GAGA

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## Idea

_GAGA_ is short for the title _Géométrie algébrique et géométrie analytique_ of the article  ([Serre 56](#Serre56)), and more generally has come to stand for the kind of results initated in this article, establishing the close relationship between [[algebraic geometry]] over the [[complex numbers]] and [[complex geometry|complex]] [[analytic geometry]], hence between [[algebraic spaces]] ([[algebraic varieties]], [[schemes]]) over the [[complex numbers]] and [[complex analytic spaces]].

For example, often one can compute [[sheaf cohomology]] either in the algebraic category or in the analytic category, obtaining the same result. This can be useful because one has more tools in the analytic category, such as [[Dolbeault cohomology|Dolbeault resolutions]]. Typically GAGA-type theorems hold for proper (compact) varieties, but not for non-proper varieties.

## Theorems

* [analytification -- Existence and fully faithfulness](analytification#Existence) (this is often referred to as "the GAGA theorem")

* [[Chow's theorem]]

* [relation between algebraic and analytic vector bundles](algebraic+vector+bundle#RelationToAnalyticVectorBundles)

* [[comparison theorem (étale cohomology)]]

## References

The original article is

* {#Serre56} [[Jean-Pierre Serre]], _Géométrie algébrique et géométrie analytique_ Université de Grenoble. Annales de l'Institut Fourier 6: 1–42, (1956) doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175

which compares the categories of projective integral $\mathbb{C}$-schemes of finite type with that of projective complex manifolds possibly with singularities. In

* [[Alexander Grothendieck]], [[SGA]] I, Exposé XII

this was generalized to a comparison of proper $\mathbb{C}$-schemes locally of finite type and complex analytic spaces.

A good modern textbook account is in

* {#Neeman07} [[Amnon Neeman]], _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__, 2007 ([publisher](http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry))

Reviews and surveys include

* Wikipedia, _[Algebraic geometry and analytic geometry](http://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry)_

* Yan Zhao, _Géométrie algébrique et géométrie analytique_, 2013 ([pdf](http://pub.math.leidenuniv.nl/~jinj/2013/efg/gaga.pdf))

* {#Demailly09} [[Jean-Pierre Demailly]], _Analytic methods in algebraic geometry_, lecture notes 2009  ([pdf](http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth.pdf))

Discussion for [[real analytic spaces]] includes

* {#Huisman02} [[Johannes Huisman]], section 2 of _The exponential sequence in real algebraic geometry and Harnack's Inequality for proper reduced real schemes_, Communications in Algebra, Volume 30, Issue 10, 2002 ([journal](https://doi.org/10.1081/AGB-120014664))

Generalization to more general [[analytic geometry]] includes

* {#ConradTemkin09} [[Brian Conrad]], M. Temkin, _Non-Archimedean analytification of algebraic spaces_, J. Algebraic Geom. 18 (2009), no. 4, 731–788 ([arXiv:0706.3441](http://arxiv.org/abs/0706.3441))

Generalization to  [[higher geometry]] ([[stacks]]) includes

* {#Lurie04} [[Jacob Lurie]], _[[Tannaka duality for geometric stacks]]_, ([arXiv:math.AG/0412266](http://arxiv.org/abs/math/0412266))

* {#Hall11} [[Jack Hall]], _Generalizing the GAGA Principle_ ([arXiv:1101.5123](http://arxiv.org/abs/1101.5123))

* {#GeraschenkoZureickBrown12} [[Anton Geraschenko]], David Zureick-Brown, _Formal GAGA for good moduli spaces_ ([arXiv:1208.2882](http://arxiv.org/abs/1208.2882))

Discussion in the context of [[moduli spaces]] includes

* Kai-Wen Lan, _Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties_ [pdf](http://math.umn.edu/~kwlan/articles/cpt-ana-alg.pdf)

[[!redirects GAGA principle]]

category: reference


galaxy

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## Definition

A galaxy is a collection of stars, compact stars, interstellar gas and dust which are bound together by [[gravity]].

## Properties

* [[galaxy rotation curve]]

* [[Tully-Fisher relation]]

## Examples

* [[Milky Way]]

* [[Messier 87]]

## Related concepts

* [[observable universe]]

* [[dark matter]]

* [[structure formation]]

## References

### General

See also

* Wikipedia, _[Galaxy](http://en.wikipedia.org/wiki/Galaxy)_

* Wikipedia, _[Galaxy clusters](https://en.wikipedia.org/wiki/Galaxy_cluster)_

### Structure formation and computer simulation

Computer simulation of galaxy [[structure formation]]:

* Julio Navarro, _Simulations of Galaxy Formation:  the State of the Art_ ([pdf](http://www.physics.ox.ac.uk/iau311/navarro_iau311.pdf))

* Mark Vogelsberger, Federico Marinacci, Paul Torrey, Ewald Puchwein, _Cosmological Simulations of Galaxy Formation_, Nature Reviews Physics volume 2, pages 42–66 (2020)  ([arXiv:1909.07976](https://arxiv.org/abs/1909.07976), [doi:10.1038/s42254-019-0127-2](https://doi.org/10.1038/s42254-019-0127-2))

* [FIRE: Feedback In Realistic Environments
Increasing the predictive power of galaxy formation simulations
](https://fire.northwestern.edu/)

  * [visualization](https://fire.northwestern.edu/visualizations/)

* Volonteri, _Simulation of the formation of a galaxy similar to our Milky Way_ ([video](https://m.youtube.com/watch?v=s-25dEcY-WU]))

* Shea Garrison-Kimmel, _Next-generation Galaxy Formation Simulations with FIRE_, 2018 ([video recording](https://youtu.be/sSkrm66uDvw))

* {#Fitts16} Fitts et al. _FIRE in the Field: Simulating the Threshold of Galaxy Formation_, Mon.Not.Roy.Astron.Soc.471:3547-3562, 2017 ([arXiv:1611.02281](https://arxiv.org/abs/1611.02281))

* {#Hopkins17} Hopkins et al. _FIRE-2 Simulations: Physics versus Numerics in Galaxy Formation_. Monthly Notices of the Royal Astronomical Society, Volume 480, Issue 1, 11 October 2018, Pages 800–863 ([arXiv:1702.06148](https://arxiv.org/abs/1702.06148), [doi:10.1093/mnras/sty1690](https://doi.org/10.1093/mnras/sty1690))

* {#ElBadry18} El-Badry et al. _Gas Kinematics in FIRE Simulated Galaxies Compared to Spatially Unresolved HI Observations_, Monthly Notices of the Royal Astronomical Society, Volume 477, Issue 2, 21 June 2018, Pages 1536–1548 ([arXiv:1801.03933](https://arxiv.org/abs/1801.03933), [doi:10.1093/mnras/sty730](https://doi.org/10.1093/mnras/sty730))

[[galactic rotation curves]] qualitatively reproduced in [Hopkins 17, Figure 4, Figure 5](#Hopkins17)

baryonic [[Tully-Fisher relation]] qualitatively reproduced in [El-Badry 18, Figure 4](#ElBadry18)

* [[Justin Read]] et al. _EDGE I: the mass-metallicity relation as a critical test of galaxy formation physics_ ([arXiv:1904.02723](https://arxiv.org/abs/1904.02723))

Simulation specifically of Milky Way-like galaxies:

* [VINTERGATAN simulation](http://www.astro.lu.se/~florent/vintergatan.php)

On high redshift galaxies discovered at the [[James Webb Space Telescope]]:

* Michael Boylan-Kolchin, *Stress Testing ΛCDM with High-redshift Galaxy Candidates*, ([arXiv:2208.01611](https://arxiv.org/abs/2208.01611))

* B. E. Robertson et al. *Discovery and properties of the earliest galaxies with confirmed distances*. ([arXiv:2212.0448](https://arxiv.org/abs/2212.04480))

[[!redirects galaxies]]

[[!redirects galaxy cluster]]
[[!redirects galaxy clusters]]


galaxy rotation curve

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## Idea

The [[graph of a function|graph]] of [[angular velocity]] in a [[galaxy]] as a [[function]] of the [[distance]] from the [[center]] is called the galaxy's _rotation curve_.

For most observed [[galaxies]] their rotation curves do not follow the predictions of standard [[gravity]] applied to the visible [[baryon|baryonic]] [[mass]] distribution in the galaxy. This has led to the speculation that either there is further invisible but gravitating [[dark matter]] making up most of the mass of galaxies, or the laws of [[gravity]] are modified on galactic scales ("[[MOND]]").

Another apparent peculiarity is that the _asymptotic_ rotation of galaxies does follow the total visible [[baryon|baryonic]] [[mass]] in a way called the _[[Tully-Fisher relation]]_.

Recent computer simulation ([FIRE](https://fire.northwestern.edu)) of [[galaxy|galactic]] [[structure formation]] using the [[standard model of cosmology|standard]] [[cold dark matter]] [[model (physics)|model]] qualitatively reproduces the peculiar [[galactic rotation curves]] that motivated dark matter (or [[MOND]], for that matter) in the first place ([Hopkins et al. 17, Figure 4, Figure 5](#Hopkins17)):

\begin{center}

\end{center}

> graphics grabbed from ([Hopkins et al. 17](#Hopkins17))

and also reproduces well the baryonic [[Tully-Fisher relation]] ([El-Badry et al. 18, Figure 4](#ElBadry18)) which used to be an issue in the [[standard model of cosmology|standard]] [[cold dark matter]] [[model (physics)|model]]:

\begin{center}

\end{center}

> graphics grabbed from ([El-Badry et al. 18](#ElBadry18))

More in detail, [[galaxy rotation curves]] exhibit a close correlation between [[angular velocity]] and total _visible_ enclosed math at any given [[radius]], called the _[[radial acceleration relation]]_ (RAR) or _[[mass-discrepancy acceleration relation]]_. This, too, is reproduced in [[cold dark matter]]-[[model (in theoretical physics)]]  computer simulation ([Santos-Santos et al. 15](#SantosSantosEtAl15), [Cintio-Lelli 15](#CintioLelli15), [Keller-Wadsley 16](#KellerWadsley16), [Ludlow et al 16](#LudlowEtAl16)). These early simulations were not found conclusive in [Lelli et al 16, section 8.2](#LelliEtAl16). But more detailed analysis ([PSF 18](#PSF18)) and more refined simulation [Dutton-Maccio-Obreja-Buck 19](#DuttonMaccioObrejaBuck19) has then been claimed to confirm the statement.

A conceptual explanation via stellar feedback is discussed in [GBFH 19](#GBFH19).

## References

### General

See also

* Wikipedia, _[Galaxy rotation curve](https://en.wikipedia.org/wiki/Galaxy_rotation_curve)_

### Structure formation and Computer simulation

Computer simulation of [[structure formation]] with [[dark matter]] on [[galaxy]]-scales:

* {#Hopkins17} Hopkins et al. _FIRE-2 Simulations: Physics versus Numerics in Galaxy Formation_. Monthly Notices of the Royal Astronomical Society, Volume 480, Issue 1, 11 October 2018, Pages 800–863 ([arXiv:1702.06148](https://arxiv.org/abs/1702.06148), [doi:10.1093/mnras/sty1690](https://doi.org/10.1093/mnras/sty1690))

* {#ElBadry18} El-Badry et al. _Gas Kinematics in FIRE Simulated Galaxies Compared to Spatially Unresolved HI Observations_, Monthly Notices of the Royal Astronomical Society, Volume 477, Issue 2, 21 June 2018, Pages 1536–1548 ([arXiv:1801.03933](https://arxiv.org/abs/1801.03933), [doi:10.1093/mnras/sty730](https://doi.org/10.1093/mnras/sty730))

On the [[mass-discrepancy acceleration relation]]:

* {#CintioLelli15} Arianna Di Cintio, Federico Lelli, _The mass discrepancy acceleration relation in a $\Lambda CDM$ context_, Monthly Notices of the Royal Astronomical Society: Letters, Volume 456, Issue 1, 11 February 2016, Pages L127–L131 ([arXiv:1511.06616](https://arxiv.org/abs/1511.06616), [doi:10.1093/mnrasl/slv185](https://doi.org/10.1093/mnrasl/slv185))

* {#SantosSantosEtAl15} Isabel M. Santos-Santos et al. _The distribution of mass components in simulated disc galaxies_ ([arXiv:1510.02474](https://arxiv.org/abs/1510.02474))

* {#KellerWadsley16} B.W. Keller, J.W. Wadsley, _$\Lambda CDM$ is Consistent with SPARC Radial Acceleration Relation_ ([arXiv:1610.06183](https://arxiv.org/abs/1610.06183))

* {#LudlowEtAl16} Aaron D. Ludlow et. al. _The Mass-Discrepancy Acceleration Relation: a Natural Outcome of Galaxy Formation in Cold Dark Matter halos_, Phys. Rev. Lett. 118, 161103 (2017) ([arXiv:1610.07663](https://arxiv.org/abs/1610.07663))

* {#PSF18} Chiara Di Paolo, Paolo Salucci, Jean Philippe Fontaine, _The Radial Acceleration Relation (RAR): the crucial cases of Dwarf Discs and of Low Surface Brightness galaxies_, ApJ 2019 ([arXiv:1810.08472](https://arxiv.org/abs/1810.08472))

* {#DuttonMaccioObrejaBuck19} Aaron A. Dutton, Andrea V. Macciò, Aura Obreja, Tobias Buck, _NIHAO XVIII: Origin of the MOND phenomenology of galactic rotation curves in a LCDM universe_ ([arXiv:1902.06751](https://arxiv.org/abs/1902.06751))

Critical comments in

* {#LelliEtAl16} Federico Lelli, Stacy S. McGaugh, James M. Schombert, Marcel S. Pawlowski, section 8.2 of _One Law To Rule Them All: The Radial Acceleration Relation of Galaxies_ ([arXiv:1610.08981](https://arxiv.org/abs/1610.08981))

A conceptual explanation via stellar feedback:

* {#GBFH19} Michael Y. Grudić, Michael Boylan-Kolchin, Claude-André Faucher-Giguère, Philip Hopkins, _Stellar feedback sets the universal acceleration scale in galaxies_ ([arxiv:1910.06345](https://arxiv.org/abs/1910.06345))

Simulation specifically of Milky Way-like galaxies:

* [VINTERGATAN simulation](http://www.astro.lu.se/~florent/vintergatan.php)

[[!redirects galaxy rotation curves]]

[[!redirects galactic rotation curves]]


Galilean group

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## Idea

A 10 dimensional Lie group, the relativistic counterpart of which is the 10 dimensional [[Poincare group]]. Both include the [[isometry group]] of [[Euclidean space]]. Moreover, both have important linear representations including actions on tangents of trajectories (i.e. velocities), which - in case of the Galilean group - shift all but one (i.e. the spatial) components like a translation.

## Related concepts

* [[classical mechanics]]

* [[translation group]]

* [[Lorentz group]], [[Poincare group]]

* [[super Euclidean group]]

* [[space group]]

  * [[wallpaper group]]

## References

* Wikipedia, _[Galilean group](http://en.wikipedia.org/wiki/Galilean_transformation#Galilean_group)_

* Wikipedia, _[Euclidean group](http://en.wikipedia.org/wiki/Euclidean_group)_

* [[David R. Hilbert]], [[Nick Huggett]], _Groups in Mind_, Philosophy of Science 73 (5):765-777 (2006) ([doi:10.1086/518635](https://philpapers.org/rec/HILGIM-2), [[HilbertHiggettGroups.pdf:file]])

[[!redirects Euclidean group]]

[[!redirects Galileo group]]

[[!redirects Galilei group]]

[[!redirects Galilelan group]]


Galileo Galilei

* [Wikipedia entry](http://en.wikipedia.org/wiki/Galileo_Galilei)

* [Stanf. Enc. of Phil. Entry](http://plato.stanford.edu/entries/galileo/)

## Quotes:
 {#Quotes}

On [[mathematics]] as the language of [[physics]]:

> {#TheBookOfnature} [[natural philosophy|Philosophy]] $[$i.e. [[natural philosophy]], i.e. [[physics]] $]$ is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of [[mathematics]], and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth. ([Galilei, Il Saggiatore, 1623](http://en.wikipedia.org/wiki/The_Assayer)).

Compare to:

* [[David Hilbert]], _[[Naturerkennen und Logik]]_, Lecture at the _Kongress der Gesellschaft Deutscher Naturforscher und Ärtze_, 1930  ([pdf](http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.pdf), [audio](http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3))

* [[Eugene Wigner]], _[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]_ 1960.

\linebreak

On the [[Milky Way]]:

> the Galaxy is nothing else than a congeries of innumerable stars distributed in clusters. To whatever region of it you direct your spyglass, an immense number of stars immediately offer themselves to view, of which very many appear rather large and very conspicuous but the multitude of small ones is truly unfathomable.

> (in _Sidereus Nuncius_ 1610, as translated in [Van Helder 1989](#vanHelder), [p. 34](http://people.reed.edu/~wieting/mathematics537/SideriusNuncius.pdf#page=35))

## Selected writings

* {#vanHelder} *Sidereus Nuncius*, 1610, Enlish translation by Albert Van Helden, University of Chicago 1989 ([pdf](people.reed.edu/~wieting/mathematics537/SideriusNuncius.pdf))

## Related entries

* [[physics]], [[natural philosophy]]

* [[mathematical physics]]

* [[Galilei group]]

* [[Milky Way]]

category: people

[[!redirects Galilei]]
[[!redirects Galileo]]


Gallina specification language

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## Idea

A formal language for specifying [[logic|logical]] [[theories]]. Gallina is used in the [[proof assistant]] system [[Coq]].

## References

* _[The Gallina specification language ](https://coq.inria.fr/distrib/current/refman/language/gallina-specification-language.html)_

[[!redirects Gallina]]


Galois category

\tableofcontents

\section{Introduction}

A Galois category is a category equipped with structure allowing for a construction of a 'fundamental group' from it, akin to the construction of the fundamental group of a topological space from the category of covering spaces of it. See [[Grothendieck's Galois theory]] for more on the latter.

\section{Details}

There are two ways to define a Galois category. We give them both below, following [SGA1](#SGA1).

\begin{defn} A _Galois category_ is a category equivalent to the [[classifying topos of a profinite group]]. \end{defn}

\begin{defn} A _Galois category_ is a [[category]] $G$ for which there exists a [[functor]] $F: G \rightarrow \mathsf{FinSet}$, where $\mathsf{FinSet}$ is the category [[FinSet]] of finite sets, such that the following hold.

1. $G$ has [[finite limit|finite limits]].
1. $G$ has [[finite colimit|finite colimits]].
1. For every arrow $f: X \rightarrow Y$ of $G$, there are objects $Z_{f}$ and $Z'_{f}$ of $G$ such that $Y$ is isomorphic to the coproduct $Z_{f} \sqcup Z'_{f}$, such that there is a [[strict epimorphism]] $u: X \rightarrow Z_{f}$, and such that the canonical [[coprojection]] functor $v: Z_{f} \rightarrow Y$ is a [[monomorphism]].
1. $F$ is [[exact functor|exact]], that is to say, preserves finite limits and finite colimits.
1. $F$ is [[conservative functor|conservative]].

\end{defn}

\begin{rmk} The original form of 2. in §4 of SGA1 is slightly weaker, and the axiom 4. was originally two axioms which together were slightly weaker than our axiom, namely that $F$ preserved finite limits, and that $F$ preserved the colimits required to exist in the weaker form of 2. However, as discussed in Remark 4.2 of SGA1, if the weaker axioms hold and the other axioms hold, then 2. and 4. as we have given them hold, so we prefer to use them for simplicity and brevity. \end{rmk}

\section{References}

* {#SGA1} _Revêtements étales et groupe fondamental (SGA 1)_, [[Alexander Grothendieck]], 1971, Springer-Verlag, vol. 224 of _Lecture notes in mathematics_.


Galois cohomology

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## Idea

_Galois cohomology_ is the [[group cohomology]] of [[Galois groups]] $G$. Specifically, for $G$ the Galois group of a [[field extension]] $L/K$, Galois cohomology refers to the group cohomology of $G$ with [[coefficients]] in a $G$-[[module]] naturally associated to $L$.

Galois cohomology is studied notably in the context of _[[algebraic number theory]]_.

## Properties

### Relation to étale cohomology

Galois cohomology of a [[field]] $k$ is essentially the
[[étale cohomology]] of the [[spectrum of a commutative ring|spectrum]]
$Spec(k)$.

See also at _[[comparison theorem (étale cohomology)]]_.

### In terms of cohesive homotopy type theory
 {#InCohesiveHomotopyTypeTheory}

We make some comments on the formulation of Galois cohomology in [[cohesive homotopy type theory]].

As discussed at _[[Galois theory]]_, the [[absolute Galois group]] $G_{Galois}$ of a [[field]] $K$ is the [[fundamental group]] of the [[spectrum of a commutative ring|spectrum]] $X \coloneqq Spec(K)$. Hence its [[delooping]] $\mathbf{B}G_{Galois}$ is the [[fundamental groupoid]]

$$
  \Pi_1(X)  \simeq \mathbf{B}G_{Galois}
  \,.
$$

In [[cohesive homotopy type theory]] there exists the [[fundamental ∞-groupoid]]-construction  -- the [[shape modality]] $\Pi$ --

$$
  X \colon Type \;\vdash \; \Pi(X) \colon Type
  \,.
$$

Moreover, by the discussion at _[[group cohomology]]_ in the section _[group cohomology - In terms of homotopy type theory](http://ncatlab.org/nlab/show/group%20cohomology#InHomotopyTypeTheory)_

1. a $G_{Galois}$-[[module]] $A$ is a $\mathbf{B}G_{Galois}$-[[dependent type]];

1. the [[group cohomology]] is the [[dependent product]] over the [[function type]]

   $$
     \vdash \; \left( \prod_{x \colon \mathbf{B}G_{Galois}} \left( * \to A \right)\right) \colon Type
     \,.
   $$

Hence, generally in [[cohesive homotopy type theory]], for $X$ a [[type]] and

$$
  x \colon \Pi(X) \;\vdash \; A(x) \colon Type
$$

a $\Pi(X)$-[[dependent type]], we can say that the corresponding _$\infty$-Galois cohomology_ is

$$
  \vdash \; \left( \prod_{x \colon \Pi(X)} \left( * \to A\right) \right)
  \colon Type
  \,.
$$

**Warning.** Currently there is not written down yet a [[model]] for [[cohesive homotopy type theory]] given by a [[cohesive (∞,1)-topos]] over a [[site]] like the [[étale site]].

## Examples

* [[Hilbert's theorem 90]]

## Related concepts

* [[algebraic number theory]]

## Literature

* [[John Tate]], _Galois cohomology_ ([pdf](http://modular.math.washington.edu/edu/2010/582e/refs/tate-galois_cohomology.pdf))

* [[Jean-Pierre Serre]], _Galois cohomology_, Springer Monographs in Mathematics (1997) doi:[10.1007/978-3-642-59141-9](https://doi.org/10.1007/978-3-642-59141-9)

  _Cohomologie galoisienne_, Lecture Notes in Mathematics, 5 (Fifth ed. 1994), Springer-Verlag, MR1324577 doi:[10.1007/BFb0108758)](https://doi.org/10.1007/BFb0108758)

* {#GilleSzamuely06} [[Philippe Gille]], [[Tamás Szamuely]], *Central Simple Algebras and Galois Cohomology*, Cambridge University Press  2006 ([doi:10.1017/CBO9780511607219](https://doi.org/10.1017/CBO9780511607219), [pdf](http://www.math.ens.fr/~benoist/refs/Gille-Szamuely.pdf))

* Grégory Berhuy, _An introduction to Galois cohomology and its applications_ ([pdf](http://www-fourier.ujf-grenoble.fr/~berhuy/fichiers/NTUcourse.pdf))

* M. Kneser, _Lectures on Galois cohomology of classical groups_ ([pdf](http://www.math.tifr.res.in/~publ/ln/tifr47.pdf))

See also:

* Wikipedia, _[Galois cohomology](http://en.wikipedia.org/wiki/Galois_cohomology)_

A generalization in the setup of [[coring]]s:

* [[Tomasz Brzezi?ski]], _Descent cohomology and corings_, Comm Algebra 36:1894-1900, 2008, [math.RA/0601491](http://arxiv.org/abs/math/0601491)

In a [[model theory|model theoretic]] context:

* [[Anand Pillay]], _Remarks on Galois cohomology and definability_, The Journal of Symbolic Logic __62__:2 (1997) 487-492 [doi](https://doi.org/10.2307/2275542)
* B. Poizat,  _Une theorie de Galois imaginaire_, J. Symb. Logic __48__  (1983) 1151-1170.


Galois connection

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# Galois connections
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## Idea
 {#Idea}

In [[order theory]] the term *Galois connection* (due to [Ore 44](#Ore44), who spelled it "connexion") can mean both: "[[adjunction]] between [[posets]]" and "[[dual adjunction]] between [[posets]]"; the former notion is sometimes called "monotone Galois connection" and the latter "antitone Galois connection". In this article the term "Galois connection" shall mean "[[dual adjunction]] between [[posets]]".

The term _Galois correspondence_ is also in use. For some authors it is synonymous to "Galois connection", others reserve it for its restriction to its [[fixed point of an adjunction|fixed points]], where it becomes an [[adjoint equivalence]].

The example that gives the concept its name is the relation between [[subgroups]] and [[subfields]] in [[Galois theory]] (see [below](#GaloisTheory)), but adjunctions between posets, hence Galois connections, appear also in many other and entirely different contexts, see further [below](#Examples).

## Definition
 {#Definition}

Given [[posets]] $A$ and $B$, a __Galois connection__ between $A$ and $B$ is a pair of order-reversing [[functions]] $f \colon A\to B$ and $g \colon B\to A$ such that $a\le g(f(a))$ and $b\le f(g(b))$ for all $a\in A$, $b\in B$.

A **Galois correspondence** is a Galois connection which is an [[adjoint equivalence]] (so $a = g(f(a))$ and $b = f(g(b))$ for all $a \in A$, $b \in B$).

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###### Proposition
Any Galois connection $f: A \to B$, $g: B \to A$ induces a Galois correspondence between $f(A)$ and $g(B)$, given by the composites $g(B) \hookrightarrow A \stackrel{f}{\to} f(A)$ and $f(A) \hookrightarrow B \stackrel{g}{\to} g(B)$.
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+-- {: .proof}
###### Proof
For any $a \in A$ of the form $a = g(b)$, we have $a \leq (g \circ f)(a)$ and also $(g \circ f)(a) = g(f(g(b))) \leq g(b) = a$ where the inequality follows from $b \leq f(g(b))$ and antitonicity of $g$. Hence $(g \circ f)(a) = a$ for all $a \in g(B)$. Similarly $(f \circ g)(b) = b$ for all $b \in f(A)$.
=--

## Examples
  {#Examples}

### Galois theory
  {#GaloisTheory}

The [[Galois theory]] normally taught in graduate-level algebra courses (and based on the work of [[Évariste Galois]]) involves a Galois connection between the intermediate [[fields]] of a [[Galois extension]] and the subgroups of the corresponding [[Galois group]].

### Galois connections induced from a relation
 {#InducedFromARelation}

Frequently Galois connections between collections of [[subsets]] ([[power sets]]) arise where $f(a)$ is "the set of all $y$ standing in some relation to every $x\in a$" and dually $g(b)$ is "the set of all $x$ standing in some relation to every $y\in b$."

Examples of this class of Galois connections include the following

* **(Zariski topology)** The [[closed subsets]] in the [[Zariski topology]] on [[affine space]] $k^n$ or on the set of [[maximal ideals]] of a [[polynomial ring]], which may be understood as the [[fixed point of an adjunction|fixed points]]  of a Galois connection between [[polynomials]] and [[affine space]]/[[maximal ideal]]. This is discussed at _[Zariski topology -- In terms of Galois connections](Zariski+topology#InTermsOfGaloisConnections)_.

* **(orthogonality classes)** Given a [[category]] $\mathcal{C}$, then on the [[poset]] of sub-[[classes]] of [[morphisms]] the operations of forming left and right classes with [[orthogonality]] [[lifting property]] constitute a Galois connection.

In fact all Galois connections between [[power sets]] arise this way, see [below](#BetweenPowerSets).

We now spell out in detail the Galois connections induced from a relation:

+-- {: .num_defn #GaloisConnectionFromRelation}
###### Definition
**(Galois connection induced from a [[relation]])**

Consider two [[sets]] $X,Y \in Set$ and a [[relation]]

$$
  E \hookrightarrow X \times Y
  \,.
$$

Define two [[functions]] between their [[power sets]] $P(X), P(Y)$, as follows.
(In the following we write $E(x, y)$ to abbreviate the formula $(x, y) \in E$.)

1. Define

   $$
     V_E
       \;\colon\;
     P(X)
       \longrightarrow
     P(Y)
   $$

   by

   $$
     V_E(S)
       \coloneqq
     \left\{
       y \in Y \vert  \underset{x \in  X}{\forall} \left( \left(x \in S\right) \Rightarrow E(x, y)
    \right) \right\}
   $$

1. Define

   $$
     I_E
       \;\colon\;
     P(Y)
       \longrightarrow
     P(X)
   $$

   by

   $$
     I_E(T) \coloneqq \left\{x \in X \vert \underset{y \in Y}{\forall} \left( \left(y \in T \right) \Rightarrow E(x, y)  \right)\right\}
   $$

=--

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

The construction in def. \ref{GaloisConnectionFromRelation} has the following properties:

1. $V_E$ and $I_E$ are [[contravariant functor|contravariant]] order-preserving in that

   1. if $S \subset S'$, then $V_E(S') \subset V_E(S)$;

   1. if $T \subset T'$, then $I_E(T') \subset I_E(T)$

1. The _[[adjunction]] law_ holds:
   $
     \left(
        T \subset V_E(S)
     \right)
       \,\Rightarrow\,
     \left(
       S \subset I_E(T)
     \right)
   $

   which we denote by writing

   $$
     P(X)
       \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
     P(Y)^{op}
   $$

1. both $V_E$ as well as $I_E$ take [[unions]] to [[intersections]].

=--

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

Regarding the first point: the larger $S$ is, the more conditions that are placed on $y$ in order to belong to $V_E(S)$, and so the smaller $V_E(S)$ will be.

Regarding the second point:  This is because both these conditions are equivalent to the condition $S \times T \subset E$.

Regarding the third point: Observe that in a poset such as $P(Y)$, we have that $A = B$ iff for all $C$, $C \leq A$ iff $C \leq B$ (this is the [[Yoneda lemma]] applied to posets). It follows that

$$
  \array{
    T \subset V_E(\bigcup_{i \in I} S_i) & \text{iff} & \bigcup_{i: I} S_i \subset I_E(T) \\
    & \text{iff} & \forall_{i: I} S_i \subset I_E(T) \\
    & \text{iff} & \forall_{i: I} T \subset V_E(S_i) \\
    & \text{iff} & T \subset \bigcap_{i: I} V_E(S_i)
}
$$

and we conclude $V_E(\bigcup_{i: I} S_i) = \bigcap_{i: I} V_E(S_i)$ by the [[Yoneda lemma]].

=--

+-- {: .num_prop #GaloisClosureOperator}
###### Proposition
**([[closure operators]] from Galois connection)**

Given a Galois connection as in def. \ref{GaloisConnectionFromRelation}, consider the [[composition|composites]]

$$
  I_E \circ V_E \;\colon\; P(X) \longrightarrow P(X)
$$

and

$$
  V_E \circ I_E \;\colon\; P(Y) \longrightarrow P(Y)
  \,.
$$

These satisfy:

1. For all $S \in P(X)$ then  $S \subset I_E \circ V_E(S)$.

1. For all $S \in P(X)$ then $V_E \circ I_E \circ V_E (S)  =  V_E(S)$.

1. $I_E \circ V_E$ is [[idempotent]] and [[covariant functor|covariant]].

and

1. For all $T \in P(Y)$ then $T \subset V_E \circ I_E(T)$.

1. For all $T \in P(Y)$ then $I_E \circ V_E \circ I_E (T) = I_E(T)$.

1. $V_E \circ I_E$ is [[idempotent]] and [[covariant functor|covariant]].

This is summarized by saying that
$I_E \circ V_E$ and $V_E \circ I_E$ are _[[closure operators]]_ ([[idempotent monads]]).

=--

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

The first statement is immediate from the adjunction law (prop. \ref{GaloisConnectionAsAdjunction}).

Regarding the second statement:
This holds  because applied to sets $S$ of the form $I_E(T)$, we see $I_E(T) \subset I_E \circ V_E \circ I_E(T)$. But applying the contravariant map $I_E$ to the inclusion $T \subset V_E \circ I_E(T)$, we also have $I_E \circ V_E \circ I_E(T) \subset I_E(T)$.

This directly implies that the function $I_E \circ V_E$. is idempotent, hence the third statement.

The argument for $V_E \circ I_E$ is directly analogous.

=--

In view of prop. \ref{GaloisClosureOperator} we say that:

+-- {: .num_defn #GaloisClosedElements}
###### Definition
**(closed elements)**

Given a Galois connection induced from a relation as in def. \ref{GaloisConnectionFromRelation}, then

1. $S \in P(X)$ is called *closed* if $I_E \circ V_E(S) = S$;

1. the _closure_ of $S \in P(X)$ is $Cl(S) \coloneqq I_E \circ V_E(S)$

and similarly

1. $T \in P(Y)$ is called *closed* if $V_E \circ I_E(T) = T$;

1. the _closure_ of $T \in P(Y)$ is $Cl(T) \coloneqq V_E \circ I_E(T)$.

=--

It follows from the properties of [[closure operators]], hence form prop. \ref{GaloisClosureOperator}:

+-- {: .num_prop #GaloisFixedPoints}
###### Proposition
**([[fixed point of an adjunction|fixed points]] of a [[Galois connection]])**

Given a Galois connection induced from a relation as in def. \ref{GaloisConnectionFromRelation}, then

1. the closed elements of $P(X)$ are precisely those in the [[image]] $im(I_E)$ of $I_E$;

1. the closed elements of $P(Y)$ are precisely those in the [[image]] $im(V_E)$ of $V_E$.

We says these are the _[[fixed point of an adjunction|fixed points]]_ of the Galois connection. Therefore the restriction of the Galois connection

$$
  P(X)
    \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
  P(Y)^{op}
$$

to these fixed points yields an [[equivalence of categories|equivalence]]

$$
  im(I_E)
    \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\simeq}
  im(V_E)^{op}
$$

now called a _[[Galois correspondence]]_.

=--

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

Given a Galois connection induced from a relation as in def. \ref{GaloisConnectionFromRelation}, then the sets of closed elements according to def. \ref{GaloisClosedElements} are closed under forming [[intersections]].

=--

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

If $\{T_i \in P(Y)\}_{i: I}$ is a collection of elements closed under the operator $K = V_E \circ I_E$, then by the first item in prop. \ref{GaloisClosureOperator} it is automatic that $\bigcap_{i: I} T_i \subset K(\bigcap_{i: I} T_i)$, so it suffices to prove the reverse inclusion. But since $\bigcap_{i: I} T_i \subset T_i$ for all $i$ and $K$ is covariant and $T_i$ is closed, we have $K(\bigcap_{i: I} T_i) \subset K(T_i) \subset T_i$ for all $i$, and $K(\bigcap_{i: I} T_i) \subset \bigcap_{i: I} T_i$ follows.

=--

## Properties

### Galois connections between power sets
 {#BetweenPowerSets}

*Every* Galois connection between full [[power sets]],

$$(f: P(X) \to P(Y)^{op}) \dashv (g: P(Y)^{op} \to P(X))$$

is of the form in def. \ref{GaloisConnectionFromRelation} [above](#InducedFromARelation): there is some [[binary relation]] $r$ from $X$ to $Y$ such that

$$f(S) = \{y: \forall_{x \in X} x \in S \Rightarrow r(x, y)\}, \qquad g(T) = \{x: \forall_{y \in Y} y \in T \Rightarrow r(x, y)\}$$

Indeed, define $r: X \times Y \to \mathbf{2}$ by stipulating that $r(x, y)$ is true if and only if $y \in f(\{x\})$. Because $f$ is a left adjoint, it takes colimits in $P(X)$ (in this case, unions) to colimits in $P(Y)^{op}$, which are intersections in $P(Y)$. Since every $S$ in $P(X)$ is a union of singletons $\{x\}$, this gives

$$f(S) = \bigcap_{x \in S} f(\{x\}) = \{y: \forall_{x \in S} r(x, y)\}$$

which is another way of writing the formula for $f$ given above. We observe that

$$T \subseteq f(S) = \{y: \forall_{x \in S} r(x, y)\}$$

if and only if

$$S \times T \subseteq r$$

(now viewing $r$ extensionally in terms of subsets). This last symmetrical expression in $S$ and $T$ means

$$S \subseteq g(T) := \{x: \forall_{y \in T} r(x, y)\}$$

which means we have a Galois connection between $f$ and $g$ under this definition; since $g$ is uniquely determined by the presence of a Galois connection with $f$, we conclude that all Galois connections between power sets arise in this way, via a relation $r$ between $X$ and $Y$.

## Related concepts

* [[modality]]
* [[adjoint modality]]
* [[adjunction]]
* Every Galois connection is an [[idempotent adjunction]].
* [[nucleus of a profunctor]]

## References

The concept is due to

* {#Ore44} Øystein Ore, _Galois connexions_ , Trans. AMS **55** (1944) pp.493-513. ([pdf](http://www.ams.org/journals/tran/1944-055-00/S0002-9947-1944-0010555-7/S0002-9947-1944-0010555-7.pdf))

Introduction is in

* M. Erné, E. Koslowski, A. Melton, G. E. Strecker, _A primer in Galois connections_ , Annals New York Academy of Sciences **704** (1993) pp.103-125. ([draft](http://www.math.ksu.edu/~strecker/primer.ps))

See also

* Wojciech Dzik, Jouni Järvinen, Michiro Kondo, _Characterising intermediate tense logics in terms of Galois connections_ ([arXiv:1401.7646](http://arxiv.org/abs/1401.7646)).

[[!redirects Galois connection]]
[[!redirects Galois connections]]
[[!redirects Galois correspondence]]
[[!redirects Galois correspondences]]


Galois cover

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The incarnation of [[principal bundle]] for a [[finite group]] in [[algebraic geometry]].

## Related concepts

* [[Galois theory]],

## References

* [[James Milne]], section 6 of _[[Lectures on Étale Cohomology]]_

[[!redirects Galois covers]]

[[!redirects Galois covering]]
[[!redirects Galois coverings]]


Galois deformation ring

[[!redirects Galois deformation rings]]

#Contents#
* table of contents
{:toc}

## Idea

A *Galois deformation ring* is the [[ring]] that [[representable functor|represents]] the [[deformation functor]] $\mathrm{Def}_{\overline{\rho}}$ that assigns to a complete [[Noetherian ring|Noetherian]] local $W(\mathbb{F})$-algebra $A$ the set of deformations (equivalence classes of lifts) of a fixed residual representation $\overline{\rho}:\Gal(\overline{F}/F)\to GL_{n}(\mathbb{F})$ to $A$.

They were first introduced by Mazur, and find application in modularity, i.e. showing that a [[Galois representation]] comes from a [[modular form]]. As such they are instrumental in the proof of the [[Taniyama-Shimura conjecture]].

## Definitions

Let $F$ be a [[number field]], let $\mathbb{F}$ be a [[finite field]], with ring of [[Witt vectors]] $W(\mathbb{F})$, and let $\overline{\rho} \colon \Gal(\overline{F}/F)\to GL_{n}(\mathbb{F})$ be a [[linear representation]]. In reference to $\mathbb{F}$ being the [[residue field]] of $W(\mathbb{F})$ we refer to $\overline{\rho}$ as a *residual representation*.

\begin{definition}
Let $A$ be a complete Noetherian $W(\mathbb{F})$-algebra. A _lift_, or *framed deformation*, of $\overline{\rho}$ is a [[Galois representation]] $\rho \colon \Gal(\overline{F}/F)\to GL_{n}(A)$ such that reduction of $\rho$ by the unique [[maximal ideal]] $\mathfrak{m}$ of $A$ gives back $\overline{\rho}$.
\end{definition}

\begin{definition}
A *deformation* is an [[equivalence class]] of lifts, where two lifts are considered equivalent if they are [[conjugation action|conjugate]] by an element of the [[kernel]] of the reduction map.
\end{definition}

\begin{definition}\label{FramedDeformationFunctor}
The *framed deformation functor * $\mathrm{Def}_{\overline{\rho}}^{\Box}$ is the functor assigns to a complete Noetherian local algebra $A$ the set of lifts of $\overline{\rho}$ to $A$.
\end{definition}

\begin{theorem}
The framed deformation functor $\mathrm{Def}_{\overline{\rho}}^{\Box}$ (Def. \ref{FramedDeformationFunctor}) is represented by a lift $\rho(R_{\overline{\rho}}^{\Box})$. We refer to the ring $R_{\overline{\rho}}^{\Box}$ as the *universal framed deformation ring*.
\end{theorem}

\begin{definition}
The *deformation functor* $\mathrm{Def}_{\overline{\rho}}^{\Box}$ is the functor assigns to a complete Noetherian local algebra $A$ the set of lifts of $\overline{\rho}$ to $A$
\end{definition}

\begin{definition}
We say that $\overline{\rho}$ is *Schur* if $\mathrm{End}_{\mathbb{F}[\mathrm{Gal}(\overline{F}/F)]}\overline{\rho}=\mathbb{F}$.
\end{definition}

\begin{theorem}
Let $\overline{\rho}$ be Schur. Then the deformation functor $\mathrm{Def}_{\overline{\rho}}$ is represented by a deformation $\rho(R_{\overline{\rho}}^{\Box})$. We refer to the ring $R_{\overline{\rho}}$ as the *universal deformation ring*.
\end{theorem}

## Tangent spaces and Galois cohomology

(...)

## Derived version

A version of Galois deformation rings in [[derived geometry]] has been developed by [[Soren Galatius]] and [[Akshay Venkatesh]].

## Related

* [[deformation theory]]

## References

* [[Toby Gee]], _Modularity Lifting Theorems: Notes for Arizona Winter School_, [pdf](http://wwwf.imperial.ac.uk/~tsg/Index_files/ArizonaWinterSchool2013.pdf)

* [[Soren Galatius]] and [[Akshay Venkatesh]], _Derived Galois Deformation Rings_, [arxiv:1608.07236](https://arxiv.org/abs/1608.07236)

[[!redirects Galois deformation rings]]


Galois extension

Classically, "Galois extension" refers to a class of [[field extension|extensions of fields]].

An extension $K\subset L$ of [[field]]s is **Galois** if it is [[normal field extension|normal]] and [[separable field extension|separable]]. In that case, the [[automorphism group]] $Aut_K(L)$ of $K$-automorphisms of $L$ is called the [[Galois group]] and often denoted by $Gal(L:K)$.

There is a famous [[Galois theory]] for such extensions.

Now there are numerous generalizations, including [[Hopf-Galois extension]]s, coalgebra Galois extensions etc.


Galois group

> see at [[Galois theory]] for more

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[field extension]] one can consider the corresponding automorphism group.  The main statement of [[Galois theory]] is that, when the [[field extension]] is Galois, this [[group]] is called the Galois group and its [[subgroup]]s correspond to subextensions of the [[field extension]].

In [[algebraic geometry]], [[Grothendieck]] defined an analogue of the Galois group called the [[etale fundamental group]] of a connected [[scheme]].

Even more generally there is an analogue of the Galois group in [[stable homotopy theory]].  In fact one can define the Galois group of any [[presentable (infinity,1)-category|presentable]] [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[stable (infinity,1)-category]], and there is an analogue of the [[Galois theory|Galois correspondence]].  In particular one gets a Galois group associated to an [[E-infinity ring spectrum]].  One recovers the Galois group of a [[scheme]] as the Galois group of its [[derived category of quasi-coherent sheaves]].

## Definition

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

Let $K\hookrightarrow L$ denote a Galois [[field extension]], then the [[automorphism group]]

$$Gal(K\hookrightarrow L):=Aut_K(L)$$

consisting just of those automorphisms of $L$ whose restriction to $K$ is the identity is called *Galois group of the field extension $K\hookrightarrow L$.

=--

Every Galois group $Gal(K\hookrightarrow L)=lim_{K\hookrightarrow E\hookrightarrow L}Gal (K\hookrightarrow E)$  is a [[profinite group|profinite]] [[topological group]] in that it is the limit of the topologically discrete Galois groups of the intermediate finite extensions between $K$ and $L$.

The just defined Galois group is the one occurring in the classical [[Galois theory]] for fields. The analog of the Galois group in [[Galois theory|Galois theory for schemes]] is a [[fundamental group]] (of a scheme) and is rarely called a ''Galois group''.

The Galois group $Gal(K\hookrightarrow K_s)$ of the separable closure of $K$ is called the *[[absolute Galois group]]* of $K$. In this case we have $Gal(K\hookrightarrow K_s)\simeq \pi_1(Spec\; K)$ is equivalent to the [[fundamental group]] of the [[scheme]] $Spec K$. In particular the notion of [[fundamental group of a topos|fundamental group (of a point of) a topos]] generalizes that of Galois group. This observation is the starting point and motivating example of [[Grothendieck's Galois theory]] and more generally of that of [[homotopy groups in an (infinity,1)-topos]].

If the scheme, moreover, is a [[group scheme]] (i.e. endowed with a group structure) modules over the Galois group, which are called [[Galois module|Galois modules]], play an important role in [[algebraic number theory]].

## Properties

### Relation to étale fundamental group
 {#RelationToEtaleFundamentalGroup}

For $K$ a [[field]], then the absolute Galois group of $K$ is equivalent to the [[étale fundamental group]]/[[algebraic fundamental group]] of the [[spectrum of a commutative ring|spectrum]] of $K$.

$$
   \pi_1(Spec(K)) \simeq Gal(K_{sep}/K)
  \,.
$$

If $K$ is a [[number field]], write $\mathcal{O}_K$ for its [[ring of integers]], so that $Spec(\mathcal{O}_K)$ is an [[arithmetic curve]]. Then

$$
  \pi_1(Spec(\mathcal{O}_K)) \simeq Gal(K_{alg}^{ur}/K)
  \,,
$$

where $K_{alg}^{ur}$ is the maximal algebraic extension of $K$ that is [[unramified]] at all non-zero [[prime ideals]] (e.g. [Lenstra 85, Example 1.12](#Lenstra85)).

See also at _[Galois theory -- Statement of the main theorem](Galois+theory#StatementOfMainTheorem)_.

### Relation to Frobenius maps
 {#RelationToFrobeniusMaps}

For $K$ a [[number field]] then the [[Frobenius maps]] induce canonical elements in the Galois group. See at _[Frobenius morphism -- As elements of the Galois group](Frobenius+morphism#AsElementsOfGaloisGroup)_.

This crucially enters the definition of [[Artin L-functions]] associated with [[Galois representations]].

## Related concepts

* [[Galois theory]]

* [[motivic Galois group]]

## References

Monographs

* Julio R. Bastida, *Field Extensions and Galois Theory*, Cambridge University Press (1984) [[doi:10.1017/CBO9781107340749](https://doi.org/10.1017/CBO9781107340749)]

* {#Lenstra85} [[Hendrik Lenstra]], _Galois theory for schemes_ , Mathematisch Instituut Universiteit van Amsterdam (1985) [[pdf](http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf)]

For the Galois group in [[stable homotopy theory]], see

* [[Akhil Mathew]], The Galois group of a stable homotopy theory, [arXiv](http://arxiv.org/abs/1404.2156).

[[!redirects Galois groups]]

category: Galois theory


Galois module

#Contents#
* table of contents
{:toc}

## Idea

A *Galois module* $V$ is a $G$-[[module]] for a [[Galois group]] $G$; i.e. it is an abelian group on which a Galois group acts in a way compatible with the abelian group structure.

If $V$ is a _[[vector space]]_ then this is a [[linear representation]] of $G$ and one speaks of _[[Galois representation]]_.

The category of $G$-modules is equivalent to the category of [[module|modules]] over the [[group algebra|group ring]] $\mathbb{Z}[G]$.

As always is the case, a group [[action]] $G\times A\to A$ can equivalently be written as $G\to Aut(A)$. This is why Galois modules are frequently called *Galois representations*.

## Properties

+-- {: .num_proposition}
###### Proposition
Let $K\hookrightarrow L$ be a [[Galois extension]] of a number field $K$.

Then the ring of integers $O_L$ of this extension is a [[Galois module]] of $Gal(K\hookrightarrow L)$.

(see also [[Hilbert-Speiser theorem]])

=--

## Examples

+-- {: .num_example}
###### Example
($l$-adic representation)

Let $l$ be a prime number. Let $Gal(k\hookrightarrow \overline k)$ be the [[absolute Galois group]] of a [[number field]] $k$. Then a morphism of groups

$$Gal(k\hookrightarrow \overline k)\to Aut (M)$$

is called an *$l$-adic representation of $Gal(k\hookrightarrow \overline k)$. Here $M$ is either a unit dimensional [[vector space]] over the algebraic closure $\overline \mathbb{Q}_l$ or a finitely generated module over the [[integral closure]] $\overline \mathbb{Z}_l$.

In particular the $l$-adic Tate-module is of this kind.
=--

+-- {: .num_example}
###### Example
($l$-adic [[Tate module]])
Let $l$ be a prime number. Let $A$ be an abelian group. The *$l$-adic Tate module* is defined to be the limit

$$T_l(A)=lim_n \;ker (l^n)$$

i.e. it is the [[directed limit|limit over the directed diagram]] $ker(p^{n+1})\to ker(p^n)$. Here the [[kernel]] $ker(p^n)$ of the multiplication-with-$p^n$ map $p^n:A\to A$ is called $p^n$-[[torsion]] of $A$.
=--

+-- {: .num_example}
###### Example
(*the* Tate-module)

Let $k_S$ denote the separable closure of $k$. Let $A$ be the group of [[root of unity|roots of unity]] of $k_s$ in $k$. Then the $l$-adic Tate-module of the absolute Galois group $Gal(k\hookrightarrow k_s)$ is called *the $l$-adic Tate module of $k$* or the *$l$-adic cyclotomic character of $k$.

It is equivalently the Tate-module of the [[multiplicative group scheme]] $\mu_k$.

The Tate-module is endowed with the structure of a $\mathbb{Z}$-module by $z(a_n)_n=((z\; modulo\; p^n)a_n)_n$.
=--

+-- {: .num_example}
###### Example
($l$-adic [[Tate module]] of an abelian variety)

Let $l$ be a prime number. Let $G$ be an [[abelian variety]] over a field $k$. Let $k_s$ denote the separable closure of $k$. The $k_s$-valued points of $G$ assemble to an abelian group.

Then there are classical results on the [[rank]] of the Tate-module $T_l(G)$: For example if the characteristic of $k$ is a prime number $p\neq l$ we have that $T_l(G)$ is a free $\mathbb{Z}_l$ module of rank $2dim(G)$.

A special case of the [[Tate conjecture]] can be formulated via Tate-modules:

Let $k$ be finitely generated over its prime field of characteristic $p\neq l$. Let $A,B$ be two abelian varieties over $k$. Then the conjecture states that

$$hom(A,B)\otimes \mathbb{Z}_p\simeq hom(T_l(A),T_l(B))$$

If $k$ is a finite field or a number field the conjecture is true.
=--

+-- {: .num_example}
###### Example
([[p-adic cohomology|l-adic cohomology]] of a smooth variety)

Let $l$ be a prime number. Let $X$ be a [[variety||smooth variety]] over a field $k$ of characteristic prime to $l$. Let $k_s$ denote the separable closure of $k$.

The $l$-adic cohomology in degree $i$ is defined to be the directed limit $lim_n\; H^i_{et}(X_{k_s}, \mathbb{Z}/l^n\mathbb{Z})$. It is a Galois module where the action is given by pullback.

More specifically, given $\sigma\in Gal(k\hookrightarrow k_s)$ it acts on the $X_{k_s}=X\otimes_k k_s$ via the second factor. This is an isomorphism, since $\sigma$ is an automorphism, and hence $\sigma^*$ on cohomology is an isomorphism.

Note that since we have a equivalence $T_l A\simeq H_{et}^1(A_{k_s}, \mathbb{Z}_l)^\vee$, we have that the [[Tate module|l-adic Tate module]] is a special case of the $l$-adic cohomology.
=--

## Related concepts

* [[class field theory]]

* Every Galois representation induces an [[Artin L-function]].

## References

* Wikipedia, _[Galois module](http://en.wikipedia.org/wiki/Galois_module)_

* [[Richard Taylor]], _Galois representations_ ([pdf](http://math.stanford.edu/~lekheng/flt/taylor-long.pdf))

Review of the fact that Galois representations encode [[local systems]] are are hence analogs in [[arithmetic geometry]] of [[flat connections]] in [[differential geometry]] includes

* {#Lovering} Tom Lovering, _Étale cohomology and Galois Representations_, 2012 ([pdf](http://tlovering.files.wordpress.com/2012/06/essay-body1.pdf))

See also at _[[function field analogy]]_.

[[!redirects Galois modules]]


Galois representation

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

#Contents#
* table of contents
{:toc}

## Idea

A _Galois representation_ is a [[linear representation]] of a [[Galois group]] (often the [[absolute Galois group]] of some [[field]]). In other words, given a [[field extension]] $E$ of $F$, and a [[vector space]] $V$ over some field $k$, a Galois representation $\rho$ is a [[group homomorphism]]

$$\rho:\Gal(E/F)\to \GL(V)$$

where $\GL(V)$ is the group of [[linear transformations]] of $V$. If $V$ is an $n$-dimensional vector space, $\GL(V)$ is the same as the [[general linear group]] $\GL_{n}(k)$. A [[Galois module]] is a generalization of a Galois representation to [[modules]] instead of vector spaces.

## Continuous Galois representations

When the Galois group is the [[absolute Galois group]] of a [[number field]] or a finite extension of the [[p-adic numbers]], and the field of scalars of the underlying vector space of the representation is a [[topological field]], one often considers continuous Galois representations (noting that the absolute Galois group is a [[topological group]]).

## Examples

An example of a continuous Galois representation can be obtained by taking the $\ell$-adic Tate module as discussed in the article on [[Galois modules]] and taking its tensor product with $\mathbb{Q}_{\ell}$.

## p-adic Galois representations

Continuous Galois representations of $\mathrm{Gal}(\overline{F}/F)$, where $F$ is a finite extension of the [[p-adic numbers]] $\mathbb{Q}_{p}$, over an underlying vector space over $E$, where $E$ is another extension of $\mathbb{Q}_{p}$ (often with some condition that $E$ is big enough) are called _p-adic Galois representations_ (in contrast to $\ell$-adic Galois representations, where the underlying vector space is over some extension of $\mathbb{Q}_{\ell}$, where $\ell$ is another prime not equal to $p$). p-adic Galois representations have a very rich theory (significantly richer and more complicated than $\ell$-adic Galois representations), and their study is part of [[p-adic Hodge theory]].

## Conjectures on Galois Representations

### Fontaine-Mazur Conjecture

The Fontaine-Mazur conjecture gives a criterion for when an $\ell$-adic Galois representation "comes from geometry", i.e. is obtained from the $\ell$-adic [[etale cohomology]] of a variety. Namely, a Galois representation comes from geometry if and only if it is unramified at almost all places, and de Rham (see [[p-adic Hodge theory]]) at the places over $\ell$ ([LiLFunctions](#LiLFunctions)).

### Langlands Correspondence

A restricted version of the global [[Langlands correspondence]] ([BuzzardMSRI](#BuzzardMSRI)) for $\GL_{n}$ states that algebraic automorphic representations of $\GL_{n}$ over $\mathbb{Q}$ are in bijection with compatible systems of $\ell$-adic Galois representations. A more general form is conjectured for number fields instead of $\mathbb{Q}$.

The [[local Langlands correspondence]] for $\GL_{n}$ states that irreducible admissible representations of $\GL_{n}(\mathbb{Q}_{p})$ are in correspondence with F-semisimple [[Weil-Deligne representations]] of $\Gal(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$. This has been proved. A more general form is also true for more general local fields instead of $\mathbb{Q}_{p}$.

For reductive groups other than $\GL_{n}$, Galois representations need to be replaced by the more general concept of [[L-parameters]] (and the statement becomes a classification, in terms of a partition into L-packets, rather than a bijection).

### Serre's Modularity Conjecture

_Serre's modularity conjecture_ states that an odd (this means the image of complex conjugation has determinant $-1$) irreducible two-dimensional Galois representation over a [[finite field]] comes from a [[modular form]], and furthermore (in its strong form) gives a recipe for the level and weight of the modular form as well. This conjecture was proved by Khare and Wintenberger in [KhareWintenberger09a](#KhareWintenberger09a) and [KhareWintenberger09b](#KhareWintenberger09b).

Modern formulations of the weight part of Serre's conjecture are stated differently, inspired by the observation by Ash and Stevens ([AshStevens86](#AshStevens86)) that a Galois representation over a finite field is modular of prime-to-$p$ level $N$ and weight $k$ if and only if the corresponding system of Hecke eigenvalues appears in the [[group cohomology]] $H^1(\Gamma(N),\Sym^{k-2}\overline{\mathbb{F}}_{p}^{2})$. This is the same as requiring that the system of Hecke eigenvalues appears in $H^{1}(\Gamma(N),V)$, where $V$ is a Jordan-Holder factor of $\Sym^{k-2}\overline{\mathbb{F}}_{p}^{2}$.

Therefore, modern formulations of the weight part of Serre's conjecture consists of associating to a Galois representation over a finite field $\mathbb{F}_{p}$ a set of _Serre weights_, which are irreducible $\overline{\mathbb{F}}_{p}$ representations of $\GL_{2}(\mathbb{F}_{p})$ (there are also generalizations to other groups). See also [GHS18](#GHS18) for more discussion of this point of view.

## Related concepts

* [[class field theory]]

* [[etale cohomology]]

* [[Galois cohomology]]

* [[p-adic Hodge theory]]

* [[Langlands correspondence]]

## References

* [[Richard Taylor]], _Galois representations_ ([pdf](http://www.math.ias.edu/~rtaylor/longicm02.pdf))

In the context of the Langlands program:

* {#BuzzardMSRI} Kevin Buzzard, _MSRI Summer School on automorphic forms_ ([web](http://wwwf.imperial.ac.uk/~buzzard/MSRI/))

The statement of the Fontaine-Mazur conjecture as stated in this article comes from

* {#LiLFunctions} Chao Li, _Arithmetic of L-Functions_ (notes taken by Pak-Hin Lee) ([pdf](http://www.math.columbia.edu/~phlee/CourseNotes/L-functions.pdf))

The proof of Serre's modularity conjecture is given in

* {#KhareWintenberger09a} Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), _Serre's modularity conjecture (I)_, Inventiones Mathematicae, 178 (3): 485–504

* {#KhareWintenberger09b} Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586

The modern formulation of the weight part of Serre's conjecture is discussed in

* {#GHS18} [[Toby Gee]], [[Florian Herzig]], [[David Savitt]], _General Serre Weight Conjectures_, J. Eur. Math. Soc. 20 (2018), no. 12, 2859-2949 ([arXiv:1509.02527](https://arxiv.org/abs/1509.02527))

The historical inspiration for the previous entry can be found in

* {#AshStevens86} Avner Ash and Glenn Stevens, _Modular forms in characteristic l and special values of their L-functions_, Duke Math. J. 53 (1986), no. 3, 849–868.

Review of the fact that Galois representations encode [[local systems]] are are hence analogs in [[arithmetic geometry]] of [[flat connections]] in [[differential geometry]] includes

* {#Lovering} Tom Lovering, _Étale cohomology and Galois Representations_, 2012 ([pdf](http://tlovering.files.wordpress.com/2012/06/essay-body1.pdf))

See also at _[[function field analogy]]_.

* [[Peter Scholze]], _Locally symmetric spaces, and Galois representations_, Harvard CMD conference 2015, [yt](https://www.youtube.com/watch?v=4NCk8ItLMR0)

[[!redirects Galois representations]]


Galois Theories

This entry is about the monograph:

* {#BorceuxJanelidze2001}

  [[Francis Borceux]] and [[George Janelidze]]:

  \linebreak

  **Galois Theories**

  \linebreak

  Camb. Stud. Adv. Math. **72**

  Cambridge University Press (2001)

  [doi:10.1017/CBO9780511619939](https://doi.org/10.1017/CBO9780511619939)

on [[Galois theory]].

> The following starts out with an edited and adapted version of a review of the book by [[Tim Porter]] that appeared in Proceedings of the Edinburgh Mathematical Society (Series 2)  (2002), 45 : 761-762.

The developments  that lead to this book are due to some deeply based analogies between some initially very different looking mathematical theories, one in algebra, another in topology.

Galois originally developed some of elements of what was to become Galois theory in an attempt to understand polynomial equations, continuing work of Abel and others.  In modern language, working over a base field,  $K$, a field extension $K\subset L$ is a Galois extension when every element of $L$ is the root of a polynomial in $K[X]$, which factors  in $L[X]$ into linear factors and all of whose roots are simple.  The group $Gal[L:K]$ of this extension is the group of field automorphisms of $L$ which fix all elements of $K$ and the classical Galois theorem asserts that when $L$, considered as a $K$-vector space, is finite dimensional, the subgroups $G\subseteq Gal[L:K]$ of the Galois group classify the intermediate field extensions $K\subseteq M\subseteq L$.  The themes to note in this classical theory are (i) splitting into simpler structures, (ii) groups of automorphisms and (iii) intermediate structures classified by subgroups.

The second theory is that of [[covering space|covering spaces]] in topology.  A covering map
$\alpha : X\to B$ is one that has the property that every point of $B$ has an
open neighbourhood whose inverse image by $\alpha$ is a disjoint union of open
subsets each of which is mapped homeomorphically onto it by $\alpha$.  Given
any reasonably 'locally nice' space, there is a universal connected covering
space, $p : \widetilde{B}\to B$, such that all connected covering spaces of
$B$ are quotients of $\widetilde{B}$.  Classification of the connected
coverings is by subgroups of the automorphism group of $p$.  Of course, this
automorphism group is isomorphic to the (Poincaré) fundamental group of $B$
(under suitable local conditions).

This topological theory of covering spaces has some similarities to Galois
theory. Again one has automorphism groups and a correspondence between
intermediate structures (this time quotients not subobjects) and perhaps some
notion of splitting - there is an open cover of $B$ over each part of which
the covering splits up as a family of isomorphic pieces.

These two theories, thus, do look vaguely similar, but automorphism groups are
very common in mathematics and even that sort of 'Galois' correspondence is
not that uncommon, so surely any similarities must not be due to anything
really deep! The story of how the deep connections between them became apparent is
quite long and here is not the place to explore it in any detail.  It involves
function spaces and Riemann surfaces as well as a lot else that is central to
modern pure maths (if you want a good source for the
theory see the beautiful book (see below) by Douady and Douady).

Borceux and Janelidze's book, _Galois theories_, traces a greatly extended mathematical path beyond that
semi-classical link between Galois theory and coverings, but starts at a fairly
elementary beginning. It describes classical Galois theory, then turns to its
extension to infinitary field extensions, to étale algebras that became the
foundation for the work of
Grothendieck on the fundamental group of schemes ([[SGA1]]). This was
at the same time Galois theory and covering space theory, although for spaces
for which there was no question of being able to define a fundamental group
using paths. (This idea was fundamental in Grothendieck's later attempts to develop a higher dimensional version of this Poincaré-Galois theory in his manuscript, [[Pursuing Stacks]].)

This  book explores the connections between these theories,
searching for further cases of the general 'scenario' and tries to strip back the
superficial structure to reveal aspects of what are the essential features of
all these theories.

The book assumes a certain
knowledge of algebra and general topology, together with some familiarity with
the __elementary__ language of category theory (categories, functors, natural
transformations, limits and adjoint functors).  Its starts
with a short trip through the theory of field extensions, then goes on to look
at Grothendieck's extension of this to algebras.

Chapter 3 handles infinitary Galois theory.  Here [[profinite space|profinite spaces]] and
[[profinite group|profinite groups]] are introduced.  They are also useful in the following chapter
where the extension of Galois theory to commutative rings (due to Magid,
see reference below) is treated.    The [[Pierce
spectrum]] and [[Stone duality]] are handled clearly and simply laying the base for
Magid's profinite Galois groupoid.

With that first layer of categorical abstraction (Janelidze's abstract
categorical Galois theory) in place, other applications can be explored.
Given the introduction to the area in this review, it may seem strange that
covering maps are only introduced in Chapter 6, but here they can be very neatly
described and handled, beautifully illustrating and enriching the earlier
abstraction.

The final chapter describes one of the most important recent advances in topos
theory, giving an introduction to the Joyal--Tierney classification of
Grothendieck toposes as sheaves on localic groupoids.  The book ends with a
look at other directions the theory has taken beyond those handled in detail.
This section is particularly valuable as it should set the scene for future
research.

It introduces many deep important
concepts of algebra and category theory, introduces, motivates and uses in an
interesting way.  As one would expect, this means that the later chapters are sometimes much
harder going than the earlier ones, but the writing and structure of the book is such that the transition is fairly gradual.

Some researchers will probably feel that some of the exercises should have been worked out in detail, but given the starting assumptions of the authors, and the reasonably good set of references, there were bound to be such omissions.

##Table of chapter titles.

Introduction;

1. Classical Galois theory;

2. Galois theory of Grothendieck;

3. Infinitary Galois theory;

4. Categorical Galois theory of commutative rings;

5. Categorical Galois theorem and factorization systems;

6. Covering maps;

7. Non-Galoisian Galois theory;

Appendix;

Bibliography;

Index.

## Related entries

* [[categorical Galois theory]]

* [[Grothendieck's Galois theory]]

## Related references

* Grothendieck et al., [[SGA1]]

* R. and A. Douady, _Algèbre et théories galoisiennes_,
  Fernand Nathan, 1977.

* A. R. Magid,  _The separable Galois theory of commutative rings_,

* Marcel Dekker, 1974.

category: reference
category: Galois theory


Galois theory

> see also at _[[Galois group]]_

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

# Contents
* table of contents
{:toc}

## Idea

Classical Galois theory classifies [[field extensions]]. It is a special case of a classification of [[locally constant sheaves]] in a [[topos]] by [[permutation representations]] of the [[fundamental groupoid]]/[[fundamental group]].

Even more generally one can define a [[Galois group]] associated to a [[presentable (infinity,1)-category|presentable]] [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[stable (infinity,1)-category]].  There is an analogue of the Galois correspondence in this setting, see [Mathew 14](#Mathew14).

## Classical Galois theory for fields
  {#ClassicalGaloisTheory}

We discuss the classical/traditional case of Galois theory, which concerns the classification of [[field extensions]]. Below in _[Galois theory for schemes](#GaloisTheoryForSchemes)_ and then in _[Galois theory in a topos](http://ncatlab.org/nlab/show/Galois+theory#GaloisInTopos)_ we discuss how this is a special case of a more general concept of Galois theory in a [[topos]].

### Galois theory of fields {#GaloisTheoryForFields}

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

We call a [[field extension]] $K \subset L$ a **Galois extension** if $K \subset L$ is algebraic and there exists a subgroup $G \subset Aut(L)$ of the [[automorphism group]] such that $K \simeq L^G$ is the field of elements that are invariant under $G$.

=--

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

If $K \subset L$ is a Galois extension, we define the **[[Galois group]]** to be

$$
  Gal(L/K) := Aut_K(L)
  \,.
$$

=--

This means that we have

$$
  K \simeq L^{Gal(L/K)}
  \,.
$$

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

Let $\bar K$ be a fixed [[algebraic closure]] of $K$. If $F \subset K[X] - \{0\}$ is any collection of non-zero [[polynomial]]s, the [[splitting field]] of $F$ over $K$ is the [[subfield]] of $\bar K$ generated by $K$ and the zeros of the polynomials in $F$.

We call $f \in K[X]- \{0\}$ **separable** if it has no multiple zero in $\bar K$.

We call $\alpha \in \bar K$ **separable over $K$** if the [[irreducible polynomial]] $f^\alpha_K$ of $\alpha$ over $K$ is separable.

A subextension $L \subset \bar K$ is called **separable over $K$** if each $\alpha \in L$ is separable over $K$.

=--

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

We call $L$ **normal** over $K$ if for each $\alpha \in L$ the polynomial $f^\alpha_K$ splits completely in linear factors in $L[X]$.

=--

+-- {: .num_theorem }
###### Theorem

Let $K$ be a [[field]] and $L$ a [[subfield]] of $\bar K$

$$
 K \subset L \subset \bar K
 \,.
$$

Denote by $I$ the set of [[subfields]] $E$ of $L$ for which $E$ is a finite Galois extension of $K$. Then $I$, when [[poset|partially ordered]]  by inclusion is a directed poset.

The following assertions are equivalent:

1. $L$ is a [Galois extension](#GaloisExtension) of $K$.

1. $L$ is [normal](#normal) and [separable](#separable) over $K$.

1. There is a set $F \subset K[X] - \{0\}$ of separable polynomials such that
   $L$ is the [splitting field](#separable) of $F$ over $K$.

1. $\coprod_{E \in I} E \simeq L$

If these conditions are satisfied, then there is a group [[isomorphism]]

$$
   Gal(L/K) \simeq {\lim_{\leftarrow}}_{E \in I} Gal(E/K)
  \,,
$$

where on the right we have the [[limit]] over the [[poset]] of [[subfield]] of the [[contravariant functor]] $E \mapsto Gal(E/K)$.

=--

Since each group $Gal(E/K)$ is finite, the above isomorphism can be used to equip $Gal(L/K)$ with a [[profinite space|profinite topology]] (i.e. take the limit in the category of [[topological groups]], where each $Gal(E/K)$ has the discrete topology), making it into a [[profinite group]].  We henceforth consider $Gal(L/K)$ as a profinite group in this way.

+-- {: .num_theorem }
###### Theorem

([[main theorem of classical Galois theory]])

Let $K \subset L$ be a [Galois extension](#GaloisExtension) of fields with [Galois group](#GaloisGroup) $G$. Then the intermediate fields of $K \subset L$ correspond bijectively to the closed subgroups of $G$.

More precisely, the maps

$$
  \{E | E\;is\;a\;subfield\;of\;L\;containing\;K\}
  \stackrel{\overset{\phi}{\to}}{\underset{\psi}{\leftarrow}}
   \{H|H\;is\;a\;closed\;subgroup\;of\;G\}
$$

defined by

$$
  \phi(E) = Aut_E(L)
$$

and

$$
  \psi(H) = L^H
$$

are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\{id_L\}$.

If $E$ corresponds to $H$, then we have

1. $K \subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$)

   $[E:K] \simeq index[G:H]$ if $H$ is open;

1. $E \subset L$ is Galois with $Gal(L/E) \simeq H$ (as [[topological group]]s);

1. for every $\sigma \in G$ we have that $\sigma[E]$ corresponds to $\sigma H \sigma^{-1}$;

1. $K \subset E$ is Galois precisely if $H$ is a [[normal subgroup]] of $G$;

   $Gal(E/K) \simeq G/H$ (as [[topological group]]s) if $K \subset E$ is Galois.

=--

This appears for instance as [Lenstra, theorem 2.3](#Lenstra).

This suggests that more fundamental than the subgroups of a Galois group $G$ are its quotients by subgroups, which can be identified with transitive $G$-sets.  This naturally raises the question of what corresponds to non-transitive $G$-sets.

### In terms of separable algebras

#### Free modules

Let $A$ be a commutative [[ring]] and $N$ a [[module]] over $A$.

A collection of elements $(w_i)_{i \in I}$ of $N$ is called a [[basis]] of $N$ (over $A$) if for every $x \in N$ there is a unique collection $(a_i)_{i \in I}$ of elements of $A$ such that $a_i = 0$ for all but finitely many $i \in I$ and $x = \sum_{i \in I} a_i w_i$.

If $N$ has a basis it is called _[[free module|free]]_ (over $A$). If $N$ is free with basis a [[finite set]] of [[cardinality]] $n$, then we say that $N$ is _free with [[rank]] $n$_ (over $A$). In this case, $N$ is a [[finitely generated]] free module.

Let $N$ be a finitely generated free $A$-module with basis $w_1, w_2, \cdots, w_n$ and let $f\colon N \to N$ be $A$-linear. Then

$$
  f(w_i) = \sum_{j = 1}^n a_{i j} w_j \;\;\;
  (1 \leq i \leq n)
$$

for certain $a_{i j} \in A$, and the [[trace]] $Tr(f)$ of $f$ is defined by

$$
  Tr(f) = \sum_{i = 1}^n a_{i i}
  \,.
$$

This is an element of $A$ that only depends on $f$, and not on the choice of basis. It is easily checked that the map $Tr : Hom_A(N,N) \to A$ is $A$-linear.

#### Separable algebras

Let $A$ be a [[ring]], $B$ an $A$-[[algebra]], and suppose that $B$ is free with finite rank $n$ as an $A$-module. For every $b \in B$ the map $mult_b\colon B \to B$ defined by $mult_b\colon x \mapsto b x$ is $A$-linear, and the [[trace]] $Tr(b)$ or $Tr_{B/A}(b)$ is defined to be $Tr(mult_b)$. The map $Tr\colon B \to A$ is easily seen to be $A$-linear and to satisfy $Tr(a) = n a$ for $a \in A$.

The $A$-[[module]] $Hom_A(B,A)$ (underlying which is the [[hom-set]] in the [[category]] of [[module]]s) is clearly free over $A$ with the same rank as $B$. Define the $A$-linear map $\phi\colon B \to Hom_A(B,A)$ by

$$
  \phi(x) : y \mapsto Tr(x y)
  \,,
$$

for $x, y \in B$.

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

If for an $A$-algebra $B$ the
the morphism $\phi$ is an [[isomorphism]] we call $B$ **separable over $A$**, or a **free separable $A$-algebra** if we wish to stress the condition that $B$ is finitely generated and free as an $A$-module.

=--

#### Separable closure

Recall the notion of [separable elements](#separable)

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

Let $K$ be a [[field]] and $\bar K$ an [[algebraic closure]] of $K$. The [[separable closure]]  $K_S$ of $K$ is defined by

$$
  K_S \simeq \{x \in \bar K | x \; is \; separable \; over \; K\}
  \,.
$$

=--

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

We have that $K_S$ is a [[subfield]] of $\bar K$ and that $K_S \simeq \bar K$ precisely if $K$ is a [[perfect field]], in particular if the [[characteristic]] of $K$ is 0.

From xyz it follows that the inclusion $K \subset K_S$ is Galois.

=--

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

The Galois group $Gal(K_S/K)$ is called the **[[absolute Galois group]]** of $K$.

=--

#### Galois theory for separable algebras

+-- {: .num_theorem}
###### Theorem

Let $K$ be a [[field]] and $\pi_1(Spec K)$ its [absolute Galois group](#AbsoluteGaloisGroup). Then there is an [[equivalence of categories]]

$$
  SAlg_K^{op} \simeq \pi_1(Spec K) Set
  \,.
$$

=--

## Galois theory for schemes
 {#GaloisTheoryForSchemes}

The classical Galois theory for fields is a special case of a general geometric/[[topos theory|topos theoretic]] statement about [[locally constant sheaves]] and the action of the [[fundamental group]] on their [[fiber]]s.

### Statement of the main theorem
 {#StatementOfMainTheorem}

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

A morphism $f : Y \to X$ of [[scheme]]s is a **finite [[étale morphism]]** if there exists a [[covering]] of $X$ by [[affine scheme|affine]] [[open subset]]s $U_i = Spec A_i$, such that

* for each $i$ the open subschemes $f^{-1}(U_i)$ of $Y$ is affine,

* and equal to $Spec B_i$, where $B_i$ is a free [separable](#SeparableAlgebra) $A_i$-algebra.

In this situation we also say that $f  : Y \to X$ is a **finite étale covering** of $X$.

A [[morphism]] from a finite étale covering $f : Y \to X$ to a finite étale covering $g : Z \to X$ is a morphism of schemes $h : Y \to Z$ such that $f = g \circ h$.

This defines the [[category]] $FEt_X$ of finite étale covers of $X$.

=--

+-- {: .num_theorem #TheoremGaloisTheoryForSchemes}
###### Theorem

Let $X$ be a [[connected]] [[scheme]]. Then there exists a [[profinite group]] $\pi_1(X)$ -- the [[fundamental group]] of $X$ --  uniquely determined up to [[isomorphism]], such that the [category of finite étale coverings](#FiniteEtMorphisms) $FEex$ is [[equivalence of categories|equivalent]] to the category $Fin \pi_1(X) Set$ of finite [[permutation representation]]s of $\pi_1(X)$ ([[finite set]]s, with the [[discrete topology]], on which $\pi_1(X)$ acts continuously).

=--

This appears for instance as [Lenstra, main theorem 1.11](#Lenstra). It is fully discussed in [[SGA]]1.

The [[profinite group]], $\pi_1(X)$, is often called the _[[étale fundamental group]]_ of the connected scheme $X$.  In [[SGA1]], Grothendieck also considers coverings with profinite fibres, and a profinitely enriched fundamental groupoid. In the above the actual group $\pi_1(X)$ depends on the choice of a _fibre functor_ given by a geometric point of $X$. Different choices of fibre functor produce isomorphic groups. Taking two such fibre functors yields a $\pi_1(X)$-torsor for either version of $\pi_1(X)$.  This is important in attacks on [[Grothendieck's section conjecture]].

+-- {: .num_example}
###### Examples

* The [[disjoint union]] of $n$ copies of $X$ corresponds, under this theorem, to a finite set of $n$ elements on which $\pi_1(X)$ acts trivially.

* The fact that for $X = Spec \mathbb{Z}$ there are no other finite étale coverings of $X$ is thus expressed by the group $\pi_1(Spec \mathbb{Z})$ being _trivial_ .

* The same is true for $\pi_1(Spec K)$, where $K$ is an [[algebraically closed field]].

* If $K$ is an arbitrary [[field]], then $\pi_1(Spec K)$ is the [[absolute Galois group]] of $K$; i.e. the [[Galois group]] of the [separable closure](#SeparableClosure) $K_S$ over $K$. In this case theorem \ref{TheoremGaloisTheoryForSchemes} is a reformulation of [classical Galois theory](#ClassicalGaloisTheory).

* In particular, if $K$ is a finite field, then $\pi_1(Spec K) \simeq \hat \mathbb{Z}$.

* Let $X = Spec A$, where $A$ is the [[ring of integers]] in an [[algebraic number field]] $K$. Let $N$ be the maximal [[algebraic extension]] of $N$ that is [[unramified]] at all non-zero prime ideals of $A$. Then $\pi_1(X)$ is the Galois group of $N$ over $K$.

* More generally, if $a \in A$, $a \neq 0$, then $\pi_1(Spec A[\frac{1}{a}])$ is the Galois group, over $K$, of the maximal algebraic extension of $K$ that is unramified at all non-zero prime ideals of $A$ not dividing $a$.

=--

### Reproducing classical Galois theory of field extensions

In this section we explain the connection between the main theorem of Galois theory for schemes, theorem \ref{TheoremGaloisTheoryForSchemes}, and [classical Galois theory](#ClassicalGaloisTheory).

We denote by $k$ a [[field]]. It is our purpose to show that the [[opposite category]] of the category of free separable $K$-algebras is equivalent to the category of finite $\pi_1(X)$-sets, for a certain profinite group $\pi_1(X)$. This is a special case of the main theorem \ref{TheoremGaloisTheoryForSchemes}, with $ X = Spec K$. In the general proof we shall use the contents of this section only for algebraically closed $K$. In that case, which is much simpler, the group $\pi_1(X)$ is trivial, so that the category of finite $\pi_1(X)$-sets is just the category of finite sets.

+-- {: .num_theorem }
###### Theorem

(...)

=--

##Grothendieck's Galois Theory

In [[SGA1]], Grothendieck introduced an abstract formulation of the above theory in terms of _Galois categories_.  A Galois category is a  category, $\mathcal{C}$, satisfying a small number of properties together with a _fibre functor_ $F: \mathcal{C}\to FinSet$, preserving those properties. The theory is more fully described in the entry on [[Grothendieck's Galois theory]].

## Galois theory in topos theory {#GaloisInTopos}

One notices that [classical Galois theory](#ClassicalGaloisTheory) has an equivalent reformulation in [[topos theory]]. That puts it into a wider general abstract context and leads to a topos-theoretic general Galois theory.

### Reformulation of classical Galois theory

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

The [[étale morphism]]s $f : Y \to X$ corresponds precisely to the [[locally constant sheaves]] on $X$ with respect to the [[étale topology]], in that it is equivalently a morphism for which there is an [[étale cover]] $\{U_i \to X\}$ such that $f$ is a [[constant sheaf]] on each $U_i$.

For $K$ a [[field]] let $Et(K)$ be its [[small site|small]] [[étale site]]. And

$$
  \mathcal{E} := Sh(Et(K))
$$

the [[sheaf topos]] over it. This topos is a

* [[local topos]];

* [[locally connected topos]];

* [[connected topos]].

Then Galois extensions of $K$ correspond precisely to the [[locally constant object]]s in $\mathcal{E}$. The full [[subcategory]] on them is the [[Galois topos]] $Gal(\mathcal{E}) \hookrightarrow \mathcal{E}$.

The Galois group is the [[fundamental group of a topos|fundamental group of the topos]].

=--

### Topos-theoretic Galois theory

Accordingly in [[topos theory]] Galois theory is generally about the classification of [[locally constant sheaves]]. The Galois group corresponds to the [[fundamental group of a topos|fundamental group of the topos]] .

(...)

### Higher topos theoretic Galois theory

In the context of [[higher topos theory]], there are accordingly higher analogs of Galois theory. According to [[shape of an (∞,1)-topos|shape theory]], any [[(∞,1)-topos]] $\mathbf{H}$ has an associated fundamental ∞-groupoid $\Pi(\mathbf{H})$, which in general is a [[pro-space]] whose $1$-truncation $\Pi_1(\mathbf{H})$ is the ordinary fundamental groupoid of the underlying [[1-topos]]. Classical topos-theoretic Galois theory states that locally constant sheaves (of sets) on a locally connected topos are equivalent to representations of $\Pi_1(\mathbf{H})$, i.e., functors $\Pi_1(\mathbf{H})\to Set$. This generalizes to higher topoi as follows:

+-- {: .num_theorem }
###### Theorem

Let $\mathbf{H}$ be a [[locally n-connected (n+1,1)-topos]], $-1\leq n\leq \infty$. Then there is an equivalence of categories

$$
\mathbf{H}^lc \simeq \Fun(\Pi_{n+1}(\mathbf{H}), n Grpd),
$$

where $\mathbf{H}^{lc}\subset\mathbf{H}$ is the subcategory of locally constant objects.

=--

This generalization of Galois theory is discussed in ([Grothendieck 75](#Grothendieck75), [Hoyois 13](#Hoyois13), [Hoyois 15](#Hoyois15)). For further discussion in the case $n=\infty$, see

* [[cohesive (infinity,1)-topos -- structures]] -- [Galois theory](http://ncatlab.org/nlab/show/cohesive+%28infinity,1%29-topos+--+structures#GaloisTheory)

##'La Longue Marche à travers la Théorie de Galois'

Between January and June 1981, Grothendieck wrote about 1600 manuscript pages of a work with the above title. The subject is the [absolute Galois group](#AbsoluteGaloisGroup), $Gal(\overline{\mathbb{Q}},\mathbb{Q})$ of the rational numbers and its **geometric** action on moduli spaces of Riemann surfaces. This will (one day be) discussed at [[Long March]]. Other entries that relate to this include [[anabelian geometry]], [[children's drawings]] (in other words _Dessins d'enfants_, which is the study of graphs embedded on surfaces, their classification and the link between this and Riemann surfaces) and the [[Grothendieck-Teichmuller group]].

The [[anabelian geometry|anabelian]] question is: _how much information about the isomorphism class of an algebraic variety, $X$ is contained in the étale fundamental group of $X$?_  Grothendieck calls varieties which are completely determined by their **étale fundamental group**, _anabelian varieties_.  His anabelian dream was to classify the anabelian varieties in all dimensions over all fields.  This can be seen to relate to questions of the étale homotopy types of varieties.

(...)

## Related entries

* [[Galois group]], [[motivic Galois group]]

* [[Galois connection]]

* [[categorical Galois theory]], [[Grothendieck's Galois theory]]

* [[Galois cover]]

* [[differential Galois theory]]

* [[Galois cohomology]]

* [[Lascar group]] (a Galois group of first order theories)

* [[fundamental theorem of covering spaces]]

* [[anabelian geometry]]

* [[geometric homotopy groups in an (∞,1)-topos]], [[fundamental group of a topos]], [[fundamental ∞-groupoid of an (∞,1)-topos]]

## References

Original articles:

* [[Michael Barr]], *Abstract Galois Theory*, J. Pure Appl. Algebra **19** (1980) 21–42 [[pdf](https://www.math.mcgill.ca/barr/papers/galois.pdf), [[Barr-AbstractGaloisI.pdf:file]]]

* [[Michael Barr]], *Abstract Galois Theory II*, J. Pure Appl. Algebra **25** 3 (1982) 227–247 \[doi:10.1016/0022-4049(82)90080-9, [pdf](https://www.math.mcgill.ca/barr/papers/galois2.pdf), [[Barr-AbstractGaloisII.pdf:file]]]

Introduction:

* {#Lenstra} [[Hendrik Lenstra]], _Galois theory for schemes_ , Mathematisch Instituut Universiteit van Amsterdam (1985) [[pdf](http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf)]

  > Some of the material above is taken from this.

* [[Ravi Vakil]], [[Kirsten Wickelgren]], _Universal covering spaces and fundamental groups in algebraic geometry as schemes_, ([pdf](http://math.stanford.edu/~vakil/files/VW2Dec1910.pdf)).

Monographs:

* {#BorceuxJanelidze2001} [[Francis Borceux]], [[George Janelidze]], _[[Galois Theories]]_, Camb. Stud. Adv. Math. 72, Cambridge: Cambridge University Press (2001)

* [[Jean-Pierre Tignol]], *Galois' Theory of Algebraic Equations*, World Scientific (2001) [[doi:10.1142/4628](https://doi.org/10.1142/4628)]

The locally simply connected case is discussed for instance in

* Marco Robalo, _Galois theory towards dessins d'enfants_, dissertation, [pdf](https://dspace.ist.utl.pt/bitstream/2295/575330/1/dissertacao.pdf)

Galois theory in a [[presentable (infinity,1)-category|presentable]] [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[stable (infinity,1)-category]] is developed in

* [[Akhil Mathew]], The Galois group of a stable homotopy theory, [arXiv](http://arxiv.org/abs/1404.2156).
{#Mathew14}

Galois theory in topos theory

* [[SGA4]], _Expose VIII, Proposition 2.1._

* {#Hoyois15} [[Marc Hoyois]], _Higher Galois theory_, J. Pure Appl. Alg. __222__:7, (2018) 1859-1877 [arXiv:1506.07155](http://arxiv.org/abs/1506.07155) [doi](https://doi.org/10.1016/j.jpaa.2017.08.010)

See also

* _[The Galois Theory Web Page](http://www.math.upenn.edu/~galois/)_

* [Wikipedia: Galois theory](http://en.wikipedia.org/wiki/Galois_theory)

* Tamás Szamuely, _Galois groups and fundamental groups_, Cambridge Studies in Adv. Math. __117__ (2009)

* {#Grothendieck75} [[Alexander Grothendieck]], letter to [[Larry Breen]] 17/2/1975 [pdf](https://agrothendieck.github.io/divers/LGB17275en.pdf), [scan](https://agrothendieck.github.io/divers/LGB17275scanen.pdf)

* [[Alexander Grothendieck]], La Longue Marche a Travers la Théorie de Galois [pdf](https://agrothendieck.github.io/galoispoincaregrothendieck/galois.pdf)

category:Galois theory
[[!redirects Galois theories]]

[[!redirects Galois extension]]
[[!redirects Galois extensions]]


Galois topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Galois topos_ formalizes the collection of [[locally constant sheaves]] that are classified by [[Galois theory]] in the connected and locally connected case.

## Definition

Let

$$
  (\Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \mathcal{S}
$$

be a [[topos]] sitting by its [[global section]] [[geometric morphism]] over a base $\mathcal{S}$.

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

For $X$ an [[object]] in $\mathcal{E}$, let $Aut_{\mathcal{E}}(X)$ be its [[automorphism group]] (in $\mathcal{S}$). Then $\Delta Aut(X)$ is canonically a [[group object]] in $\mathcal{E}$.

An [[inhabited object|inhabited]] [[object]] $X$ (the [[terminal object|terminal]] morphism $X \to *$ is an [[epimorphism]]) in  $\mathcal{E}$ is called a **Galois object** if it is a $\Delta Aut(X)$-[[torsor]]/[[principal bundle]] in $\mathcal{E}$, in that the canonical morphism

$$
  (Id,\rho) :  X \times \Delta Aut(X) \stackrel{}{\to} X \times X
$$

is an [[isomorphism]].

=--

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

Any Galois object is  a [[locally constant object]]: since $X \to *$ is epi we may take it as a cover $U = X \to *$ and then then above principality condition says that pulled back to this cover $X$ becomes constant.

=--

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

A **Galois topos** is a [[topos]] that is

* [[locally connected topos|locally connected]];

* [[connected topos|connected]];

* [[separator|separated]] by its Galois objects.

=--

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

Often a Galois topos is in addition required to be pointed.

=--

## Examples

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

For $\mathcal{E}$ [[connected topos|connected]] and
[[locally connected topos|locally connected]], the full [[subcategory]] generated by [[locally constant object]]s is a Galois topos.

=--

This appears as ([Dubuc, theorem 5.2.4](#Dubuc)).

## References

The definition appears in

* [[Alexander Grothendieck]], _[[SGA]]_ 1 (1960-61), Springer Lecture Notes in Mathematics 224 (1971).

* [[Ieke Moerdijk]], _Prodiscrete groups and Galois toposes_ Proc. Kon. Nederl. Akad. van Wetens. Series A, 92-2 (1989)

* {#Dubuc} [[Eduardo Dubuc]], _On the representation theory of Galois and Atomic Topoi_ ([arXiv:0208222](http://arxiv.org/abs/math/0208222))

[[!redirects Galois object]]


Galois type

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

#Contents#
* table of contents
{:toc}

## Idea

__Galois types__ are a generalization of types for [[abstract elementary class]]es with [[amalgamation property]].

Consider the [[type (in model theory)]] $tp(a/M)$ for a first order theory in larger uncountable saturated model $M'$ (of cardinality strictly larger than of the underlying language). By a standard result it can be identified with the orbit of $a$ under the automorphism group $Aut_M(M')$  of $M'$ fixing $M$ pointwise. This is taken as a definition for some infinitary theories in the case we have the monster model. A slightly weaker version can be rephrased in terms of amalgamation.

## Definition

Let  $(K,\lt)$ be an [[abstract elementary class]] of structures.

...

## Related entries

* [[amalgamation property]]

* [[type (in model theory)|type in model theory]]

## References

* [[Rami Grossberg]], _Classification theory for abstract elementary classes_ (section 6), Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics __302__, AMS, (2002) 165--204, [pdf](http://www.math.cmu.edu/~rami/Rami-NBilgi.pdf)
* Rami Grossberg, Monica VanDieren, _Galois-stability for tame abstract elementary classes_, Journal of Mathematical Logic __6__, No. 1 (2006) 25--49, [pdf](http://www.math.cmu.edu/~rami/tame.pdf)
[[!redirects Galois types]]


Galois' last letter

this page compiles material related to:

* [[Évariste Galois]]

  **letter to Auguste Chevallier**

  29 May 1832

  \linebreak

  scan of original: [wikipedia](https://commons.wikimedia.org/wiki/File:E._Galois_Letter.jpg)

  French original text: [web](http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament/)

  English translation: [pdf](https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100)

commonly known as *la lettre testament* or *Galois' last letter*, since, famously,  the morning after writing it the author died in a duel.

The letter announces Galois' solution to the "solvability of equations by radicals". It is commonly cited as the first printed appearance of the term *[[group]]* in modern mathematical practice (specifically: [[permutation group|group of permutations]]).

category: reference


game semantics

## Idea

In [[logic]], **game semantics** is used to provide a semantic interpretation of logical constructions in terms of strategies for opposing players (a *prover* and an *opponent*) to win a game which corresponds to some proposition.

For a brief history, see this [comment](https://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_computati_3.html#c005578) by [[Samson Abramsky]].

## Related pages

* [linear logic](linear+logic#game_semantics)

## References

* [[Wilfrid Hodges]], 2013, _Logic and Games_, ([SEP](https://plato.stanford.edu/entries/logic-games/))

A game theoretic account (explained in terms of 'dialogues') of provability was developed by [[Paul Lorenzen]], see

* [[Paul Lorenzen]], [[Kuno Lorenz]], 1978, _Dialogische Logik_, Wissenschaftliche Buchgesellschaft, Darmstadt.

For a survey of this approach see

* [[Walter Felscher]], 2002, _Dialogues as a foundation for intuitionistic logic_, ([pdf](http://content.schweitzer-online.de/static/catalog_manager/live/media_files/representation/zd_std_orig__zd_schw_orig/013/156/273/9781402002359_content_pdf_2.pdf))

* [[Martin Hyland]], _Game semantics_,  in A. Pitts and P. Dybjer, editors, Semantics and Logics of Computation, pages 131–182. Cambridge University Press, 1997. ([scanned pdf](https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/gamesemantics97scan.pdf))

* [[André Joyal]], 1977, _Remarques sur la théorie des jeux à deux personnes_, Gazette des Sciences Mathématiques du Québec 4, 46–52, translated as [Remarks on the Theory of Two-Player Games](https://bosker.files.wordpress.com/2010/12/joyal-games.pdf) by [[Robin Houston]].

* [[Andreas Blass]], Game semantics and linear logic, Annals of Pure and Applied Logic 56 (1992), 183-220.

* [[Samson Abramsky]], [[Radha Jagadeesan]], Games and full completeness for multiplicative linear logic, Journal of Symbolic Logic, 59 (1994), 543 - 574

* [[Martin Hyland]] and [[Luke Ong]], On full abstraction for PCF, Information and Computation, 163 (2000), 285-408.

* [[Paul-André Melliès]], Asynchronous games 3: An innocent model of linear logic, _Proceedings of the 10th Conference on Category Theory and Computer Science_, Copenhagen, 2004. ([pdf](https://www.irif.fr/~mellies/papers/ag3-ctcs.pdf))

* [[Robin Cockett]], [[Geoff Cruttwell]] and K. Saff, _Combinatorial Game Categories_, ([pdf](https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/CGC.pdf))

* [[John Baez]], [TWF240](http://math.ucr.edu/home/baez/week240.html) on 'Holodeck games'.

For attempts to formulate a game semantics for dependent type theory, see

* [[Matthijs Vákár]], [[Radha Jagadeesan]], [[Samson Abramsky]], _Game Semantics for Dependent Types_, ([pdf](https://users.ox.ac.uk/~magd3996/research/I%26C%20Submission.pdf))

* Norihiro Yamada, _Game Semantics for Martin-Löf Type Theory_, ([arXiv:1610.01669](https://arxiv.org/abs/1610.01669))