geometric infinity-stack

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###Context###
#### Higher geometry
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[[!include higher geometry - contents]]
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The concept of _geometric $\infty$-stack_ is the refinement to [[∞-stack]] of that of _[[geometric stack]]_.

There is an intrinsic definition which iterates that of geometric stacks and says inductively that a geometric $n$-stack is one which has an $n$-[[atlas]] and such that its [[diagonal]] is $(n-1)$-representable ([Toën-Vezzosi 04, def. 1.3.3.1](#ToenVezzosi04)).

Then there is a result which says that such geometric $n$-stacks are equivalently those represented by suitable [[Kan complex]]-objects in the given [[site]] ("[[internal infinity-groupoids]]" in the site) ([Pridham 09](Pridham09)).

(There is also a definition of "geometric $\infty$-stack" in ([Toën 00, definition 4.1.4](#Toen00)), which is however different.)

## Definition

## Properties

### Presentation by Kan-fibrant simplicial objects

A presentation of geometric $\infty$-stacks, in some generality, by suitably  Kan-fibrant [[simplicial objects]] is in ([Pridham 09](#Pridham09)). See also at _[[Kan-fibrant simplicial manifold]]_.

## Examples

* Every object in the image of $Spec : T Alg_\infty^{op} \to \mathbf{H}$ is a geometric $\infty$-stack.

* Over the [[étale site]] an [[algebraic stack]] that is a [[geometric stack]] is also a geometric $\infty$-stack.

* Every [[schematic homotopy type]] is given by a geometric $\infty$-stack.

* [[Kan-fibrant simplicial manifolds]]  serve to present geometric $\infty$-stacks in [[higher differential geometry]], the [[Lie infinity-groupoids]]

## Toen 00

> The text below follows ([Toën 00](#Toen00)). Needs to be connected to the rest of the entry.

We consider the [[higher geometry]] encoded by a [[Lawvere theory]] $T$ via [[Isbell duality]]. Write $T Alg$ for the category of [[algebras over a Lawvere theory]] and write $T Alg^{\Delta}$ for the [[(∞,1)-category]] of [[cosimplicial object|cosimplicial]] $T$-algebras .

Consider a [[site]] $T \subset C \subset T Alg^{op}$ that satisfies the assumptions described at [[function algebras on ∞-stacks]]. Then, by the discussion given there, we have a pair of [[adjoint (∞,1)-functor]]s

$$
  (\mathcal{O} \dashv Spec) :  (T Alg^{\Delta})^{op}
   \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  Sh_\infty(C) =: \mathbf{H}
  \,,
$$

where $\mathbf{H} := Sh_\infty(C)$ is the [[(∞,1)-category of (∞,1)-sheaves]] over $C$, the [[big topos]] for the [[higher geometry]] over $C$.

+-- {: .num_defn}
###### Definition

An object $X \in \mathbf{H}$ is called a **geometric $\infty$-stack** over $C$ if there is it is the [[(∞,1)-colimit]]

$$
  X \simeq {\lim_\to} K_\bullet
$$

over a [[groupoid object in an (∞,1)-category|groupoid object]] $K_\bullet : \Delta \to \mathbf{H}$ in $\mathbf{H}$ such that

1. $K_0$ and $K_1$ are in the image of $Spec : (T Alg^{\Delta})^{op} \to \mathbf{H}$;

1. the target map $d_1 : K_1 \to K_0$ is lisse.

=--

For $T$ the theory of commutative [[associative algebra]]s over a commutative ring $k$ and $C$ the [[fpqc topology]] this appears as  ([Toën 00, definition 4.1.4](#Toen00)).

+-- {: .num_prop}
###### Proposition

Geometric $\infty$-stacks are stable under [[(∞,1)-pullback]]s along morphism in the image of $Spec$.

=--

+-- {: .proof}
###### Proof

Use that in the [[(∞,1)-topos]] $\mathbf{H}$ we have [[universal colimits]] and that $Spec$ is [[right adjoint]].

=--

## Related concepts

* [[geometric stack]]

* **geometric $\infty$-stack**, [[function algebras on ∞-stacks]]

* [[Arin-Luire representability theorem]]

## References

The notion of geometric $\infty$-stack as a weak quotient of affine $\infty$-stacks is considered in section 4 of

* {#Toen00} [[Bertrand Toën]], _Affine stacks (Champs affines)_ ([arXiv:math/0012219](http://arxiv.org/abs/math/0012219))

More general theory in the context of [[derived algebraic geometry]] is in

* {#ToenVezzosi04} [[Bertrand Toën]], [[Gabriele Vezzosi]] _Homotopical Algebraic Geometry II: geometric stacks and applications_ ([arXiv:math/0404373](http://arxiv.org/abs/math/0404373))

and specifically in [[E-∞ geometry]] in

* {#Lurie} [[Jacob Lurie]], _[[Representability Theorems]]_

* {#Pridham10} [[Jonathan Pridham]], _Representability of derived stacks_ ([arXiv:1011.2189](http://arxiv.org/abs/1011.2189))

Discussion of presentation of geometric $\infty$-stacks by Kan-fibrant [[simplicial objects]] in the [[site]] is in

* {#Pridham09} [[Jonathan Pridham]], _Presenting higher stacks as simplicial schemes_, Advances in Mathematics, Volume 238, Pages 184-245 ([arXiv:0905.4044](http://arxiv.org/abs/0905.4044))

[[!redirects geometric ∞-stack]]

[[!redirects geometric ∞-stacks]]
[[!redirects geometric infinity-stacks]]


geometric integration theory > history

< [[geometric integration theory]]


geometric invariant theory

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###Context###
#### Higher geometry
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[[!include higher geometry - contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Geometric invariant theory_ studies the construction of [[moduli spaces]] / [[moduli stacks]] in terms of [[quotients]] / [[action groupoids]].  (This may be thought of as the geometric aspect ([[Isbell duality|Isbell dual]] aspect) of [[invariant theory]].)

A central aspect of the theory of ([Mumford-Fogarty-Kirwan 65](MumfordFogartyKirwan65), [Mumford 77](#Mumford77)) is -- given the [[action]] of a [[reductive group]] on a [[projective variety]] -- to characterize those points in the variety -- called the [[GIT-stable point|GIT-(semi-)stable points]] -- such that the [[quotient stack]] on these is "close" to being represented by another projective variety (in that it is for instance just an [[orbifold]]/[[Deligne-Mumford stack]]). The precise statement is recalled for instance as ([Saiz 09, theorem 2.3.6](#Saiz09)).

A standard application of this is to the discussion of [[moduli spaces of bundles]], where [[action]] is that of the [[gauge group]] and where the (semi-)stable points correspond to the (semi-)(slope-)[[stable vector bundles]] ([Saiz 09, section 2.3](#Saiz09)).

## Related concepts

* [[stability]], [[stable vector bundle]]

* [[Bridgeland stability condition]] (see [there](Bridgeland+stability+condition#SlopeStabilityOfVectorBundles))

* [[Deligne-Mumford stack]]

## References

The original texts are

*  {#MumfordFogartyKirwan65} [[David Mumford]], [[John Fogarty]], [[Frances Clare Kirwan]], _Geometric invariant theory_, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) __34__, Springer-Verlag (1965/1982) ([ISBN:978-3-540-56963-3](https://www.springer.com/gp/book/9783540569633), [pdf](https://altexploit.files.wordpress.com/2017/08/mumford-d-fogarty-j-geometric-invariant-theory-springer-1982.pdf))

* {#Mumford77} [[David Mumford]], _Stability of projective varieties_, L'Enseignement Mathématique __23__ (1977) [doi](http://dx.doi.org/10.5169/seals-48919)

A good detailed review is in

* {#Saiz09} Alfonso Zamora Saiz, _On the stability of vector bundles_, Master thesis 2009 ([[SaizStableBundles.pdf:file]])

Further surveys include

* Atanas Atanasov, _Geometric invariant theory_, 2011 ([pdf slides](http://www.math.harvard.edu/~nasko/documents/git-slides.pdf))

Further developments include

* David J. Swinarski, _Geometric Invariant Theory
and Moduli Spaces of Maps_ ([pdf](http://www.math.uga.edu/~davids/DSMScthesis.pdf))

* Jürgen Hausen, _A generalization of Mumford's geometric invariant theory_ ([pdf](http://www.math.uiuc.edu/documenta/vol-06/22.pdf))

* [[David Rydh]], _Existence and properties of geometric quotients_,
J. Algebraic Geom. 22 (2013), 629–669, [publ](http://dx.doi.org/10.1090/S1056-3911-2013-00615-3), [pdf](https://people.kth.se/~dary/quotients20120504.pdf).

Relation to [[Bridgeland stability conditions]]:

* {#King94} [[Alastair King]], _Moduli of representations of finite dimensional algebras_, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.623.649&rep=rep1&type=pdf))

* {#Engenhorst14} Jan Engenhorst, _Bridgeland Stability Conditions in Algebra, Geometry and Physics_, 2014 ([pdf](https://www.freidok.uni-freiburg.de/fedora/objects/freidok:9595/datastreams/FILE1/content))


geometric Langlands > history

< [[geometric Langlands]]


geometric Langlands correspondence

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###Context###
#### Langlands correspondence
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[[!include Langlands correspondence -- contents]]
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#### Duality
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[[!include duality - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The [[conjecture|conjectural]] _geometric Langlands correspondence_ is meant to be an [[analogy|analog]] of the [[number theory|number theoretic]] [[Langlands correspondence]] under the [[function field analogy]], hence with [[number fields]] replaced by [[function fields]] and further replaced by [[rational functions]] on [[complex curves]].  The key to this analogy is the [[Weil uniformization theorem]] which expresses the [[moduli stack of G-principal bundles]] over an [[algebraic curve]] as a [[double coset space]] of various function rings (as discussed at _[Moduli of bundles over curves](moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_) of just the kind as they appear in the number-theoretic [[Langlands correspondence]] (for instance in the [[Artin reciprocity law]] and in the definition of [[automorphic representations]]).

[[!include Langlands analogies -- table]]

The original version of the conjectured statement of _geometric Langlands duality_ (going back to [Beilinson-Drinfeld 9x](#BeilinsonDrinfeld9x) and reviewed for instance in [Frenkel 05](#Frenkel05)) asserts that for $G$ a [[reductive group]] and for $\Sigma$ an [[algebraic curve]], then there is an [[equivalence of categories|equivalence]] of [[derived categories]] of, on the one hand, [[D-modules]] on the [[moduli stack of G-principal bundles]] on $\Sigma$, and, on the other hand, [[quasi-coherent sheaves]] on the  ${}^L G$-[[moduli stack of local systems]] on $\Sigma$:

$$
  \mathcal{O}Mod(Loc_{{}^L G}(\Sigma))
    \stackrel{\simeq}{\longrightarrow}
  \mathcal{D} Mod( Bun_G(\Sigma))
$$

for ${}^L G$ the [[Langlands dual group]]. Moreover, the conjecture asserts that there is canonical such an equivalence which is a [[non-abelian cohomology|non-abelian]] analogue of the [[Fourier-Mukai transform|Fourier-Mukai]] [[integral transform]] and takes [[skyscraper sheaves]] on the left ([[categorification|categorified Dirac distributions]]) to what are called "[[Hecke eigensheaves]]" on the right.
This equivalence is in turn supposed to be a certain limit of the more general [[quantum geometric Langlands correspondence]] that identifies [[twisted D-modules]] on both sides.

For the abelian case that $G$ is a [[torus]] the above equivalence has indeed been proven, given by a [[Fourier-Mukai transform]]  ([Laumon 85](#Laumon85), [Laumon 96](#Laumon96), [Rothstein 96](#Rothstein96)), see also [below](#AbelianCase).

However, in general the above version of the conjecture is false. For instance it fails in the case $G = SL_2$ and $\Sigma= \mathbb{P}^1$ ([Lafforgue 09](#Lafforgue09)).

A refined formulation of the conjecture due to ([Arinkin-Gaitsgory 12](#ArinkinGaitsgory12)), meant to fix this failure, replaces plain quasicoherent sheaves with certain "nilpotent" [[ind-objects]] of quasicoherent sheaves and refines [[derived categories]] to [[stable (∞,1)-categories]], to make the conjecture read

$$
  (Ind(\mathcal{O}Mod(Loc_{{}^L G}(\Sigma))))_{Nilp_{Glob}}
    \stackrel{\simeq}{\longrightarrow}
  \mathcal{D} Mod( Bun_G(\Sigma))
$$

([Arinkin-Gaitsgory 12, conjecture 0.1.6](#ArinkinGaitsgory12)).
This form is called the [[categorical geometric Langlands conjecture]].

Since [[D-modules]] on [[moduli stacks of G-principal bundles]] play a central role in [[gauge theory|gauge]] [[quantum field theory]] (in particular as [[Hitchin connections]] on bundles of [[conformal blocks]] of $G$-[[Chern-Simons theory]] [[holographic principle|holographically]] dual to the [[WZW model]] [[2d conformal field theory]]) and since the [[Langlands dual group]] also appears in [[electric-magnetic duality]], it has long been suggested ([Atiyah 77](S-duality#Atiyah77)) that geometric Langlands duality has a distinguished meaning also in [[mathematical physics]] in general and in [[string theory]] in particular. One proposal for a realization of the correspondence as an incarnation of [[mirror symmetry]]/[[S-duality]] is due to ([Kapustin-Witten 06](#KapustinWitten06)), which however has not been turned into [[theorems]] yet. Another proposal for realizing the [[local Langlands correspondence|local]] correspondence via another incarnation of [[mirror symmetry]] is due to ([Gerasimov-Lebedev-Oblezin 09](#GerasimovLebedevOblezin09)).

The geometric Langlands conjecture has been motivated from the [[number theory|number theoretic]] [[Langlands correspondence]] via the [[function field analogy]] and some educated guessing, but there is to date no formalization of this analogy that would allow to regard number-theoretic and the geometric correspondence as two special cases of one "global" [[arithmetic geometry]]/[[global analytic geometry]] statement. Cautioning remarks on the accuracy of the analogy and on the rigour of the mirror-symmetric proposals may be found in ([Langlands 14](#Langlands14)). Some discussion of how to possibly start to go about making the analogy more systematic are at _[[differential cohesion and idelic structure]]_.

[[!include function field analogy -- table]]

## Properties
 {#Properties}

### Abelian case
 {#AbelianCase}

In the case where $G$ is the [[multiplicative group]], hence where all bundles in question are [[line bundles]], geometric Langlands duality is well understood and is in fact a slight variant of a [[Fourier-Mukai transform]] ([Frenkel 05, section 4.4, 4.5](#Frenkel05)).

### Relation to S-duality
 {#RelationToSDuality}

The [[Kapustin-Witten TQFT]] ([KapustinWitten 2007](#KapustinWitten06)) is supposed to exhibit geometric Langlands duality as a special case of [[S-duality]].

See also at _[KK-compactification -- Formalization](http://ncatlab.org/nlab/show/Kaluza-Klein+mechanism#Formalization)_

[[!include geometric Langlands QFT -- table]]

[[!include gauge theory from AdS-CFT -- table]]

### Relation to T-duality
 {#RelationToTDuality}

In some cases the passage between a Lie group and its [[Langlands dual group]] can be understood as a special case of [[T-duality]]. ([Daenzer-vanErp](#DaenzerErp))

## Related concepts

* [[geometric class field theory]]

* [[geometric Satake equivalence]]

* [[Hitchin fibration]]

* [[topologically twisted D=4 super Yang-Mills theory]]

[[duality in physics]], [[duality in string theory]]

* [[S-duality]]

  * [[electro-magnetic duality]]

    * [[Montonen-Olive duality]]

  * [[Seiberg duality]]

  * **geometric Langlands correspondence**

    * [[S-duality]], [[Kapustin-Witten TQFT]]

  * [[quantum geometric Langlands correspondence]]

## References

### Original
 {#ReferencesOriginal}

The conjecture goes back to

* {#BeilinsonDrinfeld9x} [[Alexander Beilinson]], [[Vladimir Drinfeld]], section 5.2.7 of _Quantization of the Hitchin system and Hecke eigensheaves_, ([pdf](http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf))

based on

* {#Laumon87} [[Gérard Laumon]], sections 5.3 and 4.3.3. of _Correspondance de Langlands géométrique pour les corps de fonctions_, Duke Math. Jour., vol. 54 (1987), 309-359

Proof in the abelian case is due to

* {#Laumon85} [[Gérard Laumon]], _Transformation de Fourier géométrique_ Preprint IHES/85/M/52 (1985)

* {#Laumon96} [[Gérard Laumon]], _Transformation de Fourier généralisée_ ([arXiv:alg-geom/9603004](http://arxiv.org/abs/alg-geom/9603004))

* [[Gérard Laumon]], _Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands_, [math.AG/0207078](http://arxiv.org/abs/math.AG/0207078)

* {#Rothstein96} M. Rothstein. _Sheaves with connection on abelian varieties_, Duke Math. J., 84(3):565–598, 1996

  _Correction to: "Sheaves with connection on abelian varieties." Duke Math. J., 87(1):205–211, 1997.

Proof that the original version of the conjecture is false in general is due to

* {#Lafforgue09} [[Vincent Lafforgue]], _Quelques calculs reliés à la correspondance de Langlands géométrique pour $\mathbb{P}^1$ (version provisoire)_ 2009 ([web](http://www.math.jussieu.fr/~vlafforg/), [pdf](http://vlafforg.perso.math.cnrs.fr/geom.pdf))

The refined version of the conjecture stated in [[derived algebraic geometry]], called the [[categorical geometric Langlands conjecture]], is due to

* {#GaitsgoryRozenblyum} [[Dennis Gaitsgory]],  [[Nick Rozenblyum]], _[[Notes on geometric Langlands]], ([web](http://www.math.harvard.edu/~gaitsgde/GL/))

* {#ArinkinGaitsgory12} [[Dima Arinkin]], [[Dennis Gaitsgory]], _Singular support of coherent sheaves, and the geometric Langlands conjecture_ ([arXiv.1201.6343](http://arxiv.org/abs/1201.6343))

Other comments on the relation to [[TQFT]] include

* {#Kapranov95} [[Mikhail Kapranov]], _Analogies between the Langlands Correspondence and Topological Quantum Field Theory_, in _Functional Analysis on the Eve of the 21st Century_ Progress in Mathematics Volume 131/132, 1995, pp 119-151

Comments on the development of the geometric duality by [[Robert Langlands]] himself:

* {#Langlands13} [[Robert Langlands]], _The Search for a Mathematically Satisfying Geometric Theory of Automorphic Forms_, Notes for a lecture at Mostow-Fest, Yale 2013 ([IAS page](http://publications.ias.edu/rpl/paper/2578), [video](http://www.youtube.com/watch?v=pfpzET8UkF4), [pdf](http://publications.ias.edu/sites/default/files/lecture_6.pdf))

* {#Langlands14} [[Robert Langlands]], _[[Problems in the theory of automorphic forms -- 45 years later]]_, talk at [Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends](https://www.maths.nottingham.ac.uk/personal/ibf/files/sc3.html), Oxford, July 5 - July 8, 2014

Langlands's doubts about or dissatifaction with the "geometric Langlands program" expressed in these talks (where he suggests that his name not be associated with the "geometric" part of the program) eventually led to

* [[Robert Langlands]], _Об аналитическом виде геометрической теории автоморфных форм_, IAS 2018 ([ias:2678](http://publications.ias.edu/rpl/paper/2678), [pdf](http://publications.ias.edu/sites/default/files/iztvestiya_3.pdf))

This in turn led to the reaction

* [[Edward Frenkel]], _Is there an analytic theory of automorphic functions for complex algebraic curves?_ ([arXiv:1812.08160](https://arxiv.org/abs/1812.08160))

See also the deformation to the [[quantum geometric Langlands correspondence]], such as

* [[Mina Aganagic]], [[Edward Frenkel]], [[Andrei Okounkov]], _Quantum $q$-Langlands Correspondence_ ([arXiv:1701.03146](https://arxiv.org/abs/1701.03146))

### Surveys and reviews

* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

* [[Ron Donagi]], [[Tony Pantev]], _Lectures on the geometric Langlands
conjecture and non-abelian Hodge theory_, 2009 ([pdf](http://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf))

  > (with an eye towards [[nonabelian Hodge theory]])

More on the relation to [[string theory]] and [[S-duality]]:

* {#Frenkel09} [[Edward Frenkel]], _Gauge Theory and Langlands Duality_, Séminaire Bourbaki no 1010, June 2009 ([arXiv:0906.2747](http://arxiv.org/abs/0906.2747))

With emphasis on the role of [[magnetic monopoles]] and [['t Hoof lines]]:

* [[Alexander B. Atanasov]], *Magnetic Monopoles, 't Hooft Lines, and the Geometric Langlands Correspondence*, 2018 ([pdf](http://abatanasov.com/Files/Thesis.pdf), [slides](http://abatanasov.com/Files/Thesis%20Presentation.pdf))

### Further resources

Collections of resources are here;

* [[David Ben-Zvi]], Geometric Langlands -- Lectures and Resources ([web](http://www.math.utexas.edu/users/benzvi/Langlands.html))

* geometric Laglands [page](http://www.math.uchicago.edu/~mitya/langlands)

Notes on two introductory lecture talks are here:

* [Pantev on Langlands I](http://golem.ph.utexas.edu/string/archives/000806.html)

  [Pantev on Langlands II](http://golem.ph.utexas.edu/string/archives/000807.html)

See also

* Ngô Bảo Châu, _Le lemme fondamental pour les algebres de Lie_, [arxiv/0806.4566](http://arxiv4.library.cornell.edu/abs/0806.4566)

* James Arthur, _The Work of Ngô Bảo Châu_, Proc. ICM Hyderabad 2010, [pdf](http://www.icm2010.org.in/wp-content/icmfiles/laudaions/fields2.pdf)

* lecture notes on an introductory talk by [[Tony Pantev]]: [Pantev on Langlands I](http://golem.ph.utexas.edu/string/archives/000806.html), [Pantev on Langlands II](http://golem.ph.utexas.edu/string/archives/000807.html)

* [[Edward Frenkel]], _Langlands correspondence for loop groups_, [description](http://math.berkeley.edu/~frenkel/NEWBOOK), [pdf](http://math.berkeley.edu/~frenkel/loop.pdf)

* [[Edward Frenkel]], a Bourbaki exposition, [pdf](http://math.berkeley.edu/~frenkel/BOOK/bourbaki.pdf)

* [[Edward Frenkel]], _Langlands duality for representations of quantum groups_, [arxiv/0809.4453](http://arxiv.org/abs/0809.4453)

### Interpretation in string theory
 {#ReferencesInterpretationInStringTheory}

#### Global

An interpretation of the global geometric Langlands correspondence as describing [[S-duality]] of [[topological twist|topologically twisted]] [[super Yang-Mills theory]], incarnated in [[mirror symmetry]] on its [[KK-compactification]] to 2d [[sigma-models]] ([[A-model]]/[[B-model]]-type) was given in

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_ , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

* {#FrenkelWitten07} [[Edward Frenkel]], [[Edward Witten]], _Geometric Endoscopy and Mirror Symmetry_ ([arXiv:0710.5939](http://arxiv.org/abs/0710.5939))

* {#Witten08} [[Edward Witten]], _Mirror Symmetry, Hitchin's Equations, And Langlands Duality_ ([arXiv:0802.0999](http://arxiv.org/abs/0802.0999))

and discussed in the bigger picture of [[S-duality]] arising as the conformal invariance of the [[6d (2,0)-superconformal QFT]] in

* {#Witten09} [[Edward Witten]], _Geometric Langlands From Six Dimensions_, in Peter Kotiuga (ed.) _A Celebration of the Mathematical Legacy of Raoul Bott_, AMS 2010 ([arXiv:0905.2720](http://arxiv.org/abs/0905.2720))

Reflections on the history (and possible future) of this insight are in

* interview with [[Edward Witten]] by [[Hirosi Ooguri]], 2014 ([pdf](http://www.ams.org/notices/201505/rnoti-p491.pdf))

An exposition of the relation to [[S-duality]] and [[electro-magnetic duality]] is in ([Frenkel 09](#Frenkel09)) and in

* [[Edward Frenkel]],  _What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?_ KITP talk 2011 ([web](http://online.kitp.ucsb.edu/online/bblunch/frenkel/))

* [[Edward Frenkel]], _Overview of the links between the Langlands program and 4D super Yang--Mills theory_, KITP talk 2010, [video page](http://online.kitp.ucsb.edu/online/duallang_m10/frenkel), notes [pdf](http://online.kitp.ucsb.edu/online/duallang_m10/frenkel/pdf/Frenkel_LanglandsQFT_KITP.pdf)

Further developments are surveyed in

* [[David Ben-Zvi]], _[[Loop Groups, Characters and Elliptic Curves]]_, 2012

Further discussion is also in

* Tamas Hausel, _Global topology of the Hitchin system_ ([arXiv:1102.1717](http://arxiv.org/abs/1102.1717), [pdf slides](http://geom.epfl.ch/files/content/sites/geom/files/Calgary0312.pdf))

* Kevin Setter, _Topological quantum field theory and the geometric Langlands correspondence_. Dissertation (Ph.D.), California Institute of Technology 2013 ([web](http://resolver.caltech.edu/CaltechTHESIS:09192012-150137728))

Discussion from the point of view of [[M-theory]] is in

* [[Meng-Chwan Tan]], _M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems_ ([arXiv:1301.1977](http://arxiv.org/abs/1301.1977))

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], _Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory_ ([arXiv:1910.01134](https://arxiv.org/abs/1910.01134))

A relation to [[T-duality]] (of the group manifolds!) is discussed in

* {#DaenzerErp} [[Calder Daenzer]], [[Erik van Erp]], _T-Duality for Langlands Dual Groups_ ([arXiv:1211.0763](http://arxiv.org/abs/1211.0763))

#### Local

Discussion of  [[local Langlands correspondence|local]] [[archimedean field|Archimedean]] Langlands duality for [[Whittaker functions]] as [[mirror symmetry]] of a suitable [[A-model]] and [[B-model]] is discussed in

* {#GerasimovLebedevOblezin09} [[Anton Gerasimov]], Dimitri Lebedev, [[Sergey Oblezin]],

  _Archimedean L-factors and Topological Field Theories I_ ([arXiv:0906.1065](http://arxiv.org/abs/0906.1065))

  _Archimedean L-factors and Topological Field Theories II_ ([arXiv:0909.2016](http://arxiv.org/abs/0909.2016))

  _Parabolic Whittaker Functions and Topological Field Theories I_ ([arXiv:1002.2622](http://arxiv.org/abs/1002.2622))

[[!redirects geometric Langlands duality]]

[[!redirects geometric Langlands program]]

[[!redirects geometric Langlands theory]]
[[!redirects geometric Langlands duality]]
[[!redirects geometric Langlands]]

[[!redirects geometric Langlands conjecture]]


geometric Langlands program > history

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#### Langlands correspondence
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[[!include Langlands correspondence -- contents]]
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#### Duality
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[[!include duality - contents]]
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See at _[[geometric Langlands correspondence]]_


geometric Langlands QFT -- table

| [[geometric Langlands correspondence]] | [[S-duality]] in [[N=4 D=4 super Yang-Mills theory]] |
|----------------------------------------|-----------------------------|
| [[Hecke transformation]]                     | [['t Hooft operator]]       |
| [[local system]]/[[flat connection]]                  | [[electric eigenbrane]] (eigenbrane of [[Wilson operator]])
| [[Hecke eigensheaf]]    | [[magnetic eigenbrane]] (eigenbrane of [['t Hooft operator]] ) |

([Kapustin-Witten 06](geometric+Langlands+correspondence#KapustinWitten06))


geometric mathematics

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### Context
#### Foundations
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[[!include foundations - contents]]
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#### Mathematics
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[[!include mathematicscontents]]
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#### Constructivism, Realizability, Computability
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[[!include constructivism - contents]]
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\tableofcontents

## Idea

*Geometric mathematics* is [[mathematics]] done in [[geometric logic]] or [[geometric type theory]] or inside of a [[Grothendieck topos]]. One has [[finite limits]] and all [[colimits]], but one does not in general have [[function sets]] or [[power sets]]. The requirement that one works inside of a [[Grothendieck topos]] instead of a [[geometric category]] (cf. [Vickers 22](#Vickers22)) is because [[geometric categories]] are in general not [[balanced categories]] and so the [[principle of unique choice]] is not satisfied inside of the category.

## Related concepts

* [[constructive mathematics]]
* [[classical mathematics]]
* [[geometric type theory]]

## References

* {#Vickers22} [[Steve Vickers]], *Generalized point-free spaces, pointwise*, ([arXiv:2206.01113](https://arxiv.org/abs/2206.01113))

* [[Steve Vickers]], *Is geometric logic constructive?* ([slides](https://www.cs.bham.ac.uk/~sjv/GeologConsSlides.pdf))

* [[Ming Ng]], *Adelic Geometry via Topos Theory*, PhD thesis ([pdf](https://puzzledoyster.github.io/publications/FINALSUBMISSION.pdf))


geometric measure theory

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#### Integration theory
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[[!include integration theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Geometric measure theory_ and _geometric integration theory_ studies various [[measures]] of subsets of [[Euclidean spaces]] and possibly of some geometric generalizations) and their geometric properties. Especially, one studies [[rectifiability]] of subsets of some lower dimensionality, to define notions like area, arc length etc. and to study [[distributions]] and [[currents]] on such spaces. Very central questions and motivations belong to the [[variational calculus|variational problems]] including the study of [[minimal surface|minimal surfaces]].

## Related concepts

* [[cohomological integration]]

* [[current (distribution theory)]]

## References

* [[Herbert Federer]], *Geometric measure theory*, Grundlehren **153**, Springer (1969, 1996) [[doi:10.1007/978-3-642-62010-2](https://doi.org/10.1007/978-3-642-62010-2)]

* [[Victor Guillemin]], [[Shlomo Sternberg]], *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]

* [[Jürgen Jost]], _The geometric calculus of variations: a short survey and a list of open problems_, Exposition. Math. __6__ (1988), no. 2, 111–143, [MR89h:58036](http://www.ams.org/mathscinet-getitem?mr=938179)

* [[Steven G. Krantz]], [[Harold Parks]], *Geometric integration theory*, Springer (2008) [[pdf](http://www.math.wustl.edu/~sk/books/root.pdf)]

* Frederick J., Jr. Almgren, Almgren's big regularity paper (book form of a 1970s preprint)

* F. J. Almgren, Jr., _Geometric measure theory and elliptic variational problems_, Proc. ICM Nice 1970, vol. 2, [djvu:350 K](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0813.0820.ocr.djvu), [pdf:649 K](http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0813.0820.ocr.pdf)

* [[Frank Morgan]], *Geometric measure theory: a beginner's guide*, Academic Press (2016) [[doi:10.1016/C2015-0-01918-9](https://doi.org/10.1016/C2015-0-01918-9)]

* [[eom]]: T.C.O'Neil (2001), [Geometric measure theory](http://eom.springer.de/G/g130040.htm)

* H. Federer, W. H. Fleming, _Normal and integral currents_, Ann. of Math. (2) __72__ 1960 458–520, [MR123260](http://www.ams.org/mathscinet-getitem?mr=123260), [doi](http://dx.doi.org/10.2307/1970227)

* Stephen H. Schanuel, _What is the length of a potato? An introduction to geometric measure theory_, in: Categories in continuum physics (Buffalo, N.Y., 1982), 118–126,
Lecture Notes in Math. __1174__, Springer 1986, [MR842922](http://www.ams.org/mathscinet-getitem?mr=842922),[doi](http://dx.doi.org/10.1007/BFb0076939)

* Rodolfo Rios-Zertuche, _The variational structure of the space of holonomic measures_, [arxiv1408.5785](http://arxiv.org/abs/1408.5785)

See also:

* wikipedia [geometric measure theory](http://en.wikipedia.org/wiki/Geometric_measure_theory)

[[!redirects geometric integration theory]]


geometric model for elliptic cohomology

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#### Cohomology
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[[!include cohomology - contents]]
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#### Functorial quantum field theory
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[[!include functorial quantum field theory - contents]]
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#Contents#
* tic
{:toc}

## Idea

There are well-known _geometric models_ for some [[cohomology theory|cohomology theories]]. For instance

* [[integral cohomology]] is modeled by [[line bundle]]s and [[bundle gerbe]]s, etc.

and

* [[K-theory]] is modeled by [[vector bundle]]s.

A geometric model for [[elliptic cohomology]] is supposed to be an analogous construction for [[elliptic cohomology]] or for [[tmf]].

It is an old idea that analogous to how [[differential K-theory]] is modeled by parallel transport in [[vector bundle]]s and hence by [[functor]]s $Bord_1(X) \to Vect$,   [[elliptic cohomology]] should somehow be modeled by functors $Bord_2(X) \to Vect$, where $Bord_2(X)$ is a 1-category of [[cobordism]]s equipped with maps to $X$.

[[Stephan Stolz]] and [[Peter Teichner]] have initiated a program studying this in the article [[What is an elliptic object?]].

The fundamental idea of this program is essentially to encode [[parallel transport]] along [[cobordism]]s with maps into a given space $X$ pretty much along the lines of [[schreiber:differential cohomology|functorial differential cohomology]]. One expects that this encodes actually the differential refinements of the corresponding [[cohomology|cohomology theories]], such as [[differential K-theory]]. However, currently in the program one divides out [[concordance]] which effectively divides out the differential information and keeps just the underlying topological information.

The fundamental result of this program so far is that there is a useful notion of 2d [[FQFT]] over $X$ such that its [[partition function]] is indeed [[topological modular form]]-valued, so that it is a candidate for a model of [[tmf]].

The following is going to be an exposition of this partial result:

* Outline of the constructions and statements

  * [[Axiomatic field theories and their motivation from topology]]

  * [[(1,1)-dimensional Euclidean field theories and K-theory]]

  * [[(2,1)-dimensional Euclidean field theories and tmf]]

* Definitions

  * [[bordism categories following Stolz-Teichner]]

  * [[(2,1)-dimensional Euclidean field theories]]

See also the entry

* [[2-spectral triple]].

## References

A recent survey of the program is

* [[Stephan Stolz]], [[Peter Teichner]], _Supersymmetric Euclidean field theories and generalized cohomology_ ([pdf](http://math.berkeley.edu/~teichner/Papers/Survey.pdf))

The program was initiated in

* [[Stephan Stolz]], [[Peter Teichner]], _[[What is an elliptic object?]]_

After the original sketch in terms of extended [[FQFT]] subsequent work concentrated on ordinary 1-categorical constructions, the goal being to make very clear, and transparent and rigorous the constructions involved and the claim that

> the [[partition function]] of a (2|1)-dimensional -Eudlidean [[FQFT]] is an [[integral modular form]].

This latest development on this is in

* [[Stephan Stolz]], [[Peter Teichner]], _Super symmetric field theories and integral modular forms_ ([pdf](http://www.nd.edu/~stolz/SusyQFT.pdf))

> **warning** [[Urs Schreiber|I]] am being told that this is by now outdated and to be replaced by an improved version, which however is apparently not available yet

[[!redirects geometric models for elliptic cohomology]]
[[!redirects geometric model for tmf]]
[[!redirects geometric models for tmf]]
[[!redirects geometric model for TMF]]
[[!redirects geometric models for TMF]]


geometric model for modal logics

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### Context
#### Modalities
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# Contents
* table of contents
{:toc}

## Models in Modal Logics

The [[semantics]] of [[modal logics]] based on [[Kripke frames]] ("geometric" in contrast to [[algebraic models for modal logics]]), often called **Kripke semantics** [[Kripke (1959)](#Kripke59), [(1963)](#Kripke63)].

In the following we say *frame* for *[[Kripke frame]]*.

### Models in Monomodal Logics

Again we look at the basic model language.

+-- {: .un_defn}
###### Definition

A _model_ for $\mathcal{L}_\omega(1)$ is a pair $\mathfrak{M} = (\mathfrak{F},V)$, where $\mathfrak{F}$ is a [[frame (modal logic)
  |frame]] (for $\mathcal{L}_\omega(1)$), that is, a non-empty set, $W$, equipped with a binary relation, $R$, and $V: Prop \to \mathcal{P}(W)$, is a function, called a _valuation_, assigning to each atomic proposition $p$ a subset of the set, $W$, of worlds.
=--

### Gloss

1. Informally we think of $V(p)$ as the set of 'worlds' in our model where $p$ is true.

1. The binary relation of the frame allows for the modelling of the truth of a modal proposition, $\Diamond \phi$ or $\Box \phi$, at some world in terms of the truth of $\phi$ at related worlds.

## Satisfaction

+-- {: .un_defn}
###### Definition

Suppose that $w$ is a state in a model $\mathfrak{M} = (W,R,V)$.  We inductively define the notion of a formula $\phi$ being satisfied in the model at state $w$, as follows:

* for $p\in Prop$, $\mathfrak{M},w \models p$ if and only if $w\in V(p)$;

* $\mathfrak{M},w \models \bot$ never;

* $\mathfrak{M},w \models \neg \phi$ if and only if it is not true that $\mathfrak{M},w \models \phi$;

* $\mathfrak{M},w \models \phi \vee \psi$ if and only if $\mathfrak{M},w \models\phi$ or $\mathfrak{M},w \models\psi$;

* $\mathfrak{M},w \models \Diamond \phi$ if and only if, for some $v \in W$ such that $ R w v$, $\mathfrak{M},v \models \phi$.
=--

### Notes

1. The terminology used in talking about 'satisfaction' tends to interpret $\mathfrak{M},v \models \phi$ as saying the formula $\phi$ is **true** in $\mathfrak{M}$ at state $v$.

2. $\mathfrak{M},w \models \square \phi$ if and only if $\forall v\in W$ $ R w n$ implies $\mathfrak{M},v \models  \phi$. (The proof is fairly routine manipulation of negations.)

3. We say a set $\Sigma$ of formulae is true at state $w\in W$ of a model $\mathfrak{M}$, written $\mathfrak{M},w \models \Sigma$, if all members of $\Sigma$ are satisfied as $w$.

4. It is convenient to extend the valuation $V$ to arbitrary formulae by setting $V(\phi) := \{w \mid \mathfrak{M},w \models \phi\}$. We then get useful interpretations such as: if for all worlds $v$, $\mathfrak{M},v \models  (\phi\to \psi)$ and _modus ponens_ holds, (i.e. we have a logic as such in which we are working,  then $V(\phi)\subseteq V(\psi)$. (This comment is deliberately slightly vague, but indicates a common type of argument that will be behind many results, where more precision is available, so think of it as a template for a result rather than a result as such.)

### What is a valuation?

In the definition of a Kripke model the valuation is all important.  It is what puts meaning onto the frame. It is very useful to push this idea around through a couple of adjunctions and reinterpretations. The process is fairly intuitive, but it pays to do things reasonably formally:

1. Note that the power set $\mathcal{P}(W)$ can also be written as $\mathbf{2}^W$ the set of functions from $W$ to $\mathbf{2}$, where we write $\mathbf{2}$ for the 'set' of classical [[truth values]], $\{\bot \to \top\}$, or $\{true,false\}$, or $\{0,1\}$ or ... .   We wrote 'set' in inverted commas for several reasons:

   * Firstly an important role here is played by the multiple structures that $\mathbf{2}$ has. It is a [[Heyting algebra]]; it has a natural [[poset]] structure; it is the [[subobject classifier]] in the [[topos]] of sets, and so on. In fact it is the source of most of the logic semantic structure within this context.  Its structure induces similar structures on the powerset, $\mathcal{P}(W)$, given by union etc, that made the semantics work above.  (See also the discussion on [[dualizing object|ambimorphic objects]] in the entry on the [[Chu construction]].)

   * Next note that although we said 'set' we could do a lot of this in other settings.  For instance we could work within a more general topos with an object of possible worlds and an object of propositions. We would need the extra [[Heyting algebra]] structure on what would there probably be written as $\Omega$, and our negation interpretation would be more subtle.

   * Finally we could categorify things.  For the moment we will leave this aside, but note the discussion at [[truth value]].

2.  For convenience we will write $P$ for $Prop$, then a valuation is $V: P\to \mathbf{2}^W$, and using  co[[currying]] this corresponds to $V: P\times W\to \mathbf{2}$.  That, of course, corresponds to a subset of $P\times W$.  That subset consists of all pairs, $(p,w)$, for which $w\in V(p)$ so  interprets as the set of pairs in which the proposition $p$ is 'true' in world $w$.

3.  We could re[[currying|curry]] after transposing $P$ and $W$, to get $V$ to correspond to $\tilde{V}: W \to \mathcal{P}(P)$, so that $\tilde{V}(w)$ is the set of propositions true about the world $w$.

4.  Another useful direction is to see this as giving a binary [[Chu construction|Chu space]]. (To be investigated later.)

## Models in Multimodal Logics

Again we look at the basic model language this time with $n$ unary modal operators.

+-- {: .un_defn}
###### Definition

A _model_ for $\mathcal{L}_\omega(n)$ is a pair $\mathfrak{M} = (\mathfrak{F},V)$, where $\mathfrak{F}$ is a frame (for $\mathcal{L}_\omega(n)$) and $V: Prop \to \mathcal{P}(W)$, is a function, called a _valuation_, assigning to each atomic proposition $p$ a subset of the set, $W$, of worlds.
=--

## Satisfaction in the multimodal case

This is virtually the same as in the monomodal case, except for the last case which involves the $n$ modal operators. (We give it in full repeating the earlier cases for convenience.)

+-- {: .un_defn}
###### Definition

Suppose that $w$ is a state in a model $\mathfrak{M} = (W,R,V)$.  We inductively define the notion of a formula $\phi$ being satisfied in the model at state $w$, as follows:

* for $p\in Prop$, $\mathfrak{M},w \models p$ if and only if $w\in V(p)$;

* $\mathfrak{M},w \models \bot$ never;

* $\mathfrak{M},w \models \neg \phi$ if and only if it is not true that $\mathfrak{M},w \models \phi$;

* $\mathfrak{M},w \models \phi \vee \psi$ if and only if $\mathfrak{M},w \models\phi$ or $\mathfrak{M},w \models\psi$;

* for each $i = 1, \ldots, n$, $\mathfrak{M},w \models \Diamond_i \phi$ if and only if, for some $v \in W$ such that $ R_i w v$, $\mathfrak{M},v \models \phi$.
=--

Satisfaction on a class of models  is related to [[validity]] of modal formulae. In this way a class of frames can determine a logic and then the logic determines a class of frames.  The axiomatisation of that class/logic is then an interesting challenge.

## Related concepts

* [[modal type theory]]

* [[Kripke frame]], [[Kripke semantics]]

* [[algebraic model for modal logic]]

## References

The concept originates with:

* {#Kripke59} [[Saul A. Kripke]], *A Completeness Theorem in Modal Logic*, The Journal of Symbolic Logic **24** 1 (1959) 1-14 [[doi:10.2307/2964568](https://doi.org/10.2307/2964568), [jstor:2964568](https://www.jstor.org/stable/2964568), [pdf](https://www.filosoficas.unam.mx/~morado/Cursos/17Modal/Kripke1959.pdf)]

* {#Kripke63} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi*, Mathematical Logic Quaterly **9** 5-6 (1963) 67-96 [[doi:10.1002/malq.19630090502](https://doi.org/10.1002/malq.19630090502)]

* [[Saul A. Kripke]], *Semantical Considerations on Modal Logic*, Acta Philosophical Fennica **16** (1963) 83-94 [[pdf](http://saulkripkecenter.org/wp-content/uploads/2019/03/Semantical-Considerations-on-Modal-Logic-PUBLIC.pdf)]

* {#Kripke65} [[Saul A. Kripke]], *Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi*, in *The Theory of Models* (Proceedings of the 1963 International Symposium at Berkeley) Studies in Logic and the Foundations of Mathematics (1965) 206-220 [[doi:10.1016/B978-0-7204-2233-7.50026-5](https://doi.org/10.1016/B978-0-7204-2233-7.50026-5)]

Textbook accounts:

* {#BlackburnDeRijkeVenema01} [[Patrick Blackburn]], [[Maarten de Rijke]], [[Yde Venema]], *Modal Logic*, Cambridge Tracts in Theoretical Computer Science **53**, Cambridge University Press  (2001) [[doi:10.1017/CBO9781107050884](https://doi.org/10.1017/CBO9781107050884)]

* [[Valentin Goranko]], [[Martin Otto]], §1 in: *Model Theory of Modal Logic*, in Section 5 in: *Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 249-329 [[pdf](https://www2.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf),  [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/), [publisher page](https://www.sciencedirect.com/bookseries/studies-in-logic-and-practical-reasoning/vol/3/suppl/C)]

Generalization beyond [[Kripke frames]]:

* [[Dov Samet]], §6 in: *S5 knowledge without partitions*, Synthese **172** (2010) 145–155 [[doi:10.1007/s11229-009-9469-0](https://doi.org/10.1007/s11229-009-9469-0)]

[[!redirects geometric model for modal logic]]
[[!redirects geometric model for modal logics]]
[[!redirects geometric models for modal logic]]
[[!redirects geometric models for modal logics]]

[[!redirects Kripke semantics]]

[[!redirects geometric model of modal logic]]
[[!redirects geometric models of modal logic]]


Geometric Models for Elliptic Cohomology

> this is a sub-entry of [[A Survey of Elliptic Cohomology]], see there for background and context.

**definition**

$d$-dimensional Riemanninan field theories are symmetric monoidal functors $RCob_d \to TopVect$ from $d$-dimensional Riemannian bordisms to [[topological vector space]]s.

A field theory is very similar to a [[representation]] of a group. Only where a representation of a [[group]] $G$ is a functor from the [[delooping]] $\mathbf{B}G = {*}//G$ of $G$ to [[Vect]], an [[FQFT]] is a representation of a more complicated domain category.

**how does [[topology]] enter?**

for $X$ some [[topological space]] there is also a [[symmetric monoidal category]]

$$
  RCob_d(X)
$$

of Riemannian bordisms equipped with a continuous map to $X$.

Notice that $d RRCob_d(X)$ does depend covariantly on $X$. This means that $Fun^\otimes(RCob_d(X), TopVect)$ is contravariant in $X$.

When special structure is around, however, we also have a push-forward of such functors along morphisms.

**Example: push-forward to the point**: for $X$ as above and $X \to {*}$ the unique map to the [[point]] heuristically we want a map

$$
  d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
$$

notice that this push-forward is not an [[adjoint functor]]. Instead, it is a map that comes from integration over fibers. In particular it will change the degree of [[cohomology theory|cohomology theories]].

heuristically the pushforward

$$
    d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
$$

acts on field theories $E_X$ over $X$

$$
  E_X \mapsto E
$$

by the assignment

$$
  E(Y^{d-1}) \mapsto
  \Gamma\left(
  \array{
     E_X(Y)
     \\
     \downarrow
     \\
     Maps(Y,X)
  }
  regarded as a vector bundle
  \right)
$$

for instance when $E_X(Y) = \mathbb{C}$ then $E(Y) = \Gamma(Maps(Y,X);\mathbb{C})$. This is clearly reminiscent of the pushforward of a sheaf along a continuous functions and suggests that $d RFT(X)$ should be looked at as a sheaf on $X$. It is however not so, since if $X=X_1\cup X_2$ then an object $Y$ in $RCob_d(X)$ (i.e., a $(d-1)$-dimensional closed manifold with a map to $X$) cannot in general be reconstructed from $RCob_d(X_1)$ and $RCob_d(X_2)$. On the other hand, such a reconstruction is possible if one allows objects in $RCob_d(X_i)$ to have $(d-2)$-dimensional boundaries. This point of view leads to [[extended topological quantum field theory]].

[[!redirects Geometric models for elliptic cohomology -- Axiomatic field theories and their motivation from topol]]

category: reference


Geometric Models for Elliptic Cohomology > history

< [[Geometric Models for Elliptic Cohomology]]


geometric morphism

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###Context###
#### Topos Theory
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[[!include topos theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

For $X$ and $Y$ [[topological spaces]], a [[continuous function]] $X \to Y$ induces (in particular) two [[functors]]

* the [[direct image]] $f_* : Sh(X) \to Sh(Y)$

* the [[inverse image]] $f^* : Sh(Y) \to Sh(X)$

between the corresponding  [[Grothendieck topos|Grothendieck topoi]] of [[sheaf|sheaves]] on $X$ and $Y$. These are such that:

* $f^*$ is [[adjoint functor|left adjoint]] to $f_*$, so $f^*$ preserves all small [[colimits]] and $f_*$ preserves all small [[limits]].

* furthermore, $f^*$ is [[exact functor|left exact]] in that it preserves finite [[limits]].

Morever, if $X$ and $Y$ are
[[sober space|sober]] [[topological spaces]]
every pair of functors with these properties comes uniquely from a continuous map $X \to Y$ (see the theorem below).

A _geometric morphism_ between arbitrary [[topos|topoi]] is the direct generalization of this situation.

Another motivation of the concept comes from the fact that a [[functor]] such as $f^*$ that preserves finite [[limits]] and arbitrary [[colimits]] (since it is a [[adjoint functor|left adjoint]]) necessarily preserves all constructions in [[geometric logic]].  See also [[classifying topos]].

## Definition

+-- {: .num_defn}
###### Definition

If $E$ and $F$ are [[topos|toposes]], a **geometric morphism** $f:E\to F$ consists of a pair of [[adjoint functors]] $(f^*,f_*)$

$$
  f_* : E \to F
$$
$$
  E \leftarrow F : f^*
  \,,
$$

such that the [[left adjoint]] $f^*:F \to E$ preserves [[finite limits]].

We say that

* $f_*$ is the [[direct image]]

* $f^*$ is the [[inverse image]]

of the geometric morphism.

=--

If moreover the [[inverse image]] $f^*$ has also a [[left adjoint]] $f_! : E \to F$, then $f$ is an [[essential geometric morphism]].

+-- {: .num_remark}
###### Remark

Since [[Grothendieck topos|Grothendieck toposes]] satisfy the (dual) hypotheses of Freyd's special [[adjoint functor theorem]], any functor $f^*$ between Grothendieck toposes which preserves all small [[colimit]]s must have a [[right adjoint]].  Therefore, a geometric morphism $f : E \to F$ between Grothendieck toposes could equivalently be defined as a functor $E \leftarrow F : f^* $ preserving [[finite limits]] and all [[small set|small]] [[colimits]].

=--

+-- {: .num_remark}
###### Remark

In view of its definition in terms of a pair of [[adjoint functor]]s, the direction of a geometric morphism is a convention. However, with the other convention it would better be called an **algebraic morphism**.

See [[Isbell duality]] for more on this [[duality]] between [[algebra]] and [[geometry]].

=--

See also ([Johnstone, p. 162/163](#Johnstone)).

## Properties

We discuss some general properties of geometric morphisms. The

* [Relation to homomorphisms of locales](#RelationToHomomorphismOfLocales)

also serves as a motivation or justification of the notion of geometric morphism. The

* [Relation to homomorphisms of sites](#RelationToMorphismsOfSites)

is a fairly straightforward generalization of that situation, reflecting the passage from (sheaf-) [[(0,1)-topos]]es to general [[topos|(1,1)-toposes]].

A somewhat subtle point about geometric morphisms of toposes is that there is also another sensible notion of topos [[homomorphism]]s: [[logical morphism]]s. In

* [Relation to logical morphism](#RelationToLogicalMorphisms)

aspects of the relation between the two concepts are discussed.

The reader wishing to learn about geometric morphisms systematically might want to first read the section on _[Geometric morphisms between presheaf toposes](#BetweenPresheafToposes)_ below, as much of the following discussion makes use of a few basic facts discussed there.

### Relation to homomorphisms of locales
 {#RelationToHomomorphismOfLocales}

The definition of geometric morphisms may be motivated as being a [[categorification]] of the definition of morphisms of [[locale]]s.

Recall that

+-- {: .num_defn #LocaleHomomorphisms}
###### Definition

A [[homomorphism]] of [[locale]]s

$$
  f : X \to Y
$$

is dually a morphism of [[frame]]s (the "[[frame of opens|frames of open subsets]]" of $X$ and $Y$, respectively)

$$
  \mathcal{O}(X) \leftarrow \mathcal{O}(Y) : f^*
  \,.
$$

This, in turn, is a [[functor]] (of [[posets]]) that

1. preserves finite [[limit]]s (called [[meet]]s in this context);

1. preserves arbitrary (small) [[colimit]]s (called [[join]]s in this context).

=--

Such a preservation of finite limits and arbitrary colimits is precisely what characterizes the [[inverse image]] part of a geometric morphism, and hence by the [[adjoint functor theorem]] already characterizes the full notion of geometric morphisms. Since a [[locale]] may equivalently be thought of as a [[(0,1)-topos]], this means that geometric morphisms are direct generalization of the notion of locale homorphisms to 1-toposes.

The following says this in more precise fashion.

+-- {: .num_defn #DirectImageInducedByLocaleMorphism}
###### Definition

For $f : X \to Y$ a homomorphism of [[locale]]s, let

$$
  f_* : Sh(X) \to Sh(Y)
$$

be the functor between their [[sheaf topos]]es that sends a sheaf $F : \mathcal{O}(X)^{op} \to Set$ to the composite

$$
  f_* F : \mathcal{O}(Y)^{op} \stackrel{f^*}{\longrightarrow}
    \mathcal{O}(X)^{op} \stackrel{F}{\longrightarrow}
  Set
  \,,
$$

where $f^*$ is the corresponding frame morphism as in def. \ref{LocaleHomomorphisms}.

=--

+-- {: .num_prop}
###### Proposition

The functor $f_*$ in def. \ref{DirectImageInducedByLocaleMorphism} is the [[direct image]] part of a geometric morphism of sheaf toposes

$$
  (f^* \dashv f_*) : Sh(X) \stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}}
   Sh(Y)
  \,.
$$

Moreover, the corresponding [[inverse image]] functor $f^*$ does restrict on [[representable functor|representable]]s to the frame morphism that we also denoted $f^*$.

=--

In ([Johnstone](#Johnstone)) this appears as lemma C1.4.1 and theorem C1.4.3.

+-- {: .proof}
###### Proof

Since a morphism of [[frame]]s is a morphism of [[site]]s, as discussed there, this follows from the corresponding propositions in the section [Morphisms of sites and geometric morphisms](site#MorphismsOfSitesAndGeometricMorphisms).

=--

+-- {: .num_prop}
###### Proposition

The construction $X \mapsto Sh(X)$ extends to a [[2-functor]]

$$
  Sh : Locale \hookrightarrow Topos
$$

from the [[category]] [[Locale]] of [[locales]] to the [[2-category]] [[Topos]] of [[topos]]es and geometric morphisms between them

=--

See also at [[locale]] the section [relation to toposes](locale#RelationToToposes).

### Relation to morphisms of sites
 {#RelationToMorphismsOfSites}

See at [[morphism of sites]] the section _[Relation to geometric morphisms](morphism+of+sites#RelationToGeometricMorphisms)_.

### Relation to logical morphisms
 {#RelationToLogicalMorphisms}

+-- {: .num_prop}
###### Proposition

Every geometric morphism whose [[direct image]] is a [[logical morphism]] is an [[equivalence of categories|equivalence]].

=--

This is a restatement of [this proposition](logical+functor#LogicalMorphismsRightAdjointToCartesianFunctors) at [[logical morphism]]. See there for a proof.

But [[inverse images]] can be nontrivial logical morphisms:

+-- {: .num_prop}
###### Proposition

 The [[inverse image]] of an [[etale geometric morphism]] is a [[logical morphism]].

=--

Generally, a geometric morphism with logical inverse image is called an
[[atomic geometric morphism]]. See there for more details.

### Structure preserved by geometric morphisms
 {#StructurePreserved}

The [[inverse image]]s of geometric morphisms preserves the structure of toposes in the sense of their characterization as _categories with [[finite limit]]s that are [well-powered](indexed+category#WellPoweredness) [[indexed categories]] with respect to the canonical indexing over themselves.

This appears in ([Johnstone](#Johnstone)) as remark B2.2.7 based on example B1.3.17 and prop. B1.3.14. See at [[indexed category]] the section [Well-poweredness](indexed+category#WellPoweredness),

### Surjection/embedding factorization

Every geometric morphism factors, essentially uniquely, as a [[geometric surjection]] followed by a [[geometric embedding]]. See _[[geometric surjection/embedding factorization]]_ for more on this.

## Special classes of geometric morphisms

There are various special cases and types of classes of geometric morphisms. For instance

* [[global section|global sections]]

* [[geometric embedding]]

* [[locally connected geometric morphism]]

  * [[essential geometric morphism]]

  * [[connected geometric morphism]]

* [[etale geometric morphism]]

* [[open geometric morphism]], [[closed geometric morphism]]

* [[proper geometric morphism]]

* [[local geometric morphism]]

* [[bounded geometric morphism]]

* [[base change]]

* [[localic geometric morphism]]

* [[hyperconnected geometric morphism]]

* [[cohesive topos|cohesive morphism]]

* [[atomic geometric morphism]]

* [[dominant geometric morphism]]

The following subsections describe some of these in more detail.

### Between presheaf toposes
 {#BetweenPresheafToposes}

Let $C$ and $D$ be any two [[categories]]. We write $C^{op}$ and $D^{op}$ for their [[opposite categories]] and $[C, Set]$, $[D, Set]$ for the corresponding [[presheaf toposes]] over $C^{op}$ and $D^{op}$, respectively.

+-- {: .num_prop }
###### Proposition

Every [[functor]] $f : C \to D$ induces an ([[essential geometric morphism|essential]], even) geometric morphism

$$
  f := (f^* \dashv f_*)
  :
  [C,Set]
    \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}}}
  [D, Set]
  \,,
$$

where $f^* = (-) \circ f$ is the functor given by precomposition presheaves with $f$.

Moreover, for $\eta : f \Rightarrow g : C \to D$ a [[natural transformation]] between two such functors there is an induced [[geometric transformation]] $(f^* \dashv f_*) \Rightarrow (g^* \dashv g_*)$. This is compatible with composition in that it makes forming [[presheaf topos]]es a [[2-functor]]

$$
  [-,Set] : Cat \to Topos
$$

from the [[2-category]] [[Cat]] to the [[2-category]] [[Topos]].

=--

This appears as ([Johnstone, example A4.1.4](#Johnstone)).

+-- {: .proof}
###### Proof

Since [[categories of presheaves]] have all [[limit]]s and [[colimit]]s, the left and right [[Kan extension]]s $Lan_f$ and $Ran_f$  along $f$ exists, and form with $f^*$ an [[adjoint triple]]

$$
  [C,Set]
    \stackrel{\overset{Lan_f}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{Ran_f}{\longrightarrow}}}
  [D, Set]
  \,.
$$

Hence $f_! \simeq Lan_f$ and $f_* \simeq Ran_f$. Notice that [[left adjoint]]s and [[right adjoint]]s to a functor are, if they exist, unique up to unique [[isomorphism]].

=--

Next we consider extra [[property]] on $C$, $D$ and $f$ such that $f^*$ induces also a second geometric morphism, going the other way round. This plays a role for the discussion of [morphisms of sites](#RelationToMorphismsOfSites). For that reason we pass now from $C$ and $D$ to their [[opposite categories]] hence consider genuine [[presheaves]] on $C$ and $D$.

+-- {: .num_prop }
###### Proposition

Let $C$ and $D$ be categories with [[finite limit]]s and let $f : C \to D$ be a finite-limit [[preserved limit|preserving]] functor.

Then in the [[adjoint triple]]

$$
  (f_! \dashv f^* \dashv f_*)
  :
  [C^{op},Set]
    \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}}}
  [D^{op}, Set]
$$

the left [[Kan extension]] $f_!$ also preserves finite limits and hence in this case $f^*$ is also the [[direct image]] of a geometric morphism going the other way round:

$$
  (f_! \dashv f^* ) : [D^{op},Set] \to [C^{op}, Set]
  \,.
$$

=--

This appears as ([Johnstone, example A4.1.10](#Johnstone)).

+-- {: .proof}
###### Proof

Recall that for $F : C^{op} \to Set$ a functor, the left [[Kan extension]] $f_! F : D^{op} \to Set$ is computed over each object $d \in D$ by the [[colimit]]

$$
  (f_! F)(d) =
  \lim_\to
  \left(
     (d/f)^{op} \stackrel{U}{\to} C^{op} \stackrel{F}{\to} Set
  \right)
$$

where $(d/f)$ is the [[comma category]] and

$$
  U : (d/f) \to C
$$

is the evident forgetful functor. This is natural in $F$ and so $(f_! -)(d)$ is the functor

$$
  (f_! -)(d) : [C^{op}, Set] \stackrel{U^*}{\to}
   [(d/f)^{op}, Set]
   \stackrel{\lim_\to}{\to}
   Set
  \,.
$$

By the above argument $U^*$ has a [[left adjoint]] (the left [[Kan extension]] along $U$) hence itself preserves all [[limit]]s.

It then suffices to observe (see below) that by the fact that $f$ preserves finite limits we have that the categories $(d/f)^{op}$ are [[filtered categories]]. Then by the fact (see there) that [[filtered colimit]]s commute with finite limits, it follows that also $\lim_\to$ preserves finite limits, and hence $(f_! -)(d)$ does. Since colimits of presheaves are computed objectwise, this shows that $f_!$ preserves finite limits. This completes the proof.

Here is an explicit desciption of the filteredness of the [[comma category]] $(d/f)^{op}$ for any object $f$.

We check the axioms on a [[filtered category]]:

* _non-emptiness_ : There is an object in $(d/f)^{op}$: since $f$ by assumption preserves the [[terminal object]],  take the terminal morphism $(d \to f(*) = *)$;

* _connectedness_ : for any two objects $(d \stackrel{h_1}{\to} f(c_1))$ and $(d \stackrel{h_2}{\to} f(c_2))$ form the [[product]] $c_1 \times c_2$ and use that $f$ preserves this to produce the object $(d \stackrel{f(h_1), f(h_2)}{\to} f(c_1) \times f(c_2) \simeq f(c_1 \times c_2))$. Then the image under $f$ of the two projections provides the required [[span]]

  $$
    \array{
      && d
      \\
      & {}^{\mathllap{f(h_1)}}\swarrow & \downarrow^{(f(h_1),f(h_2))} & \searrow^{\mathrlap{f(h_2)}}
      \\
      f(c_1) &\stackrel{f(p_1)}{\leftarrow}&
      f(c_1 \times c_2)
      &
      \stackrel{f(p_2)}{\to}
      & f(c_2)
    }
    \,.
  $$

* finally, for

  $$
    \array{
       && d
       \\
       & \swarrow && \searrow
       \\
       f(c_1) && \stackrel{\overset{h_1}{\to}}{\underset{h_2}{\to}}& & f(c_2)
    }
  $$

  two [[parallel morphism]], let $eq(h_1,h_2)$ be the [[equalizer]] of the underlying morphism in $C$. Since $f$ preserves equalizers we have an object $(d \to f(eq(h_1,h_2)))$ and a morphism to $(d \to f(c_1))$ that equalizes the above two morphisms.

=--

### Surjections and embeddings

A geometric morphism $f : E \to F$ is a **surjection** if $f^*$ is [[faithful functor|faithful]].  It is an **[[geometric embedding|embedding]]** if $f_*$ is [[full and faithful functor|fully faithful]].

+--{: .un_prop}
###### Proposition
Up to equivalence, every [[geometric embedding|embedding]] of toposes is of the form
$$
  Sh_j(E) \to E
  \,,
$$
where $Sh_j(E)$ is the topos of [[sheaf|sheaves]] with respect to a [[Lawvere-Tierney topology]] $j : \Omega \to \Omega$ on $E$.
=--

This means in particular that fully faithful geometric morphisms into [[Grothendieck topos|Grothendieck topoi]] are an equivalent way of encoding a [[Grothendieck topology]].

+--{: .un_prop}
###### Proposition
Up to equivalence, every surjection of topoi is of the form
$$ E \to E_G $$
where $E_G$ is the category of coalgebras for a finite-limit-preserving [[comonad]] on $E$.
=--

Every geometric morphism $f:E\to F$ factors, uniquely up to equivalence, as a surjection followed by an embedding.  There are two ways to produce this factorization: either construct $E_G$ where $G= f^*f_*$ is the comonad induced by the adjunction $f^*\dashv f_*$, or construct $Sh_j(F)$ where $j$ is the smallest Lawvere-Tierney topology on $F$ such that $f$ factors through $Sh_j(F)$.  In fact, surjections and embeddings form a 2-categorical [[orthogonal factorization system]] on the 2-category of topoi.

### Global sections and constant sheaves

For every [[Grothendieck topos]] $E$, there is a geometric morphism

$$
  \Gamma \;\colon\; E  \stackrel{\leftarrow}{\to} Set : const
$$

called the [[global sections]] functor or *[[terminal geometric morphism]]*. It is given by the [[hom-set]] out of the [[terminal object]]

$$
  \Gamma(-) = Hom_E({*}, -)
$$

and hence assigns to each object $A\in E$ its set of [[global element]]s $\Gamma(A) = Hom_E(*,A)$. If we think of $A$ as a [[sheaf]], then $\Gamma(A)$ is the set of **global sections**.

The [[left adjoint]] $const \;\colon\; Set \to E$ of the global section functor is the canonical [[Set]]-[[copower|tensoring]] functor

$$
  \otimes \;\colon\; Set \times E \to E
$$

applied to the [[terminal object]]

$$
  const = (-)\otimes {*} : Set \to E
$$

which sends a set $S$ to the [[coproduct]] of $|S|$ copies of the terminal object

$$
  S \otimes {*} = \coprod_{s \in S} {*}
  \,.
$$

This is called the **constant object** of $E$ on the set $S$. Notably when $E$ is a [[Grothendieck topos|sheaf topos]] this is the **[[constant sheaf]]** on $S$.

The [[left adjoint]]ness is just the defining property of the [[copower|tensoring]]

$$
  Hom_E(const S, A) \simeq Hom_E(S \otimes {*},A)
  \simeq
  Hom_{Set}(S, Hom_E(*,A))
  \,.
$$

This left adjoint preserves [[product]]s, using that colimits in a  topos are stable by base change (see [[commutativity of limits and colimits]])

$$
   \left( \coprod_{s_1 \in S_1} *\right)
   \times
   \left( \coprod_{s_2 \in S_2} *\right)
   =
   \coprod_{s_1 \in S_1}  \left(* \times \left( \coprod_{s_2 \in S_2} *\right)\right)
   =
   \coprod_{s_1 \in S_1}  \left( \coprod_{s_2 \in S_2} *\right)
   =
   \coprod_{s_1 \in S_1} \coprod_{s_2 \in S_2} *
   =
   \coprod_{s \in S_1 \times S_2} *
$$

and it preserves [[equalizer]]s and therefore [[limit]]s. So it is left exact and we do have a geometric morphism.

### Point of a topos

For $E$ a topos, a geometric morphism

$$
  x : Set \to E
$$

is called a [[point of a topos]].

### Change-of-base

For $E$ any [[topos]] and $k : B \to A$ any morphism in $E$ there is the [[base change|change-of-base]] functor of [[over category|over categories]]
$$
  k^* : (E/A) \to (E/B)
$$
by [[pullback]]. As described at [[dependent product]] this functor has both a [[left adjoint]] $\coprod_k : E/B \to E/A$ as well as a [[right adjoint]] $\prod_k : E/B \to E/A$.  Therefore
$$
  (\Pi_k, k^*) : E/B \leftrightarrow E/B
$$

is a geometric morphism. Hence $(\coprod_k \dashv k^* \dashv \prod_k)$ is an [[essential geometric morphism]].

### Sheafification

A [[category of sheaves]] is a [[geometric embedding]] into a presheaf topos

$$
  Sh(C) \hookrightarrow PSh(C)
  \,.
$$

### Geometric morphisms of sheaf topoi
 {#sheaftopoi}

Geometric morphisms between [[localic topoi]] are equivalent to [[continuous maps]] of [[locales]], which in turn are equivalent to continuous maps of [[topological spaces]] if you restrict to [[sober spaces]].

Unrolling this: For $X$ a [[topological space]], write $Sh(X) := Sh(Op(X))$ as usual for the [[topos]] given by the [[category of sheaves]] on the [[category of open subsets]] $Op(X)$ with the standard [[coverage]]

+-- {: .un_lemma}
###### Lemma

For every continuous map $f : X \to Y$ of [[sober space|sober]] [[topological spaces]] with the induced [[functor]] $f^{-1} : Op(Y) \to Op(X)$ of [[sites]], the [[direct image]]

$$
  f_* : Sh(X) \to Sh(Y)
$$

and the [[inverse image]]

$$
  f^* : Sh(Y) \to Sh(X)
$$

constitute a geometric morphism

$$
   f : Sh(X) \to Sh(Y)
$$

(denoted by the same symbol, by convenient abuse of notation).

This map $Hom_{Top}(X,Y) \to GeomMor(Sh(X),Sh(Y))$ is an bijection of sets.

=--

+-- {: .proof}
###### Proof

That the induced pair $(f^*, f_*)$ forms a geometric morphism is (or should eventually be) discussed at [[inverse image]].

We now show that every geometric morphism of sheaf toposes arises this way from a continuous function, at least up to isomorphism.  (In fact, more is true: the category of geometric morphisms $Sh(X)\to Sh(Y)$ is equivalent to the poset of continuous functons $X\to Y$ with the [[specialization ordering]] ([[Elephant]], Proposition C.1.4.5.).  We follow [MacLane-Moerdijk, page 348](#MacLaneMoerdijk).

One reconstructs the continuous map $f : X \to Y$ from a geometric morphism $f : Sh(X) \to Sh(Y)$ as follows.

Write ${*} = Y \in Sh(Y)$ for the sheaf on $Op(Y)$ constant on the singleton set, the [[terminal object]] in $Sh(Y)$.

Notice that since the [[inverse image]] $f^*$ preserves finite [[limits]], every [[subobject]] $U_Y \hookrightarrow {*}$ is taken by $f^*$ to a subobject $U_X \hookrightarrow X$, obtained by applying $f^*$ to the [[pullback]] diagram

$$
  \array{
    U_Y &\to& {*} = Y
    \\
    \downarrow && \downarrow
    \\
    {*} = Y &\to& \Omega
  }
$$

that characterizes the subobject $U_Y$ in the [[topos]].

But, as the notation already suggests, the subobjects of $X,Y$ are just the open sets, i.e. the representable sheaves.

This yields a _function_ $f^* : Obj(Op(Y)) \to Obj(Op(X))$ from open subsets to open subsets.  By assumption, this preserves finite limits and arbitrary colimits, i.e. finite intersections and arbitrary unions of open sets.  In other words, it is a [[frame|frame homomorphism]], and thus can be regarded as a morphism $X\to Y$ of [[locales]].

We can now use this to define a function $\bar f : X \to Y$ of the sets underlying the topological spaces $X$ and $Y$ by setting

$$
  (\bar f(x) = y)
  \Leftrightarrow
  \forall V \ni y: x \in f^*(V)
  \,.
$$

This yields a well defined function for the following reasons (which for the moment we spell out in the case where $Y$ is [[Hausdorff space|Hausdorff]], although the result should hold ---and furthermore, hold [[constructive mathematics|constructively]]--- whenever $Y$ is sober):

* there is at most one $y$ satisfying this equation: if $y_1 \neq y_2$ both satisfy it, there are, by assumption of $Y$ being [[Hausdorff space|Hausdorff]], neighbourhoods $V_1 \ni y_1$ and $V_2 \ni y_2$ such that (using that $f^*$ preserves limits hence intersections) $f^*(V_1) \cap f^*(V_2) = f^*(V_1 \cap V_2) = \emptyset$, which contradicts the assumption.

* there is at least one $y$ satisfying this equation: again by contradiction: if there were none then every $y \in Y$ has a neighbourhood $V_y$ with $x \not\in f^*(V_y)$, so that similarly to above we conclude with $x \not\in \cup_{y \in Y} f^*(V_y) = f^*(\cup_y V_y) = f^*(Y) = X$ again a contradiction.

+-- {: .query}
Am I right that what we are really need of our space here is not necessarily that it be Hausdorff but simply that it be [[sober space|sober]]?  (Then the nonconstructive aspects of the argument ---which is what made me look at this--- come in only because the theorem that a Hausdorff space must be sober is not constructively valid.)  ---Toby

[[Mike Shulman]]: Yes, that's exactly right.  All the complication defining $\bar f$ above is just an unrolled way of saying that geometric morphisms between [[locale|localic]] topoi are equivalent to continuous maps of locales, which are equivalent to continuous functions if you have sober spaces.  I think that should be clarified.

_Toby_:  OK, I added a paragraph at the beginning of the example to clarify this.  I still need to rewrite the argument immediately above to apply to sober spaces.  (Everything else seems to go through exactly the same.)
=--

So our function $\bar f : X \to Y$ is well defined and satisfies $\bar f^{-1}(U_Y) = f^*(U_Y)$ for every open set $U_Y \in Obj(Op(Y))$. In particular it is therefore a continuous map.

It remains to check that this map reproduces the geometric morphism that we started with. For that we compute its [[direct image]] on any sheaf $A \in Sh(X)$ as

$$
  \begin{aligned}
    \bar f_*(A) : U_Y &\mapsto A(\bar f^{-1}(U_Y))
    \\
    & \simeq Hom_{Sh(X)}(\bar f^{-1}(U_Y),A)
    \\
    & = Hom_{Sh(X)}(f^* V, E)
    \\
    & \simeq Hom_{Sh(X)}(V, f_* E)
    \\
    & \simeq (f_* A)(U_Y)
  \end{aligned}
$$

=--

+-- {: .num_cor}
###### Corollary

The points $x \in X$ of the topological space $X$ are in canonical bijection with the points of $Sh(X)$ in the sense of [[point of a topos]].

=--

## Related concepts

* [[étale map]]

* [[logical morphism]]

* **geometric morphism**

  * [[geometric logic]], [[geometric theory]]

* [[(∞,1)-geometric morphism]]

* [[natural homotopy]]

## References

Geometric morphisms are the topic of section VII of

* [[Saunders MacLane]] and [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ .
 {#MacLaneMoerdijk}

Embeddings and surjections are discussed in section VII.4.

Geometric morphisms are defined in section A4 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

The special classes of geometric morphisms are discussed in section C3.

[[!redirects geometric morphisms]]


geometric nerve of a bicategory

[[!redirects Geometric nerve of a bicategory]]
[[!redirects Duskin nerve]]
[[!redirects Duskin nerves]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The (unitary) _geometric nerve_ is a natural  [[nerve]] operation on [[bicategories]]. It is a [[functor]] from [[BiCat]] to [[sSet]]. This is also sometimes called the _Duskin nerve_. The  notion is implicit in work by R. Street (1987). The direct approach was used by Duskin in work at about the same time, as explained in both articles. (Duskin's article directly on the idea was published in 2002.)

The construction, thus, yields a functor:

$$
  N : BiCat_{NLax} \to sSet
  \,.
$$

extending the ordinary nerve construction on the category of small categories, where morphisms of BiCat are _normal lax 2-functors_: these are the lax 2-functors which strictly preserve identities.

Special cases of the construction relate to earlier constructions relating to the [[homotopy coherent nerve]], see below for more detail.

## Definition

We may think of the [[simplex category]] $\Delta$ as the [[full subcategory]] of [[Cat]] on the categories free on non-empty finite linear graphs. This gives the canonical inclusion $\Delta \hookrightarrow Cat$ that defines the ordinary nerve of categories.

There is also the canonical embedding of categories into bicategories. Combined this gives the inclusion

$$
  \Delta \hookrightarrow Cat \hookrightarrow BiCat
  \,.
$$

The bicategorical nerve is the nerve induced from that. So for $C$ a bicategory we have

$$
  N(C) : [k] \mapsto BiCat_{NLax}(\Delta[k], C)
  \,.
$$

There are also an oplax version and two non-normalized versions.

## Properties

* The [[simplicial set]]s in the image of the geometric nerve are 3-[[coskeletal]].

* The geometric nerve identifies precisely [[bigroupoid]]s with 2-[[hypergroupoid]]s: those [[Kan complexes]] for which the horn fillers in dimension $\geq 3$ are _unique_ . (Theorem 8.6 in ([Duskin 2002](#Duskin02)))

(This shows in particular that bigroupoids model all [[homotopy 2-type]]s.)

* The nerve is a full and faithful functor $BiCat_{NLax}\to sSet$.

## Example

Any [[strict 2-category]] determines both a 'bicategory' in the above sense (since a 'strict' thing is also a 'weak' one)  and a simplicially enriched category.  The latter is found by taking the nerve of each 'hom-category'.  The Duskin nerve of a 2-category is the same as the [[homotopy coherent nerve]] of the corresponding $sSet$-category.  This can also be applied to [[2-groupoid]]s and, thus, results in a classifying space construction for [[crossed module]]s.

## Picturing the Duskin nerve
Following ([Johnson–Yau, Section 5.4](#JohnsonYau20)), one may picture the Duskin nerve $N(\mathcal{C})$ of a bicategory $(\mathcal{C},1^{\mathcal{C}},\circ_{\mathcal{C}},\alpha^{\mathcal{C}},\lambda^{\mathcal{C}},\rho^{\mathcal{C}})$ as follows:

1. The $0$-simplices of $N(\mathcal{C})$ are the objects of $\mathcal{C}$;

2. The $1$-simplices of $N(\mathcal{C})$ are the $1$-morphisms of $\mathcal{C}$;

3. The $2$-simplices of $N(\mathcal{C})$ are quadruples $(i,j,k,\theta)$ as in the diagram

   [[2-simplex-of-NC.svg:pic]]
   where $A,B,C\in\mathrm{Obj}(\mathcal{C})$, $i,j,k\in\mathrm{Mor}_1(\mathcal{C})$ and $\theta\colon j\circ i\Rightarrow k$ is a $2$-morphism of $\mathcal{C}$;
4. The $3$-simplices of $N(\mathcal{C})$ are $14$-tuples
   $$(A_{0},A_{1},A_{2},A_{3},f_{01},f_{02},f_{03},f_{12},f_{13},f_{23},\theta_{012},\theta_{013},\theta_{023},\theta_{123})$$
   as in the diagram
   [[3-simplex-of-NC.svg:pic]]
   such that we have an equality
   [[3-simplex-of-NC-equality-of-pasting-diagrams.svg:pic]]
   of pasting diagrams in $\mathcal{C}$;
5. The $n$-simplices of $N(\mathcal{C})$ consist of

   * A collection $\{A_{i}\}_{0\leq i\leq n}$ of objects of $\mathcal{C}$,
   * A collection $\{f_{ij}\colon A_{i}\longrightarrow A_{j}\}_{0\leq i\lt j\leq n}$ of $1$-morphisms of $\mathcal{C}$, and
   * A collection $\{\theta_{ijk}\colon f_{jk}\circ f_{ij}\Rightarrow f_{ik}\}_{0\leq i\lt j\lt k\leq n}$ of $2$-morphisms of $\mathcal{C}$

   such that, for each $i,j,k\in\mathbb{N}$ with $0\leq i\lt j\lt k\leq n$, we have an equality
   [[n-simplex-of-NC-equality-of-pasting-diagrams.svg:pic]]
   of pasting diagrams in $\mathcal{C}$;
6. The degeneracy map
   $$\mathrm{s}^{0}_{0}\colon N_{0}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C})$$
   of $N(\mathcal{C})$ in degree $0$ is the map sending a $0$-simplex $A$ of $N(\mathcal{C})$ (i.e. an object $A$ of $\mathcal{C}$) to the $1$-simplex $\mathrm{id}_{A}\colon A\to A$.
7. The degeneracy maps
   $$
   \mathrm{s}^{1}_{0} \colon N_{1}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{s}^{1}_{1} \colon N_{1}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
   $$
   of $N(\mathcal{C})$  in degree $1$ are the maps described as follows: given a $1$-simplex $\sigma=(A\xrightarrow{f}B)$ of $N(\mathcal{C})$, we have
   [[deg-1-degeneracies-of--duskin-nerve.svg:pic]]
8. The degeneracy maps in degree $2$
   $$
   \mathrm{s}^{2}_{0} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{s}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{s}^{2}_{2} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   of $N(\mathcal{C})$  in degree $2$ are the maps described as follows: given a $2$-simplex
   [[2-simplex-duskin-sigma.svg:pic]]
   of $N(\mathcal{C})$, we have
   [[deg-2-degeneracies-duskin-new.svg:pic]]
9. The face maps
   $$
   \mathrm{d}^{1}_{0} \colon N_{1}(\mathcal{C})\longrightarrow N_{0}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{1}_{1} \colon N_{1}(\mathcal{C})\longrightarrow N_{0}(\mathcal{C}),
   $$
   of $N(\mathcal{C})$ in degree $1$ are given by
   $$\mathrm{d}^{1}_{0}(A\xrightarrow{f}B)=B$$
   $$\mathrm{d}^{1}_{1}(A\xrightarrow{f}B)=A$$
10. The face maps
   $$
   \mathrm{d}^{2}_{0} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),
   $$
   of $N(\mathcal{C})$ in degree $2$ are described as follows: given a $2$-simplex
   [[2-simplex-duskin-sigma.svg:pic]]
   of $N(\mathcal{C})$, we have
   [[face-maps-deg-2-duskin.svg:pic]]
11. The face maps
   $$
   \mathrm{d}^{3}_{0} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{3}_{1} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{3}_{2} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{3}_{3} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
   $$
   of $N(\mathcal{C})$ in degree $3$ are described as follows: given a $3$-simplex
   [[3-simplex-duskin.svg:pic]]
   of $N(\mathcal{C})$, we have
   [[face-maps-deg-3-duskin.svg:pic]]
12. The face maps
   $$
   \mathrm{d}^{4}_{0} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{4}_{1} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{4}_{2} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{4}_{3} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{4}_{4} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
   $$
   of $N(\mathcal{C})$ in degree $4$ are described as follows: given a $4$-simplex $\sigma$ of $N(\mathcal{C})$ as in the diagram
   [[4-simplex-duskin.svg:pic]]
   we have
   \begin{imagefromfile}
       "file_name": "face-maps-deg-4-duskin-new2.svg"
   \end{imagefromfile}

## Related concepts

* [[nerve]]

* **geometric nerve of a bicategory**

* [[geometric nerve of a tricategory]]

* [[∞-nerve]]

* [[homotopy coherent nerve]]

## References

* [[Ross Street]], _The algebra of oriented simplexes_, Journal of Pure and Applied Algebra, Volume 49, Issue 3, December 1987, Pages 283–335

* {#Duskin02} [[John Duskin]], _Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories_, Theory and Applications of Categories **9** 10 (2002) 198–308 [[tac:9-10](http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html)]

* V. Blanco, M. Bullejos, E. Faro, _A Full and faithful Nerve for 2-categories_, Applied
Categorical Structures, Vol 13-3, 223-233, 2005. (See also [arxiv](http://arxiv.org/abs/math/0406615)).

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_ ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055)).

[[!redirects bicategory nerve]]
[[!redirects nerve of bicategories]]
[[!redirects nerves of bicategories]]

[[!redirects nerve of a bicategory]]


geometric nerve of a tricategory

[[!redirects Geometric nerve of a tricategory]]
[[!redirects Street nerve]]
[[!redirects Street nerves]]
[[!redirects tricategory nerve]]
[[!redirects nerve of tricategories]]
[[!redirects nerve of a tricategory]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _geometric nerve_ is a natural  [[nerve]] operation on [[tricategories]], associating to every [[tricategory]] $\mathcal{C}$ a [[simplicial set]] $\mathrm{N}^{\mathsf{S}}_{\bullet}(\mathcal{C})$.

Also called the _Street nerve_ of $\mathcal{C}$, this notion, like the [[Duskin nerve]] of a [[bicategory]], is implicit in  ([ [[Ross Street]]'s work on [[oriental]]s](#Street1987)). While this construction was announced in ([Duskin, 2002](#Duskin2002)) as to appear in a (then) forthcoming paper, the latter never appeared. Instead, the notion was developed by Cegarra–Heredia in ([Cegarra–Heredia, 2012](#CegarraHeredia2012)).

Roughly, the [[Street]] nerve of $\mathcal{C}$ may be thought of as the [[simplicial set]] $\mathrm{N}^{\mathsf{S}}_{\bullet}(\mathcal{C})$ whose $n$-simplices are [[lax functors]] from the locally discrete [[tricategory]] associated to $[n]$ to $\mathcal{C}$ satisfying a variety of unitality conditions. In detail, however, the definition is more involved, as one faces problems with the definition of [[lax functor]]s between [[tricategories]] found in ([Gordon–Powers–Street](#GPS1995)); see ([Definition 5.1.1 of Cegarra–Heredia](#CegarraHeredia2012)).

There are other nerve constructions for tricategories besides the Street nerve. One of them is the **Grothendieck nerve**, which to every [[tricategory]] $\mathcal{C}$ associates a "[[pseudosimplicial bicategory]]" $\mathrm{N}^\mathsf{G}_\bullet(\mathcal{C})\colon\Delta^\op\longrightarrow\mathsf{Bicats}$; meaning a pseudofunctor from the locally discrete tricategory associated to $\Delta^\op$ to the tricategory $\mathsf{Bicats}$. Its bicategory of $n$-simplices has as objects $n$-tuples of composable morphisms of $\mathcal{C}$.

One also has the **Segal nerve** $\mathrm{N}^\mathsf{Segal}_\bullet(\mathcal{C})\colon\Delta^\op\longrightarrow\mathsf{Bicats}$ of $\mathcal{C}$, which is a simplicial bicategory, and a kind of "rectification" of $\mathrm{N}^\mathsf{G}_\bullet(\mathcal{C})$. When $\mathcal{C}$ is the locally discrete tricategory associated to a bicategory $\mathcal{B}$, the Segal nerve of $\mathcal{C}$ agrees with the $2$-nerve of $\mathcal{B}$ introduced in ([Lack–Paoli, 2006](#LackPaoli2006)).

All of these nerve constructions are equivalent in the sense that their classifying spaces are homotopy equivalent to each other. For more details, see ([Cegarra–Heredia, 2012](#CegarraHeredia2012)).

## Properties

* The Street nerve identifies precisely nerves of [[trigroupoid]]s with $3$-[[hypergroupoid]]s: those [[Kan complexes]] for which the horn fillers in dimension $\geq 4$ are _unique_; see ([Carrasco, 2014](#Carrasco2014)).

## Picturing the Street nerve

Let ($\mathcal{C}$,$\mathsf{Hom}_{\mathcal{C}}(-,-)$,$\otimes$,$1^\mathcal{C}$,$\alpha$,$\alpha^{\bullet}$,$\phi$,$\phi^{\bullet}$,$\lambda$,$\lambda^{\bullet}$,$\eta$,$\eta^{\bullet}$,$\rho$,$\rho^{\bullet}$,$\epsilon$,$\epsilon^{\bullet}$,$\mathbf{\pi}$,$\mathbf{\lambda}$,$\mathbf{\mu}$,$\mathbf{\rho}$) be a tricategory, where

* ($\alpha$,$\alpha^{\bullet}$,$\phi$,$\phi^{\bullet}$) is the associator adjoint equivalence of $\mathcal{C}$,
* ($\lambda$,$\lambda^{\bullet}$,$\eta$,$\eta^{\bullet}$) is the left unitor adjoint equivalence of $\mathcal{C}$,
* ($\rho$,$\rho^{\bullet}$,$\epsilon$,$\epsilon^{\bullet}$) is the right unitor adjoint equivalence of $\mathcal{C}$, and
* $\mathbf{\pi}$,$\mathbf{\lambda}$,$\mathbf{\mu}$,$\mathbf{\rho}$ are the pentagonator, left, middle, and right $2$-unitors of $\mathcal{C}$.

(Below we also write these with "$^\mathcal{C}$" superscripts, and write $\mathbf{\pi}$,$\mathbf{\lambda}$,$\mathbf{\mu}$,$\mathbf{\rho}$ in blackboard bold font.)

The **Street nerve** of the tricategory $\mathcal{C}$ is then the simplicial set $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ where

1. The $0$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are the objects of $\mathcal{C}$;

2. The $1$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are the $1$-morphisms of $\mathcal{C}$;

3. The $2$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are quadruples $(i,j,k,\theta)$ as in the diagram

   [[2-simplex-of-NC.svg:pic]]
   where $A,B,C\in\mathrm{Obj}(\mathcal{C})$, $i,j,k\in\mathrm{Mor}_1(\mathcal{C})$ and $\theta\colon j\circ i\Rightarrow k$ is a $2$-morphism of $\mathcal{C}$;

4. The $3$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are $15$-tuples
   $$(A_{0},A_{1},A_{2},A_{3},f_{01},f_{02},f_{03},f_{12},f_{13},f_{23},\theta_{012},\theta_{013},\theta_{023},\theta_{123},\Gamma_{0123})$$

   as in the diagram
   [[3-simplex-of-the-street-nerve.svg:pic]]

5. The $4$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are $28$-tuples

   * ($A_{0}$,$A_{1}$,$A_{2}$,$A_{3}$,$f_{01}$,$f_{02}$,$f_{03}$,$f_{04}$,$f_{12}$,$f_{13}$,$f_{14}$,$f_{23}$,$f_{24}$,$f_{34}$,$\theta_{012}$,$\theta_{013}$,$\theta_{014}$,$\theta_{023}$, $\theta_{024}$,$\theta_{123}$,$\theta_{124}$,$\theta_{134}$,$\theta_{234}$,$\Gamma_{0123}$,$\Gamma_{0124}$,$\Gamma_{0134}$,$\Gamma_{0234}$,$\Gamma_{1234}$)

   with objects, $1$-morphisms, and $2$-morphisms as in the diagram
   [[4-simplex-duskin.svg:pic]]
   and $3$-morphisms as in the diagrams
   \begin{imagefromfile}
       "file_name": "4-simplex-street.svg",
       "width":     "800"
   \end{imagefromfile}
   such that the diagram
   \begin{imagefromfile}
       "file_name": "street-4th-oriental.svg",
       "width":     "800"
   \end{imagefromfile}
   corresponding to [[Street]]'s fourth [[oriental]], commutes. (See [ [here] ](https://ncatlab.org/nlab/files/street-nerve-fourth-oriental.pdf) for a zoomable PDF).

6. The $n$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$, similarly to the $4$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$, consist of

   * A collection $\{A_{i}\}_{0\leq i\leq n}$ of objects of $\mathcal{C}$,
   * A collection $\{f_{ij}\colon A_{i}\longrightarrow A_{j}\}_{0\leq i\lt j\leq n}$ of $1$-morphisms of $\mathcal{C}$,
   * A collection $\{\theta_{ijk}\colon f_{jk}\circ f_{ij}\Rightarrow f_{ik}\}_{0\leq i\lt j\lt k\leq n}$ of $2$-morphisms of $\mathcal{C}$, and
   * A collection $\{\Gamma_{ijkl}\}_{0\leq i\lt j\lt k\lt l\leq n}$ of $3$-morphisms of $\mathcal{C}$

   such that, for each $i,j,k,l\in\N$ with $0\leq i\lt j\lt k\lt l\leq n$, the diagram corresponding to [[Street]]'s fourth [[oriental]] above commutes.
7. The degeneracy map
   $$\mathrm{s}^{0}_{0}\colon \mathrm{N}^{\mathsf{S}}_{0}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C})$$
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $0$ is the map sending a $0$-simplex $A$ of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ (i.e. an object $A$ of $\mathcal{C}$) to the $1$-simplex $\mathrm{id}_{A}\colon A\to A$.

8. The degeneracy maps
   $$
   \mathrm{s}^{1}_{0} \colon \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{s}^{1}_{1} \colon \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
   $$
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $1$ are the maps described as follows: given a $1$-simplex $\sigma=(A\xrightarrow{f}B)$ of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$, we have
   [[deg-1-degeneracies-of--duskin-nerve.svg:pic]]

9. The degeneracy maps in degree $2$
   $$
   \mathrm{s}^{2}_{0} \colon \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{s}^{2}_{1} \colon \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C}),
   $$
   $$
   \mathrm{s}^{2}_{2} \colon \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C}),
   $$
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$  in degree $2$ are the maps described as follows: given a $2$-simplex
   [[2-simplex-duskin-sigma.svg:pic]]
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$, we have (page author's note: please take the following $3$-morphisms with a grain of salt; I'm quite unsure about whether they are correct or not. In any case, note that we _must_ use the left, middle, and right $2$-unitors of $\mathcal{C}$ here---this is where they appear in the Street nerve!)
   \begin{imagefromfile}
       "file_name": "deg-2-degeneracies-street-1-new.svg",
       "width":     "800"
   \end{imagefromfile}
   \begin{imagefromfile}
       "file_name": "deg-2-degeneracies-street-2.svg",
       "width":     "800"
   \end{imagefromfile}
   \begin{imagefromfile}
       "file_name": "deg-2-degeneracies-street-3-new.svg",
       "width":     "800"
   \end{imagefromfile}
   For the details regarding these pastings, see [ [this PDF] ](https://ncatlab.org/nlab/files/pastings-in-street-nerve.pdf).

10. The face maps
   $$
   \mathrm{d}^{1}_{0} \colon \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{0}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{1}_{1} \colon \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{0}(\mathcal{C}),
   $$
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $1$ are given by
   $$\mathrm{d}^{1}_{0}(A\xrightarrow{f}B)=B$$
   $$\mathrm{d}^{1}_{1}(A\xrightarrow{f}B)=A$$

11. The face maps
   $$
   \mathrm{d}^{2}_{0} \colon \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{2}_{1} \colon \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{2}_{1} \colon \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{1}(\mathcal{C}),
   $$
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $2$ are described as follows: given a $2$-simplex
   [[2-simplex-duskin-sigma.svg:pic]]
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$, we have
   [[face-maps-deg-2-duskin.svg:pic]]

12. The face maps
   $$
   \mathrm{d}^{3}_{0} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{3}_{1} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{3}_{2} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
   $$
   $$
   \mathrm{d}^{3}_{3} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
   $$
   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $3$ are described as follows: given a $3$-simplex $\sigma$ of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ as in the diagram
   [[3-simplex-of-the-street-nerve.svg:pic]]
   we have
   [[face-maps-deg-3-duskin.svg:pic]]

## Related concepts

* [[nerve]]

* [[geometric nerve of a bicategory]]

* **geometric nerve of a tricategory**

* [[∞-nerve]]

* [[homotopy coherent nerve]]

## References

* {#CegarraHeredia2012} Antonio M. Cegarra and Benjamín A. Heredia, _Geometric Realizations of Tricategories_. Algebraic & Geometric Topology 14, no. 4 (2014): 1997-2064. [ [arXiv:1203.3664] ](https://arxiv.org/abs/1203.3664)

* {#Carrasco2014} Pilar Carrasco, _Nerves of Trigroupoids as Duskin-Glenn’s $3$-Hypergroupoids_, Applied Categorical Structures 23.5 (2015): 673–707.

* {#LackPaoli2006} Stephen Lack and Simona Paoli, _2-nerves for bicategories_, Journal of $K$-Theory 38.2 (2008): 153–175. [ [arXiv:0607271] ](https://arxiv.org/abs/math/0607271).

* {#Street1987} [[Ross Street]], _The algebra of oriented simplexes_, Journal of Pure and Applied Algebra, Volume 49, Issue 3, December 1987, Pages 283–335. [pdf](https://ncatlab.org/nlab/files/Street-AlgebraOfOrientedSimplexes.pdf)

* {#GPS1995} [[Robert Gordon]], [[John Power]], [[Ross Street]], _Coherence for tricategories_, Mem. Amer. Math Soc. 117 (1995) no 558.

* {#Duskin2002} [[John Duskin]], _Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories_ ([tac](http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html)), Theory and Applications of Categories, Vol. 9, No. 10, 2002, pp. 198–308.
{#Duskin}


geometric origin of inhomogeneous media

# The geometric origin of inhomogeneous media
* table of contents
{: toc}

## Summary

Just as geometry is encoded in the metric tensor and manifests itself via the [[Hodge star operator]], so too do the [[electromagnetic constitutive equations]]. For example, in linear media we have the simple constitutive relations

$$
  E = \frac{1}{\epsilon} D \quad\text{and}\quad B = \mu H.
$$

between the electric field $E$ and the magnetic field $B$.

In 4d, we have

$$
  F = B + E\wedge d t
$$

The 4d constitutive relation is

$$
  G = \star F
  \,,
$$

which under assumptions of linearity gives

$$
  \star F = -\eta(D-H\wedge d t)
  \,,
$$

where $\eta = \sqrt{\frac{\mu}{\epsilon}}$. This may be written in a form that more closely mimics the tradition relations via

$$
  \star(v d t\wedge E) = \frac{1}{\epsilon} D\quad\text{and}\quad\star B = \mu H\wedge v d t
  \,,
$$

where $v = \frac{1}{\sqrt{\mu\epsilon}}$ (Note: $v = c$ in vacuum).

What this means is the the electromagnetic properties of matter can be interpreted geometrically and are encoded in the
[[Hodge star operator]]. Conversely, it means that geometrical properties of matter can be interpreted electromagnetically.

## References

For more details see page 111 of [[Eric Forgy]]\'s [dissertation](http://ncatlab.org/ericforgy/show/Dissertation).


geometric point

#Contents#
* table of contents
{:toc}

##Idea

The notion of _geometric point_ refers to a certain kind of [[morphism]] between [[schemes]] in [[algebraic geometry]]; that is, to a specialized sort of [[generalized element]] of a scheme.

While a [[point]] in a [[topological space]], $X$,  can be thought of as a [[continuous function]] from a singleton space to $X$, in [[algebraic geometry]] the 'spaces' come with more structure as they are [[schemes]] and singletons correspond to the [[prime spectrum|spectra]] of [[fields]].  Category-theoretically, one may think of *any* morphism into $X$ as a [[generalized element|generalized point]] of $X$, but when doing geometry it is often appropriate to restrict to a subclass of these to consider as the (less generalized) "points".

## Definition

Suppose a scheme $S$ is defined over  a field $k$, so is equipped with a morphism to $Spec (k)$.

+-- {: .num_defn}
###### Definition

A **geometric point** $\xi$ in $S$ is a [[morphism]] from the [[spectrum of a commutative ring|spectrum]] $Spec(K)$ to $S$ where $K$ is an [[algebraic closure]]/[[separable closure]] of $k$.

=--

+-- {: .num_remark}
###### Remark

In general the set of geometric points of a scheme is different from the set of ordinary points of its underlying topological space.

=--

## Related concepts

* [[singular point of an algebraic variety]]

* [[étale fundamental group]]

* [[generalized element]]

* [[geometric fiber]]

* [[section conjecture]]

##References

* [[EGA]]

* [[SGA1]]

* [[James Milne]], section 4 of _[[Lectures on Étale Cohomology]]_

[[!redirects geometric points]]