergel-bimodules/)
[The geometry of Soergel bimodules](http://sbseminar.wordpress.com/2007/08/23/the-geometry-of-soergel-bimodules/)
###Finite dimensional algebras###
[Gale duality and linear programing](http://sbseminar.wordpress.com/2008/04/06/gale-duality-and-linear-programing/)
[Gale and Koszul duality, together at last](http://sbseminar.wordpress.com/2008/07/14/gale-and-koszul-duality-together-at-last/)
[Koszul algebras and Koszul duality](http://sbseminar.wordpress.com/2007/11/01/koszul-algebras-and-koszul-duality/)
###Linear equivalence###
[Laplacian spectra and linearly equivalent G-sets](http://sbseminar.wordpress.com/2007/09/19/laplacian-spectra-and-linearly-equivalent-g-sets/)
[Zeta function relations and linearly equivalent group actions](http://sbseminar.wordpress.com/2007/08/29/zeta-function-relations-and-linearly-equivalent-group-actions/)
###Social choice theory###
[Why Arrow's theorem is a scam](http://sbseminar.wordpress.com/2008/02/17/why-arrows-theorem-is-a-scam/)
[Stable marriages](http://sbseminar.wordpress.com/2008/01/21/stable-marriages/)
category: people
[[!redirects BWebster:Ben Webster]]
|
Ben Williams
|
https://ncatlab.org/nlab/source/Ben+Williams
|
* [webpage](http://www.math.ubc.ca/~tbjw/)
category: people
|
Benedict Gross
|
https://ncatlab.org/nlab/source/Benedict+Gross
|
[website](http://www.math.harvard.edu/~gross/)
category: people
|
Benedikt Ahrens
|
https://ncatlab.org/nlab/source/Benedikt+Ahrens
|
* [webpage](https://benediktahrens.net/)
## Selected writings
On the [[UniMath project]]:
* [[Benedikt Ahrens]], _UniMath: its origins, present, and future_ (2017) [[pdf](https://unimath.github.io/bham2017/UniMath_origins-present-future.pdf), [[Ahrens-UniMathHistory.pdf:file]]]
On [[2-dimensional type theory]]:
* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, August 2022, No. 12, Pages 1–14, ([doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334))
* [[Benedikt Ahrens]], [[Paige Randall North]], [[Michael Shulman]], [[Dimitris Tsementzis]], *The Univalence Principle* ([arXiv:2102.06275](https://arxiv.org/abs/2102.06275))
On [[univalence]] in [[homotopy type theory]]:
* [[Benedikt Ahrens]], [[Paige Randall North]], *Univalent foundations and the equivalence principle*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[arXiv:2202.01892](https://arxiv.org/abs/2202.01892), [doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]
On [[2-dimensional type theory]]:
* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, 12 (2022) 1–14, [[doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334)]
On [[univalent bicategory|univalent]] [[bicategories]] in [[homotopy type theory]]:
* [[Benedikt Ahrens]], [[Dan Frumin]], [[Marco Maggesi]], [[Niels van der Weide]], _Bicategories in Univalent Foundations_, Mathematical Structures in Computer Science **31** 10 (2022) 1232-1269 [[arXiv:1903.01152](https://arxiv.org/abs/1903.01152), [doi:10.1017/S0960129522000032](https://doi.org/10.1017/S0960129522000032)&rbrack
On [[monoidal category|monoidal]] [[univalent categories]]:
* [[Kobe Wullaert]], [[Ralph Matthes]], [[Benedikt Ahrens]], *Univalent Monoidal Categories* [[arXiv:2212.03146](https://arxiv.org/abs/2212.03146)]
## Related entries
* [[internal category in homotopy type theory]]
* [[UniMath project]]
category: people
|
Benedikt Fluhr
|
https://ncatlab.org/nlab/source/Benedikt+Fluhr
|
* [Homepage](https://bfluhr.com/)
|
Bengt Nilsson
|
https://ncatlab.org/nlab/source/Bengt+Nilsson
|
* [webpage](http://fy.chalmers.se/~tfebn/)
## Selected writings
On [[Kaluza-Klein compactification]] in [[supergravity]]:
* {#DuffNilssonPope86} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Kaluza-Klein supergravity_, Physics Reports Volume 130, Issues 1–2, January 1986, Pages 1-142 ([spire:229417](https://inspirehep.net/record/229417), <a href="https://doi.org/10.1016/0370-1573(86)90163-8">doi:10.1016/0370-1573(86)90163-8</a>)
On the [[partition function]] of the [[superstring]] ([[heterotic string theory|heterotic string]] and [[type II string theory|type II string]]) as a [[modular form]] with values in the [[Chern character]] of the [[background field|background]] [[field strengths]] ("character-valued partition function", then also called the *[[elliptic genus]]*/*[[Witten genus]]*) and relation to [[Green-Schwarz anomaly cancellation]]:
* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], *Heterotic string-loop calculation of the anomaly cancelling term*, Nuclear Physics B Volume 289, 1987, Pages 609-627 (<a href="https://doi.org/10.1016/0550-3213(87)90397-X">doi:10.1016/0550-3213(87)90397-X</a>)
* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (<a href="https://doi.org/10.1016/0550-3213(88)90468-3">doi:10.1016/0550-3213(88)90468-3</a>)
On [[DBI-action]] and [[Green-Schwarz action functional]] for [[D-branes]]:
* [[Martin Cederwall]], Alexander von Gussich, [[Aleksandar Mikovic]], [[Bengt Nilsson]], Anders Westerberg, _On the Dirac-Born-Infeld Action for D-branes_, Phys.Lett.B390:148-152, 1997 ([arXiv:hep-th/9606173](https://arxiv.org/abs/hep-th/9606173))
* {#CGNSW96} [[Martin Cederwall]], Alexander von Gussich, [[Bengt Nilsson]], Per Sundell, Anders Westerberg, _The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity_, Nucl.Phys. B490 (1997) 179-201 ([arXiv:hep-th/9611159](http://arxiv.org/abs/hep-th/9611159))
category: people
|
Beni Yoshida
|
https://ncatlab.org/nlab/source/Beni+Yoshida
|
* [institute page](https://perimeterinstitute.ca/people/beni-yoshida)
## Selected writings
An observation on classical [[error correcting codes]] preconceiving aspects of [[holographic tensor network]] [[quantum error correcting codes]]:
* [[Beni Yoshida]], *Information storage capacity of discrete spin systems*, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https://arxiv.org/abs/1111.3275))
Introducing the [[HaPPY code]] (a [[quantum error correcting code]] whose generating [[tensor network]] exhibits [[holographic entanglement entropy]]):
* {#PYHP15} [[Fernando Pastawski]], [[Beni Yoshida]], [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))
On [[AdS/CFT in condensed matter physics]]:
* Mike Blake, Yingfei Gu, [[Sean A. Hartnoll]], Hong Liu, [[Andrew Lucas]], Krishna Rajagopal, [[Brian Swingle]], [[Beni Yoshida]], *Snowmass White Paper: New ideas for many-body quantum systems from string theory and black holes* [[arXiv:2203.04718](https://arxiv.org/abs/2203.04718)]
category: people
|
Benjamin Antieau
|
https://ncatlab.org/nlab/source/Benjamin+Antieau
|
* [website](http://www.math.washington.edu/~bantieau/)
## Selected writings
On [[prismatic cohomology]] applied to [[algebraic K-theory]]:
* {#AKN22} [[Benjamin Antieau]], [[Achim Krause]], [[Thomas Nikolaus]], _On the K-theory of $\mathbb{Z}/p^{n}$ -- Announcement_ [[arXiv:2204.03420](https://arxiv.org/abs/2204.03420)[
category: people
|
Benjamin Assel
|
https://ncatlab.org/nlab/source/Benjamin+Assel
|
* [webpage](https://www.perimeterinstitute.ca/people/benjamin-assel)
## Selected writings
On [[Coulomb branch]] [[singularities]] in 3d [[super QCD]]:
* [[Benjamin Assel]], [[Stefano Cremonesi]], _The Infrared Physics of Bad Theories_, SciPost Phys. 3, 024 (2017) ([arXiv1707.03403](https://arxiv.org/abs/1707.03403))
category: people
|
Benjamin Balsam
|
https://ncatlab.org/nlab/source/Benjamin+Balsam
|
* [webpage](http://www.math.sunysb.edu/~balsam/)
## related $n$Lab entry
* [[Turaev-Viro model]]
category: people
|
Benjamin Blander
|
https://ncatlab.org/nlab/source/Benjamin+Blander
|
#related $n$Lab entries#
* [[model structure on simplicial presheaves]]
category: people
|
Benjamin Grinstein
|
https://ncatlab.org/nlab/source/Benjamin+Grinstein
|
* [webpage](http://leewick.ucsd.edu/~ben/)
## Selected writings
On [[flavor physics]]:
* [[Benjamin Grinstein]], _TASI-2013 Lectures on Flavor Physics_ ([arXiv:1501.05283](https://arxiv.org/abs/1501.05283))
On [[flavor physics]] in view of the [[flavour anomalies]] seen at [[LHCb]] and other experiments
* [[Benjamin Grinstein]], _A path to flavor_, talk at _[Implications of LHCb measurement and future prospects](https://indico.cern.ch/event/769902/)_ CERN 2019 ([pdf](https://indico.cern.ch/event/769902/contributions/3582540/attachments/1926501/3193107/Grinstein-high-res.pdf), [[GrinsteinFlavor2019.pdf:file]], [indico:3582540](https://indico.cern.ch/event/769902/contributions/3582540))
category: people
[[!redirects Ben Grinstein]]
|
Benjamin Hennion
|
https://ncatlab.org/nlab/source/Benjamin+Hennion
|
* [webpage](http://guests.mpim-bonn.mpg.de/hennion/)
## related $n$Lab entries
* [[tangent complex]]
* [[formal neighbourhood of the diagonal]]
category: people
[[!redirects Hennion]]
|
Benjamin M. Mann
|
https://ncatlab.org/nlab/source/Benjamin+M.+Mann
|
[[!redirects B. M. Mann]]
## Selected writings
On the [[moduli space of Yang-Mills monopoles]] and [[rational maps]] to [[complex projective spaces]]:
* [[Charles P. Boyer]], [[B. M. Mann]], *Monopoles, non-linear $\sigma$-models, and two-fold loop spaces*, Commun. Math. Phys. 115, 571–594 (1988) ([arXiv:10.1007/BF01224128](https://doi.org/10.1007/BF01224128))
On [[homotopy type of spaces of rational maps]]:
* [[Fred Cohen]], [[Ralph Cohen]], [[B. M. Mann]], [[R. J. Milgram]], _The topology of rational functions and divisors of surfaces_, Acta Math (1991) 166: 163 ([doi:10.1007/BF02398886](https://doi.org/10.1007/BF02398886))
* [[Benjamin M. Mann]], [[R. James Milgram]], *Some spaces of holomorphic maps to complex Grassmann manifolds*, J. Differential Geom. 33(2): 301-324 (1991) ([doi:10.4310/jdg/1214446318](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-2/Some-spaces-of-holomorphic-maps-to-complex-Grassmann/10.4310/jdg/1214446318.full))
* [[Charles P. Boyer]], [[B. M. Mann]], [[J. C. Hurtubise]], [[R. J. Milgram]], *The topology of the space of rational maps into generalized flag manifolds*, Acta Mathematica. 1994;173(1):61-101 ([doi:10.1007/BF02392569](https://projecteuclid.org/journals/acta-mathematica/volume-173/issue-1/The-topology-of-the-space-of-rational-maps-into-generalized/10.1007/BF02392569.full))
category: people
|
Benjamin Pierce
|
https://ncatlab.org/nlab/source/Benjamin+Pierce
|
* [website](http://www.cis.upenn.edu/~bcpierce/)
## Selected writings
On [[subtypes]] (with an early discussion of what came to be called *[[lenses (in computer science)]]* motivated by [[object-oriented programming]]):
* {#HofmannPierce96} [[Martin Hofmann]], [[Benjamin Pierce]], *Positive Subtyping*, Information and Computation **126** 1 (1996) 11-33 [[doi:10.1006/inco.1996.0031](https://doi.org/10.1006/inco.1996.0031)]
Introducing the terminology of [[lenses (in computer science)]]:
* Aaron Bohannon, [[Benjamin C. Pierce]], Jeffrey A. Vaughan, *Relational lenses: a language for updatable views*, Proceedings of Principles of Database Systems (PODS) (2006) 338-347 [[doi:10.1145/1142351.1142399](https://doi.org/10.1145/1142351.1142399), [pdf](https://www.cis.upenn.edu/~bcpierce/papers/dblenses-pods.pdf)]
* {#FosterGreenwaldMoorePierceSchmitt07} J. N. Foster, M. B. Greenwald, J. T. Moore, [[Benjamin C. Pierce]], A. Schmitt, _Combinators for bidirectional tree transformations: A linguistic approach to the view-update problem_, ACM Transactions on Programming Languages and Systems **29** 3 (2007) 17-es [[doi:x10.1145/1232420.1232424](https://doi.org/10.1145/1232420.1232424)]
category: people
[[!redirects Benjamin C. Pierce]]
|
Benjamin Schumacher
|
https://ncatlab.org/nlab/source/Benjamin+Schumacher
|
* [Wikipedia entry](https://en.wikipedia.org/wiki/Benjamin_Schumacher)
## Selected writings
Early discussion of [[quantum computation]] and introducing the terminology *[[q-bit]]*:
* [[Benjamin Schumacher]], *Quantum coding*, Phys. Rev. A **51** (1995) 2738 $[$[doi:10.1103/PhysRevA.51.2738](https://doi.org/10.1103/PhysRevA.51.2738)$]$
On [[quantum systems]] and [[quantum information theory]]:
* [[Benjamin Schumacher]], [[Michael Westmoreland]], *Quantum Processes, Systems, and Information*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511814006](https://doi.org/10.1017/CBO9780511814006)]
category: people
|
Benjamin Steinberg
|
https://ncatlab.org/nlab/source/Benjamin+Steinberg
|
[[!redirects Ben Steinberg]]
[[!redirects Ben Steinberg]]
[[!redirects B. Steinberg]]
* [Homepage](http://www.sci.ccny.cuny.edu/~benjamin/)
* [Blog](http://bensteinberg.wordpress.com/author/bsteinbg/)
category:people
|
Benjamin-Ono equation
|
https://ncatlab.org/nlab/source/Benjamin-Ono+equation
|
* thesis by Alexandre Landry [pdf](http://archimede.bibl.ulaval.ca/archimede/fichiers/27192/27192.pdf)
related entries: [[integrable model]]
|
Benno van den Berg
|
https://ncatlab.org/nlab/source/Benno+van+den+Berg
|
* [website](https://staff.fnwi.uva.nl/b.vandenberg3/)
## Selected writings
On [[infinity-groupoid]]/[[omega-groupoid]]-[[structure]] on [[types]] in [[homotopy type theory]]:
* [[Benno van den Berg]], [[Richard Garner]], *Types are weak omega-groupoids*, Proceedings of the London Mathematical Society **102** 2 (2011) 370-394 [[arXiv:0812.0298](https://arxiv.org/abs/0812.0298), [doi:10.1112/plms/pdq026](https://doi.org/10.1112/plms/pdq026)]
On a [[topos]] for [[continuous logic]]:
* Daniel Figueroa, [[Benno van den Berg]], _A topos for continuous logic_ ([arXiv:2107.10543](https://arxiv.org/abs/2107.10543))
On [[objective type theory]]:
* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))
category: people
|
Benny Lautrup
|
https://ncatlab.org/nlab/source/Benny+Lautrup
|
* [Wikipedia entry](https://en.wikipedia.org/wiki/Benny_Lautrup)
## related $n$Lab entries
* [[Nakanishi-Lautrup field]], [[antighost field]]
category: people
|
Beno Eckmann
|
https://ncatlab.org/nlab/source/Beno+Eckmann
|
* [English Wikipedia entry](http://en.wikipedia.org/wiki/Beno_Eckmann)
## Selected writings
On [[internalization]], [[H-spaces]], [[monoid objects]], [[group objects]] in [[algebraic topology]]/[[homotopy theory]] and introducing the [[Eckmann-Hilton argument]]:
* [[Beno Eckmann]], [[Peter Hilton]], Theorem 1.12: *Structure maps in group theory*, Fundamenta Mathematicae 50 (1961), 207-221
([doi:10.4064/fm-50-2-207-221](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/2/94854/structure-maps-in-group-theory))
* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories I multiplications and comultiplications_, Math. Ann. 145, 227–255 (1962) ([doi:10.1007/BF01451367](https://doi.org/10.1007/BF01451367))
* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories III primitive categories_, Math. Ann. **150** 165–187 (1963) ([doi:10.1007/BF01470843](https://doi.org/10.1007/BF01470843))
On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:
* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _Seminar on triples and categorical homology theory_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]
[[!redirects B. Eckmann]]
[[!redirects Eckmann]]
category:people
|
Benoit Fresse
|
https://ncatlab.org/nlab/source/Benoit+Fresse
|
__[Benoit Fresse](http://math.univ-lille1.fr/~fresse/)__ is a mathematician at Lille. One of his interests concerns the theory of [[operads]].
## Selected writings:
On [[Lie theory]] of [[formal groups]] over an [[operad]]:
* _Lie theory of formal groups over an operad_, J. Alg. __202__, 455--511, 1998, [doi](http://dx.doi.org/10.1006/jabr.1997.7280)
* _Modules over operads and functors_, Springer LNM __1967__, 2009, x+308 pp. [MR2010e:18009](http://www.ams.org/mathscinet-getitem?mr=2494775)
On the [[homotopy theory]] of [[operads]] and [[Grothendieck-Teichmüller groups]]:
* {#FresseHOGTG} [[Benoit Fresse]], *Homotopy of Operads and Grothendieck-Teichmüller Groups (Volumes 1 & 2)*, Mathematical Surveys and Monographs **217** (2017) [[ISBN:978-1-4704-3480-9](https://bookstore.ams.org/surv-217/), [website](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/), [pdf (draft of Part I)](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.648.9110&rep=rep1&type=pdf)]
> Fresse lists two main aims of the book project: 1. the development of a [[rational homotopy theory]] for operads, and 2. to explain the relationship between operads and [[Grothendieck-Teichmüller groups]].
On the [[bar-cobar construction]] for [[dg-Hopf algebras]]:
* [[Benoit Fresse]], _The universal Hopf operads of the bar construction_ ([arXiv:math/0701245](https://arxiv.org/abs/math/0701245))
category: people
|
Benoit Jubin
|
https://ncatlab.org/nlab/source/Benoit+Jubin
|
My web page is [here](http://math.berkeley.edu/~jubin/).
category: people
[[!redirects Benoît Jubin]]
|
Benoît Valiron
|
https://ncatlab.org/nlab/source/Beno%C3%AEt+Valiron
|
* [webpage](https://www.monoidal.net/)
* [InSpire page](https://inspirehep.net/authors/2012244)
## Selected writings
Early consideration of [[quantum programming languages]] as [[quantum lambda-calculus]] invoking [[linear type theory|linear types]]:
* [[Benoît Valiron]], *A functional programming language for quantum computation with classical control*, MSc thesis, University of Ottawa (2004) [[doi:10.20381/ruor-18372](http://dx.doi.org/10.20381/ruor-18372), [pdf](https://ruor.uottawa.ca/bitstream/10393/26790/1/MR01625.PDF)]
* {#SelingerValiron04} [[Peter Selinger]], [[Benoît Valiron]], *A lambda calculus for quantum computation with classical control*, Proc. of TLCA 2005 [[arXiv:cs/0404056](https://arxiv.org/abs/cs/0404056), [doi:10.1007/11417170_26](https://doi.org/10.1007/11417170_26)]
* {#SelingerValiron09} [[Peter Selinger]], [[Benoît Valiron]], *Quantum Lambda Calculus*, in: *Semantic Techniques in Quantum Computation*, Cambridge University Press (2009) 135-172 [[doi:10.1017/CBO9781139193313.005](https://doi.org/10.1017/CBO9781139193313.005), [pdf](https://www.mscs.dal.ca/~selinger/papers/qlambdabook.pdf)]
* Pablo Arrighi, Alejandro Díaz-Caro, [[Benoît Valiron]], *A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus*, EPTCS **88** (2012) 1-15 [[arXiv:1012.4032](https://arxiv.org/abs/1012.4032), [doi:10.4204/EPTCS.88.1](https://doi.org/10.4204/EPTCS.88.1)]
> [[Selinger (2016)](Quipper#Selinger16):] When the [QPL workshop series](https://www.mathstat.dal.ca/~selinger/qpl/) was first founded, it was called "*Quantum Programming Languages*". One year I wasn't participating, and while I wasn't looking they changed the name to "*Quantum Physics and Logic*" --- same acronym!
> Back in those days in the early 21st century we were actually trying to do programming languages for quantum computing $[$[Selinger & Valiron 2004](#SelingerValiron04)$]$, but the sad thing is: In those days nobody really cared. $[...]$
> Now it's 15 years later and several of these parameters have changed: There has been a renewed interest, from government agencies and also from companies who are actually building quantum computers. $[...]$.
> So now people are working on quantum programming languages *again*.
Implementation as an [[embedded programming language]] in [[Haskell]]:
* Richard Eisenberg, [[Benoît Valiron]], Steve Zdancewic, *Typechecking Linear Data: Quantum Computation in Haskell* [[[EisenbergValironZdancewic-LinearData.pdf:file]]]
Introducing the [[functional programming language|functional]] [[quantum programming language]] [[Quipper]]:
* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices 48(6):333-342, 2013 ([arXiv:1304.3390](https://arxiv.org/abs/1304.3390))
* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))
* [[Jonathan Smith]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: Concrete Resource Estimation in Quantum Algorithms_, QAPL 2014 ([arXiv:1412.0625](https://arxiv.org/abs/1412.0625))
On [[linear algebra]] ([[finite-dimensional vector spaces]] over [[finite fields]]) as [[categorical semantics]] for [[linear logic]]/[[linear type theory]]:
* [[Benoît Valiron]], [[Steve Zdancewic]], *Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi*, in: *Proc. of ICTAC'14*, Lecture Notes in Computer Science **8687**, Springer (2014) 442-459 [[arXiv:1406.1310](https://arxiv.org/abs/1406.1310), [doi:10.1007/978-3-319-10882-7_26](https://doi.org/10.1007/978-3-319-10882-7_26), slides:[pdf](https://www.cs.bham.ac.uk/~drg/bll/steve.pdf), [[Zdancewic-LinearLogicAlgebra.pdf:file]]]
Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the [[Quipper]]-language:
* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 [[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf), [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)]
On [[dynamic lifting]] (of [[quantum measurement]]-results) in [[quantum programming]]/[[quantum computing]]:
* [[Dongho Lee]], Sebastien Bardin, [[Valentin Perrelle]], [[Benoît Valiron]], *Formalization of a Programming Language for Quantum Circuits with Measurement and Classical Control*, talk at *[Journées Informatique Quantique 2019](https://quantcert.github.io/GT-IQ'19/)* (Nov 2019) [[pdf](https://quantcert.github.io/GT-IQ%2719/slides/Lee.pdf), [[LBPV_QCWithClassicalControl.pdf:file]]]
* [[Dongho Lee]], [[Valentin Perrelle]], [[Benoît Valiron]], Zhaowei Xu, *Concrete Categorical Model of a Quantum Circuit Description Language with Measurement*, Leibniz International Proceedings in Informatics **213** (2021) 51:1-51:20 [[arXiv:2110.02691](https://arxiv.org/abs/2110.02691), [doi:10.4230/LIPIcs.FSTTCS.2021.51](https://doi.org/10.4230/LIPIcs.FSTTCS.2021.51)]
See also:
* Kostia Chardonnet, Marc de Visme, Benoît Valiron, Renaud Vilmart, *The Many-Worlds Calculus* [[arXiv:2206.10234](https://arxiv.org/abs/2206.10234)]
## Related entries
* [[software verification]]
category: people
[[!redirects Benoit Valiron]]
|
Benson Farb
|
https://ncatlab.org/nlab/source/Benson+Farb
|
* [personal page](http://www.math.uchicago.edu/~farb/)
* [institute page](https://mathematics.uchicago.edu/people/profile/benson-farb/)
* [Wikipedia page](https://en.wikipedia.org/wiki/Benson_Farb)
## Selected writings
Introducing the notion of *[[FI-modules]]* ([[FinInj|$FinSet_{inj}$]]-[[representations]]):
* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], *FI-modules and stability for representations of symmetric groups*, Duke Math. J. **164** 9 (2015) 1833-1910 [[arXiv:1204.4533](https://arxiv.org/abs/1204.4533), [doi:10.1215/00127094-3120274](https://doi.org/10.1215/00127094-3120274)]
* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], [[Rohit Nagpal]], *FI-modules over Noetherian rings*, Geom. Topol. **18** (2014) 2951-2984 [[arXiv:1210.1854](https://arxiv.org/abs/1210.1854), [doi:10.2140/gt.2014.18.2951](https://doi.org/10.2140/gt.2014.18.2951)]
and the related notion of [[representation stability]]:
* {#ChurchFarb13} [[Thomas Church]], [[Benson Farb]], *Representation theory and homological stability*, Advances in Mathematics
**245** (2013) 250-314 [[doi:10.1016/j.aim.2013.06.016](https://doi.org/10.1016/j.aim.2013.06.016)]
* [[Benson Farb]], *Representation Stability* [[arXiv:1404.4065](https://arxiv.org/abs/1404.4065)]
category: people
[[!redirects Benson S. Farb]]
[[!redirects Benson Stanley Farb]]
|
Beppe Metere
|
https://ncatlab.org/nlab/source/Beppe+Metere
|
My web page is [here](https://gmetere.github.io/).
category: people
## Selected writings
* A. S. Cigoli, S. Mantovani, [[G. Metere]], *On pseudofunctors sending groups to 2-groups*, Mediterr. J. Math. 20, 17 (2023). [[arXived version](https://arxiv.org/abs/2112.08695)]
* A. S. Cigoli, S. Mantovani, [[G. Metere]], [[E. M. Vitale]], *Fibered aspects of Yoneda's regular span*, Advances in Mathematics, 360 (2020). [[arXived version](https://arxiv.org/abs/1806.02376)]
* S. Mantovani, [[G. Metere]], *Normalities and commutators*, Journal of Algebra 324 (2010), no. 9. [[arXived version](https://arxiv.org/abs/1002.1372)]
[[!redirects G. Metere]]
[[!redirects Giuseppe Metere]]
|
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