étale cohomology

[[!redirects etale cohomology]]

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

#Contents#
* table of contents
{:toc}

## Idea ##

Traditional _étale cohomology_ (e.g. [Deligne 77](#Deligne77)) is the [[abelian sheaf cohomology]] for [[sheaf|sheaves]] on the [[étale site]] of a [[scheme]] -- which is an analog of the [[category of open subsets]] of a [[topological space]] $X$ , or rather the analog of the category of [[étale spaces]] over $X$, with [[finite set|finite]] [[fibers]].

A certain [[inverse limit]] over étale cohomology groups for different [[coefficients]] yields [[ℓ-adic cohomology]], which is a [[Weil cohomology theory]].

More generally, there is étale [[generalized cohomology theory]] with [[coefficients]] in [[sheaves of spectra]] on the [[étale site]] ([Jardine 97](#Jardine97)). Still more generally, there is étale generalized cohomology on the [[étale (∞,1)-site]] ([Antieau-Gepner 12](#AntieauGepner12), [Lurie](#Lurie)).

## Definition ##

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

Given a [[scheme]] $X$ of finite type, the small [[étale site]] $X_{et}$ is the [[category]] whose [[objects]] are [[étale morphisms]] $Spec R \to X$ and whose morphisms $(f:Spec R\to X)\to (f':Spec R'\to X)$ are morphisms $\alpha: Spec(R)\to Spec(R')$ of schemes completing triangles: $f'\circ\alpha=f$ (notice that the morphisms between étale morphisms are automatically étale). This category naturally carries a [[Grothendieck topology]] that makes it a [[site]], the [[étale site]].

For $A \in Sh(X_{et}, Ab)$ an [[abelian sheaf]] on $X$, the **étale cohomology** $H_{et}^\bullet(X,A)$ of $X$ with [[coefficients]] in $A$  is the [[abelian sheaf cohomology]] with respect to this site.

=--

## Basic properties
 {#BasicProperties}

The following are some basic properties of étale [[cohomology groups]] for various standard choices of [[coefficients]].

### Relation to Zariski cohomology
 {#RelationZariskiEtaleCohomology}

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

A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every étale [[sheaf]] is a Zariski sheaf, but not necessarily conversely.

=--

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

For $X$ a [[scheme]], the inclusion

$$
  \epsilon \;\colon\; X_{Zar} \longrightarrow X_{et}
$$

of the [[Zariski site]] into the [[étale site]] is indeed a [[morphism of sites]]. Hence there is a [[Leray spectral sequence]] which computes étale cohomology in terms of Zariski cohomology

$$
  E^{p,q}_2 = H^p(X_{Zar}, R^q \epsilon^\ast \mathcal{F})
  \Rightarrow
  E^{p+q} = H^{p+q}(X_{et}, \mathcal{F})
  \,.
$$

=--

This is originally due to ([[Grothendieck]], [[SGA]] 4 (Chapter VII, p355)). Reviews include ([Tamme, II 1.3](#Tamme)).

### With coefficients in coherent modules
 {#WithCoefficientsInCoherentModules}

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

For $N$ a [[quasi-coherent sheaf]] of $\mathcal{O}_X$-[[modules]] and $N_{et}$ the induced étale sheaf (by the discussion at [étale topos -- Quasicohetent sheaves](etale+topos#QuasiCoherentModules)),
then the [[edge morphism]]

$$
  H^p_{Zar}(X, N)
  \longrightarrow
  H^p_{et}(X,N_{et})
$$

of the [[Leray spectral sequence]] of remark \ref{LerayForInclusionOfZariskiIntoEtale} is an  [[isomorphism]] for all $p$, itentifying the [[abelian sheaf cohomology]] on the [[Zariski site]] with [[coefficients]] in $N$ with the étale cohomology with coefficients in $N_{et}$.

Moreover, for $X$ affine we have

$$
  H^p_{et}(X, N_{et}) \simeq 0
  \,.
$$

=--

This is due to ([[Grothendieck]], [[FGA]] 1). See also for instance ([Tamme, II (4.1.2)](#Tamme)).

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

By the discussion at _[[edge morphism]]_ it suffices to show that

$$
  R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0
  \,.
$$

By the discussion at _[[direct image]]_ (also at _[[abelian sheaf cohomology]]_), $R^q \epsilon_\ast N$ is the [[sheaf]] on the [[Zariski topology]] which is the [[sheafification]] of the [[presheaf]] given by

$$
  U \mapsto H^q(X_{et}|U, N)
  \,,
$$

hence it is sufficient that this vanishes, or rather, by locality ([[sheafification]]) it suffices to show this vanishes for $X = U = Spec(A)$ an affine [[algebraic variety]].

By the existence of [cofinal affine étale covers](etale+site#CofinalAffineCovers) the [[full subcategory]] $X_{et}^{a} \hookrightarrow X_{at}$ with the induced [[coverage]] is a [[dense subsite]] of affines. Therefore it suffices to show the statement there.
Moreover, by the finiteness condition on [[étale morphisms]]  every cover of $X_{et}^{a}$ may be refined by a finite cover, hence by an affine covering map

$$
  Spec(B) \longrightarrow Spec(A)
  \,.
$$

It follows (by a discussion such as e.g. at [[Sweedler coring]]) that the corresponding [[Cech cohomology]] complex

$$
  N_{et}(Spec(A)) \to C^0(\{Spec(B) \to Spec(A)\}, N_{et}) \to C^1(\{Spec(B) \to Spec(A)\}, N_{et}) \to \cdots
$$

is of the form

$$
  0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots
  \,.
$$

known as the [[Amitsur complex]].

Since $A \to B$ is a [[faithfully flat morphism]] it follows by the [[descent theorem]] that this is [[exact sequence|exact]], hence that the cohomology indeed vanishes.

=--

### With coefficients in a cyclic group
 {#WithCoefficientsInACyclicGroup}

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

If $X = Spec(A)$ is an affine [[reduced scheme]] of [[characteristic]] a [[prime number]] $p$, then its [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/p\mathbb{Z}$ is

$$
  H^q(X, (\mathbb{Z}/p\mathbb{Z})_X)
  \simeq
  \left\{
    \array{
      A/(F - id)A & if\; q = 1
      \\
      0 & if \; q \gt 0
    }
  \right.
  \,.
$$

=--

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

Under the given assumptions, the [[Artin-Schreier sequence]] (see there) induces a [[long exact sequence in cohomology]] of the form

$$
  \begin{aligned}
    0 & \to
    H^0(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^0(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^0(X_{et}, \mathcal{O}_X)
   \\
    & \to
     H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^1(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^1(X_{et}, \mathcal{O}_X)
   \\
    & \to
     H^2(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^2(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^2(X_{et}, \mathcal{O}_X) \to \cdots
  \end{aligned}
  \,,
$$

where $F(-) = (-)^p$ is the [[Frobenius endomorphism]].
By prop. \ref{CohomologyWithCoeffsInCoherentModules}
the terms of the form $H^{p \geq 1}(X, \mathcal{O}_X)$ vanish,
and so from [[exact sequence|exactness]] we find an [[isomorphism]]

$$
  H^0(X_{et}, \mathcal{O}_X)/(F-id)(H^0(X_{et}, \mathcal{O}_X))
    \stackrel{\simeq}{\to}
  H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
  \,,
$$

hence the claimed isomorphism

$$
  A/(F-id)(A)
    \stackrel{\simeq}{\to}
  H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
  \,.
$$

By the same argument all the higher cohomology groups vanish, as claimed.

=--

### With coefficients in the multiplicative group

the étale cohomology groups with [[coefficients]] in the [[multiplicative group]] $\mathbb{G}_m$ in the first few degrees go by special names:

* $H^0_{et}(-, \mathbb{G}_m)$: [[group of units]];

* $H^1_{et}(-, \mathbb{G}_m)$: [[Picard group]] ([[Hilbert's theorem 90]], [Tamme, II 4.3.1](#Tamme));

* $H^2_{et}(-, \mathbb{G}_m)$: [[Brauer group]];

### With coefficients in groups of roots of unity

* [[Kummer sequence]]

([Tamme, II, 4.4](#Tamme))

(...)

## Main theorems
 {#MainTheorems}

The following are the main theorems characterizing properties of étale cohomology. Together these theorems imply that étale cohomology, in its variant as [[l-adic cohomology]], is a [[Weil cohomology theory]].

### Proper base change theorem

* [[proper base change theorem]]

([Milne, section 17](#Milne))

### Comparison theorem: Relation to singular cohomology

* [[comparison theorem (étale cohomology)]]

([Milne, section 21](#Milne))

### Künneth formula
 {#KünnethFormula}

([Milne, section 22](#Milne))

### Cycle map
 {#CycleMap}

([Milne, section 23](#Milne))

### Poincaré duality
 {#PoincareDuality}

([Milne, section 24](#Milne))

### Lefschetz fixed-point formula

[Künneth formula](#KünnethFormula) + [cycle map](#CycleMap) + [Poincaré duality](#PoincareDuality) $\Rightarrow$ [[Lefschetz fixed-point formula]]

([Milne, section 25](#Milne))

\section{Relation to motivic cohomology}

We have the following, which is Theorem 10.2 in [MazzaVoevodskyWeibel2006](#MazzaVoevodskyWeibel2006).

\begin{thm} Let $X$ be a smooth, separated [[scheme]] over a [[field]] $k$. Let $n$ be an integer prime to the characteristic of $k$. Then for every integer $p$ and every integer $q \geq 0$, we have that $H^{p,q}_{L}(X, \mathbb{Z} / n)$, the étale (or Lichtenbaum) motivic cohomology of $X$, is isomorphic to $H^{p}_{et}\left(X, \mathbb{Z}/n(q) \right)$, the [[étale cohomology]] of $X$ with coefficients in the $q$-th [[Tate twist]] of $\mathbb{Z}/n$. \end{thm}

The inverting of smashing with the [[Tate sphere]] as opposed to just with $S^{1}$ in the construction of the [[stable motivic homotopy category]] allows the $q$ part of [[motivic cohomology]] to be represented in it (by the motivic Eilenberg-MacLane spectrum), as opposed to just the $(p,0)$ part. Combining this with the above theorem, we see that inverting the Tate sphere as opposed to just $S^{1}$ allows the Tate twists of étale cohomology to be represented.

## Related concepts

* [[basics of étale cohomology]]

* [[étale morphism]], [[étale site]], étale cohomology

* [[étale (∞,1)-site]], [[étale topos]]

* [[étale homotopy]]

* [[Weil conjecture]]

* [[continuous étale cohomology]]

## References

### History, motivation and original accounts

Étale cohomology was conceived by [[Artin]], [[Deligne]], [[Alexander Grothendieck|Grothendieck]] and [[Verdier]] in 1963. It was used by Deligne to prove the [[Weil conjectures]]. Some useful (and also funny) remarks on this are in the beginning of

* [[Spencer Bloch]], Review of [[Milne]]'s _[[Étale Cohomology]]_ ([pdf](http://www.ams.org/bull/1981-04-02/S0273-0979-1981-14894-1/S0273-0979-1981-14894-1.pdf); publisher's [book page](http://www.worldscibooks.com/mathematics/7773.html))

See also

* MathOverflow _[Étale cohomology — Why study it?](http://mathoverflow.net/questions/6070/etale-cohomology-why-study-it)_

The classical references include [[SGA]], esp.

* [[Pierre Deligne]] et al., _Cohomologie étale_ , Lecture Notes in Mathematics __569__, Springer-Verlag, 1977.
{#Deligne77}

* [[James Milne]], _[[Étale Cohomology]]_, Princeton Mathematical Series __33__, 1980. xiii+323 pp.

See also

* [[Barry Mazur]], _Notes on étale cohomology of number fields_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 6 no. 4 (1973), p. 521-552  ([Numdam](http://www.numdam.org/item?id=ASENS_1973_4_6_4_521_0), [pdf](http://modular.math.washington.edu/edu/2010/582e/refs/mazur-notes_on_etale_cohomology_of_number_fields_original.pdf))

### Reviews and modern accounts

* [[Günter Tamme]], _[[Introduction to Étale Cohomology]]_, 1994
 {#Tamme}

A modern textbook, though largely based on the material in SGA is

* Lei Fu, _Étale cohomology theory_, Nankai Tracts in Math. __13__, World Sci. 2011; ([toc pdf](http://www.worldscibooks.com/etextbook/7773/7773_toc.pdf); Preface [pdf](http://www.worldscibooks.com/etextbook/7773/7773_preface.pdf); chap. 1 Descent theory [pdf](http://www.worldscibooks.com/etextbook/7773/7773_chap01.pdf))

Lecture notes include

* [[James Milne]], _[[Lectures on Étale Cohomology]]_ ([html](http://www.jmilne.org/math/CourseNotes/lec.html), [pdf](http://www.jmilne.org/math/CourseNotes/LEC.pdf))
 {#Milne}

* [[Aise Johan de Jong]], _Étale cohomology_ 2009, in _[[The Stacks Project]]_ ([pdf](http://math.columbia.edu/~pugin/Teaching/Etale_files/EtaleCohomology.pdf))
{#deJong}

* [[Evan Jenkins]], _Étale cohomology seminar_  ([web](http://math.uchicago.edu/~ejenkins/etale/))

* Donu Arapura, _An introduction to Étale cohomology_ ([pdf](http://www.math.purdue.edu/~dvb/preprints/etale.pdf))

* Antoine Ducros, _Étale cohomology of schemes and analytic spaces_, ([pdf](http://www.math.jussieu.fr/~ducros/Cohetale.pdf))

See also

* Edgar Costa, _Étale cohomology_ ([pdf](https://dspace.ist.utl.pt/bitstream/2295/686086/1/tese.pdf))

* Thomas H. Geisser, _Weil-etale motivic cohomology_, [K-th archive](http://www.math.illinois.edu/K-theory/0565)

Discussion of [[generalized cohomology theory]] on the [[étale site]] but with [[coefficients]] in [[sheaves of spectra]] is in

* [[Rick Jardine]], _Generalized Étale cohomology theories_, 1997 Progress in mathematics volume 146
 {#Jardine97}

Discussion of generalized étale cohomology over the [[étale (∞,1)-site]] (hence in [[higher topos theory]]/[[higher algebra]]) is in

* [[Benjamin Antieau]], [[David Gepner]], _Brauer groups and étale cohomology in derived algebraic geometry_ ([arXiv:1210.0290](http://arxiv.org/abs/1210.0290))

 {#AntieauGepner12}

* [[Jacob Lurie]], _Descent theorems_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf))
  {#Lurie}

\subsection{Other}

* {#MazzaVoevodskyWeibel2006} Mazza, [[Vladimir Voevodsky|Voevodsky]], [[Chuck Weibel|Weibel]], _Lecture notes on motivic cohomology_, Clay Mathematics Monographs 2. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. xiv, 216 p. (2006). [zentralblatt](https://zbmath.org/?q=an%3A1115.14010) [pdf](http://www.claymath.org/library/monographs/cmim-2-voevodsky-cov1.pdf)

[[!redirects étale cohomology]]

[[!redirects generalized étale cohomology]]
[[!redirects generalized étale cohomologies]]

[[!redirects generalized etale cohomology]]
[[!redirects generalized etale cohomologies]]


étale cover

[[!redirects etale cover]]

#Contents#
* table of contents
{:toc}

## Definition

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

An **étale cover** of an algebraic [[scheme]] $X$ is a set
$\{p_i : U_i \to X\}$ of [[étale morphism]]s locally of finite type
which are jointly surjective in the sense that $X$ equals the union of
set-theoretic images:

$$
  X = \union_i p_i(U_i).
$$

=--

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

The condition of being locally of finite type is just strengthening the variant of the  notion of étale: in the case of non-Noetherian schemes Grothendieck requires instead that étale morphisms be locally of finite presentation; for the purpose of [[étale topology]] locally of finite type is required.

=--

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

The [[étale site]] has [[covering]]s given by the étale covers.

=--

## Properties

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

Every étale cover is a [[cover]] in the [[fpqc topology]].

=--

This appears for instance as ([tag 03PH](https://stacks.math.columbia.edu/tag/03PH) of the stacks project).

## References

* [The Stacks project](https://stacks.math.columbia.edu)

[[!redirects etale covers]]

[[!redirects étale cover]]
[[!redirects étale covers]]

[[!redirects etale covering]]
[[!redirects etale coverings]]

[[!redirects étale covering]]
[[!redirects étale coverings]]


étale geometric morphism

[[!redirects etale geometric morphism]]

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

#Contents#
* table of contents
{:toc}

## Idea

A [[sheaf]] $F$ on (the [[site]] [[category of open subsets|of open subsets]] of) a [[topological space]] $X$ corresponds to an [[étalé space]] $\pi_F : Y_F \to X$.  This space $Y_F$ has itself a [[sheaf topos]] associated to it, and the map $Y_F \to X$ induces a [[geometric morphism]] of [[sheaf topos]]es

$$
  \pi_F : Sh(Y_F) \to Sh(X)
  \,.
$$

Due to the special nature of $Y_F$, the topos on the left is equivalent to the [[slice topos]] $Sh(X)/F$, and the projection morphism above factors through a canonical standard [[geometric morphism]] $Sh(X)/F \to Sh(X)$

$$
  \pi_F : Sh(Y_F) \stackrel{\simeq}{\to} Sh(X)/F \to Sh(X)
  \,.
$$

And conversely, every [[local homeomorphism]] $Y \to X$ of [[topological space]]s corresponds to a [[geometric morphism]] of sheaf toposes of this form.

This motivates calling a [[geometric morphism]]

$$
  \mathcal{X} \to \mathcal{Y}
$$

a **local homeomorphism of toposes** or **étale geometric morphism** if it factors as an equivalence followed by a projection out of an overcategory topos.

If the topos is a locally ringed topos, or more generally a [[structured (∞,1)-topos]], it makes sense to require additionally that the local homeomorphism is compatible with the extra structure.

## Definition

For $\mathbf{H}$ a [[topos]] (or [[(∞,1)-topos]], etc.) and for $X \in \mathbf{H}$ an [[object]], the [[overcategory]] $\mathbf{H}_{/X}$ is also a topos ($(\infty,1)$-topos, etc), the _[[slice topos]]_ ([[slice (∞,1)-topos]], ...).

The canonical projection $\pi_! : \mathbf{H}_{/X} \to \mathbf{H}$ is part of an [[essential geometric morphism|essential]] (in fact, [[locally connected geometric morphism|locally connected]]/ [[locally ∞-connected (∞,1)-topos|locally ∞-connected]]) geometric morphism:

$$
  \pi = (\pi_! \dashv \pi^* \dashv \pi_*) :
  \mathbf{H}_{/X}
   \stackrel{\overset{\pi_!}{\longrightarrow}}{\stackrel{\overset{\pi^*}{\longleftarrow}}{\underset{\pi_*}{\longrightarrow}}}
  \mathbf{H}
  \,.
$$

This is the [[base change geometric morphism]] for the terminal morphism $X \to *$.

### For toposes

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

A [[geometric morphism]] $\mathbf{K} \to \mathbf{H}$ is called a
**local homeomorphism of toposes**, or an **étale geometric morphism**, if it is equivalent to such a projection--- in other words, if it factors by geometric morphisms as $\mathbf{K} \stackrel{\simeq}{\to} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ for some $X \in \mathbf{H}$ .

=--

### For structured toposes

If the [[(∞,1)-topos]]es in question are [[structured (∞,1)-toposes]], then this is refined to the following

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

A morphism $f : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}})$ of [[structured (∞,1)-topos]]es is an **étale morphism** if

1. the underlying morphism of $(\infty,1)$-toposes is an étale geometric morphism;

1. the induced map $f^* \mathcal{O}_\mathcal{Y} \to \mathcal{O}_\mathcal{X}$
   is an equivalence.

=--

This is [[Structured Spaces|StSp, Def. 2.3.1]].

## Examples

If $\mathbf{H}$ is a [[localic topos]] $Sh(S)$ over a [[topological space]] $S$ we have that $X \in Sh(S)$ corresponds to an [[étalé space]] over $X$ and $\mathbf{H}/X \to \mathbf{H}$ to an [[étale map]].

If $\mathcal{G}$ is a [[geometry (for structured (∞,1)-toposes)]]
then for $f : U \to X$ an _admissible_ morphism in $\mathcal{G}$,
the induced morphism of [[structured (∞,1)-topos]]es

$$
  Spec^\mathcal{G} U \to Spec^{\mathcal{G}} X
$$

is an étale geometric morphism of structured $(\infty,1)$-toposes.

This is [[Structured Spaces|StrSp, example 2.3.8]].

## Properties

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

The [[inverse image]] of an étale geometric morphism is a [[cartesian closed functor]].

=--

See at _[[cartesian closed functor]]_ for proof.

Therefore

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

An étale geometric morphism is a cartesian [[Wirthmüller context]].

=--

+-- {: .num_prop}
###### Proposition
**(recognition of étale geometric morphisms)**

A [[geometric morphism]] $(f^* \dashv f_*) : \mathbf{K} \to \mathbf{H}$ is étale precisely if

1. it is [[essential geometric morphism|essential]];

1. $f_!$ is a [[conservative functor]];

1. For every [[diagram]] $X \to Y \leftarrow f_! Z$ in $\mathbf{H}$ the
   induced diagram

   $$
     \array{
       f_!(f^* X \times_{f^* Y} Z) &\to& f_! Z
       \\
       \downarrow && \downarrow
       \\
       X &\to& Y
     }
   $$

   is a [[pullback]] diagram.

=--

For [[(∞,1)-topos]]es this is [[Higher Topos Theory|HTT, prop. 6.3.5.11]].

+-- {: .num_prop #RecoveringAToposFromItsSlices}
###### Proposition
**(Recovering a topos from its etale overcategory)

For $\mathbf{H}$ an $(\infty,1)$-topos we have

$$
  \mathbf{H} \simeq ((\infty,1)Topos/\mathbf{H})_{et}
  \,,
$$

where $((\infty,1)Topos/\mathbf{H})_{et} \subset (\infty,1)Topos/\mathbf{H}$ is the full [[sub-(∞,1)-category]] of the [[over-(∞,1)-category]] on the etale geometric morphisms $\mathbf{K} \to \mathbf{H}$.

=--

This is [[Higher Topos Theory|HTT, remark 6.3.5.10]].

## Related concepts

* [[slice topos]], [[slice (∞,1)-topos]]

## References

The notion of local homeomorphisms of toposes is page 651 (chapter C3.3) of

* [[Peter Johnstone]], _[[Elephant|Sketches of an Elephant]]_ .

The notion of étale geometric morphisms between [[(∞,1)-topos]]es is introduced in section 6.3.5 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .

Discussion of the refinement to [[structured (∞,1)-topos]]es is in section 2.3 of

* [[Jacob Lurie]], _[[Structured Spaces]]_ .

[[!redirects etale geometric morphism]]
[[!redirects etale geometric morphisms]]
[[!redirects étale geometric morphisms]]
[[!redirects locally homeomorphic geometric morphism]]
[[!redirects local homeomorphism of toposes]]
[[!redirects local homeomorphism of topoi]]
[[!redirects local homeomorphisms of toposes]]
[[!redirects local homeomorphisms of topoi]]


étale groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A [[topological groupoid]] or [[Lie groupoid]] $C$ is called an **étale groupoid** if the [[domain|source]]-map $s : Mor C \to Obj C$ is a [[local homeomorphism]] or [[local diffeomorphism]], respectively, and hence exhibits the space of morphisms as an [[étale space]] over the space of objects.

=--

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

In the situation of def. \ref{PlainDefinition} it follows that all the other structure maps ([[codomain|target]], [[identity]], [[composition]]) are also local homeomorphisms, resp. local diffeomorphisms.

This means that an étale groupoid is equivalently an [[internal groupoid]] in the [[category]] whose objects are [[topological spaces]]/[[smooth manifold]]s and whose [[morphisms]] are local homeomorphisms/diffeomorphisms.

=--

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

Definition \ref{PlainDefinition} is not invariant under the general notion of [[equivalence]] of Lie groupoids, the equivalence between them regarded as [[smooth groupoids]], specifically as [[differentiable stacks]] ("[[Morita equivalence]]").

But it does make sense to take an **étale smooth groupoid** to be a [[smooth groupoid]]/[[differentiable stack]] which is equivalent, as such, to, hence is presented by an étale Lie groupoid as in def. \ref{PlainDefinition}.
This notion has been called **folitation groupoid** in ([Crainic-Moerdijk 00](#CrainicMoerdijk)).

=--

The following characterizes foliation groupoids

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

For a [[Lie groupoid]] $\mathcal{G}_\bullet$ the following are equivalent

1. $\mathcal{G}$ is a foliation groupoid, hence is equivalent, as a [[differentiable stack]] to an étale groupoid as in def. \ref{PlainDefinition};

1. The [[Lie algebroid]] $(\mathfrak{g},\mathcal{G}_0)$ which corresponds to $\mathcal{G}$ under [[Lie differentiation]] has an [[injection|injective]] [[anchor map]];

   hence the [[orbits]] of $\mathcal{G}$ form the [[leaves]] of a [[foliation]], the foliation whose leaves are tangent to the vectors in the image of this anchor map;

1. All [[isotropy groups]] of $\mathcal{G}_\bullet$ are [[discrete groups]].

=--

This is ([Crainic-Moerdijk 00, theorem 1](#CrainicMoerdijk)).

## Properties

### Cohomology and homology

In the literature one finds, roughly speaking, two different approaches to the study of étale groupoids. One approach is based on the construction of the convolution algebras associated to an étale groupoid, in the spirit of Connes' [[noncommutative geometry]], and involves the study of [[cyclic homology|cyclic]] and [[Hochschild homology]] and [[Hochschild cohomology|cohomology]] of these algebras. The other approach uses methods of algebraic topology such as the construction of the [[classifying space]] of an
étale groupoid and its [[abelian sheaf cohomology|(sheaf) cohomology groups]].

### Relation to Haefliger groupoids
 {#RelationToHaefligerGroupoids}

For $X_\bullet$ an étale groupoid, there is a canonical morphism

$$
  X_\bullet \longrightarrow \mathcal{H}(X_0)
$$

to the [[Haefliger groupoid]], example \ref{HaefligerGroupoid}, of its manifold of objects. The [[kernel]] of this map is the _ineffective part_ of $X_\bullet$. If the kernel vanishes, then $X$ is called an [[effective Lie groupoid]].

(e.g. [Carchedi 12, section 2.2](Carchedi12))

### Characterization by convolution Hopf algebroids (Gelfand duality)
 {#GelfandDuality}

The [[groupoid convolution algebra]] $C^\ast(\mathcal{G}_\bullet)$ of a [[Lie groupoid]] with its canonical [[atlas]] remembered has the structure of a [[Hopf algebroid]].
In ([Mrčun 99](#Mrcun99), [Kališnik-Mrčun 07](#KalisnikMrcun07)) étale Lie groupoids are characterized dually by their Hopf algebroids  (a refinement of [[Gelfand duality]] to [[noncommutative topology]]).

### Characterization by site of manifolds and étale maps
 {#CharacterizationBySiteOfManifolds}

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

The [[2-category]] of [[étale stacks]] with [[étale maps]] between them is equivalent to the [[2-topos]] over the [[site]] of [[smooth manifolds]] with [[local diffeomorphisms]] between them

=--

([Carchedi 12, theorem 3.4, corollary 3.3](Carchedi12))

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

A [[smooth stack]] is an étale stack precisely if it is in the [[essential image]] of the [[left Kan extension]] along the non-full inclusion of [[sites]]

$$
  SmthMfd^{et} \to SmthMfd
$$

of [[smooth manifolds]], with [[local diffeomorphisms]] on the left and all [[smooth functions]] on the right.

=--

([Carchedi 12, theorem 3.5, corollary 3.4](Carchedi12))

In particular:

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

A [[smooth stack]] is an [[effective Lie groupoid|effective]] étale stack precisely if under the prolongation of prop. \ref{EtaleStackAsProlongations} it is equivalent to the image of a [[sheaf]] (i.e. of a [[0-truncated]] stack).

=--

([Carchedi 12, corollary 4.1, corollary 4.2](Carchedi12))

### Formalization in differential cohesion
 {#FormalizationInDifferentialCohesion}

See at _[[differential cohesion]]_ the section _[Etale objects](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#EtaleObjects)_.

## Examples

Étale groupoids arise naturally as models for [[leaf spaces]] of [[foliations]], for [[orbifold]]s, and for [[orbit spaces]] of [[discrete group]] [[actions]].

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

Every [[topological space]] may be regarded as an étale groupoid with only identity morphisms.

=--

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

For $X$ a [[topological space]] and $\Gamma$ a [[discrete group]] with a continuous [[action]] $X \times \Gamma \to X$ on $X$, the [[action groupoid]] $X//\Gamma$ is étale.

=--

+-- {: .num_example #HaefligerGroupoid}
###### Example

The [[Haefliger groupoid]] $\Gamma^q$ has the [[Cartesian space]] $\mathbb{R}^q$ as its space of objects. A [[morphism]] $x \to y$ is a [[germ]] of a [[diffeomorphism]] $(\mathbb{R}^q,x) \to (\mathbb{R}^q, y)$.

This groupoid, and its [[geometric realization]] play a central role in [[foliation theory]].

=--

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

Every [[orbifold]] is an étale [[Lie groupoid]].

=--
## Related concepts

* [[étale space]]

* [[orbifold]], [[Deligne-Mumford stack]]

* [[étale ∞-groupoid]]

* [[proper Lie groupoid]], [[effective Lie groupoid]]

## References

A standard textbook account is section 5.5. of

* [[Ieke Moerdijk]], [[Janez Mrčun]], _[[Introduction to foliations and Lie groupoids]]_, Cambridge (2003)

The relation between étale groupoid and [[foliations]] is analyzed in detail in

* [[Marius Crainic]], [[Ieke Moerdijk]], _Foliation groupoids and their cyclic homology_ ([arXiv:math/0003119](http://arxiv.org/abs/math/0003119))
 {#CrainicMoerdijk}

See also at _[[orbifold]]_ for basic and introductory literature.

Further discussion of étale groupoids and their properties includes

* [[Marius Crainic]], [[Ieke Moerdijk]], _A Homology Theory for Étale groupoids_ ([journal](http://www.math.uiuc.edu/K-theory/0284/))

* [[David Carchedi]], _Sheaf Theory for Étale Geometric Stacks_ ([arXiv:1011.6070](http://arxiv.org/abs/1011.6070))

* {#Carchedi12} [[David Carchedi]], section 2.2, section 3 of _Étale Stacks as Prolongations_ ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))

The [[groupoid convolution algebra|convolution]]-[[Hopf algebroids]] of étale Lie groupoids have been characterized in

* [[Janez Mrčun]], _On spectral representation of coalgebras and Hopf algebroids_ ([arXiv:math/0208199](http://arxiv.org/abs/math/0208199))
 {#Mrcun99}

* [[Jure Kališnik]], [[Janez Mrčun]], _Equivalence between the Morita categories of etale Lie groupoids and of locally grouplike Hopf algebroids_ ([arXiv:math/0703374](http://arxiv.org/abs/math/0703374))
 {#KalisnikMrcun07}

[[!redirects étale groupoids]]

[[!redirects etale groupoid]]
[[!redirects etale groupoids]]

[[!redirects étale Lie groupoid]]
[[!redirects étale Lie groupoids]]

[[!redirects etale Lie groupoid]]
[[!redirects etale Lie groupoids]]

[[!redirects foliation groupoid]]
[[!redirects foliation groupoids]]

[[!redirects étale stack]]
[[!redirects étale stacks]]

[[!redirects etale stack]]
[[!redirects etale stacks]]


étale homotopy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _étale homotopy_ can be understood as a vast generalization of the following classical fact.

The [[nerve theorem]] says that for $X$ a [[paracompact topological space]] and $\{U_i \to X\}$ a [[good cover]] of $X$ by open subsets, then the [[simplicial set]] obtained from the [[Cech nerve]] of the covering by degreewise contracting all connected components to a point, presents the [[homotopy type]] of $X$.

If $X$ here is more generally a [[locally contractible space]] there is in general no notion of "good" enough open cover anymore. Instead, one can consider the above kind of construction for _all_ [[hypercovers]] and take the [[limit]] over the resulting simplicial sets. The classical theorem by [Artin-Mazur](#ArtinMazur) states that this still gives the [[homotopy type]] of $X$.

The construction itself, however, makes sense for arbitrary topological spaces and in fact for arbitrary [[sites]].

In the literature, particularly the _[[étale site]]_ is often considered and "étale homotopy" is often implicitly understood to take place over this site.

But the concept is much more general. In particular, one can understand the construction of the limit over contractions of [[hypercovers]] as a presentation of naturally defined [[(∞,1)-functors]] in [[(∞,1)-topos theory]].

Notably, if the given site is a a [[locally ∞-connected site]], then the étale homotopy construction computes precisely the [[derived functor]] that presents the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]. Many constructions in the literature can be understood as being explicit realizations of this simple general concept. Detailed discussion of this is at _[[geometric homotopy groups in an (∞,1)-topos]]_.

Even more generally, étale homotopy give the notion of [[shape of an (∞,1)-topos]]. (...)

## Examples
 {#Examples}

### Chevalley groups and Galois groups

For the special case of [[fundamental groups]], the concept of étale homotopy groups also goes by the name of _[[Chevalley fundamental groups]]_.

The étale fundamental group of a [[scheme]] is its [[absolute Galois group]]. See at _[Galois theory -- Statement of the main result](Galois+theory#StatementOfMainTheorem)_.

### Étale contractibility

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

For $k$ a [[field]] of [[characteristic]] 0, then the [[affine line]] $\mathbb{A}^1_k$ is étale [[contractible homotopy type|contractible]]. This is no longer the case in [[positive number|positive]] [[characteristic]].

=--

([HSS 13, section 1](#HSS13))

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

Let $k$ be an [[algebraically closed field]] of [[positive number|positive]] [[characteristic]]. Then the only [[smooth variety]] over $k$ which is étale [[contractible homotopy type|contractible]] is the point $Spec(k)$. In fact this is the only smooth variety which is [[n-connected object of an (infinity,1)-topos|2-connected]].

=--

([HSS 13, theorem 1](#HSS13))

## Related concepts

* [[Čech homology]]

* [[algebraic fundamental group]]

* [[anabelian geometry]]

* [[étale topos]], [[pro-étale topos]]

* [[étale cohomology]]

## References

### General

Original articles include

* [[Michael Artin]], [[Barry Mazur]], _Homotopy of varieties in the etale topology_, Proceedings of a Conference on Local Fields, Driebergen 1966, Springer.

* [[Michael Artin]], [[Barry Mazur]], _Étale Homotopy_, Springer Lecture Notes in Mathematics 100, Berlin (1969)
  {#ArtinMazur}

* [[Eric Friedlander]], _Fibrations in étale homotopy theory [(numdam)](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1973__42_/PMIHES_1973__42__5_0/PMIHES_1973__42__5_0.pdf)_, Publ. Math.
Inst. des Haut. Études Scient., 42, (1973), 5 – 46.

The modern perspective from the point of view of [[model structure on simplicial presheaves|model structures on simplicial presheaves]] is in

* {#Isaksen01} [[Daniel Isaksen]], _Étale realization of the $\mathbb{A}^1$-homotopy theory of schemes_, 2001 ([K-theory archive](http://www.math.uiuc.edu/K-theory/0495/))

* [[G. Quick]], _Stable étale realization and étale cobordism_, Adv. in Math.,
214 (2007), 730–760.

and fully abstractly from the point of view of  [[(∞,1)-topos]]-theory ([[shape of an (∞,1)-topos]]) in

* {#Hoyois13a} [[Marc Hoyois]], _Higher Galois theory_, ([pdf](http://math.mit.edu/~hoyois/papers/highergalois.pdf))

and ([Hoyois 13b, section 1](#Hoyois13b)).

and in, where a comparison theorem over the complex numbers is also proven:

* {#Carchedi15} [[Dave Carchedi]], _On the étale homotopy type of higher stacks_ ([arXiv:1511.07830](http://arxiv.org/abs/1511.07830))

An introduction is in

* Tomer Schlank, Alexei Skorobogatov, _A very brief introduction to étale homotopy_. In: "Torsors, étale homotopy and applications to rational points". LMS Lecture Note Series 405, Cambridge University Press, 2013. ([pdf](https://www.ma.imperial.ac.uk/~anskor/SCHLANK-SKOROBOGATOV.pdf))

Lecture notes on the étale fundamental group are in

* [[James Milne]], section 4 of _[[Lectures on Étale Cohomology]]
Generalization to [[simplicial schemes]] is discussed in

* [[Eric Friedlander]], 1982, _Étale homotopy of simplicial schemes_ , volume 104 of Annals of Mathematics Studies , Princeton University Press, Princeton, N.J.

More on this is in

* Michael Misamore, _Étale homotopy types and bisimplicial hypercovers_, Homology, Homotopy and Applications, Vol. 15 (2013), No. 1, pp.27-49. ([web](http://www.intlpress.com/HHA/v15/n1/a2/))

### Examples and applications

* {#Hoyois13b} [[Marc Hoyois]], _The étale-symmetric Künneth theorem_, 2013 ([pdf](http://math.mit.edu/~hoyois/papers/etalekunneth.pdf))

Discussion in [[positive characteristic]] is in

* {#HSS13} Armin Holschbach, Johannes Schmidt, [[Jakob Stix]], _Étale contractible varieties in positive characteristic_ ([arXiv:1310.2784](http://arxiv.org/abs/1310.2784))

Étale homotopy type of [[moduli stacks of curves]] is discussed in

* Paola Frediani, Frank Neumann, _Étale homotopy types of moduli stacks of algebraic curves with symmetries_, K-Theory 30: 315-340, 2003 ([arXiv:math/0404387](http://arxiv.org/abs/math/0404387))

[[!redirects étale homotopy theory]]

[[!redirects etale homotopy]]
[[!redirects etale homotopy theory]]

[[!redirects etale homotopy group]]
[[!redirects étale homotopy group]]
[[!redirects etale homotopy groups]]
[[!redirects étale homotopy groups]]

[[!redirects étale fundamental group]]
[[!redirects étale fundamental groups]]
[[!redirects etale fundamental group]]
[[!redirects etale fundamental groups]]

[[!redirects étale homotopy type]]
[[!redirects étale homotopy types]]

[[!redirects etale homotopy type]]
[[!redirects etale homotopy types]]


étale infinity-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _étale ∞-groupoid_ is meant to be an [[∞-groupoid]]-analog to an [[étale groupoid]].

For details see at _[[V-manifold]]_.

## Examples

* classifying étale stack for [[symplectic forms]]: [Carchedi 12, Example 2](#Carchedi12)

* classifying étale stack for [[complex structures]]: [Carchedi 12, p. 38](#Carchedi12)

## References

A formalization of the [[petit topos|petit]] [[(∞,1)-toposes]] corresponding to étale ∞-groupoids is in

* [[Jacob Lurie]], _[[Structured Spaces]]_.

A characterization of étale ∞-groupoids as objects in a [[big topos|big]] [[(∞,1)-topos]] is given in

* {#Carchedi12} [[David Carchedi]], _Étale Stacks as Prolongations_, Advances in Mathematics Volume 352, 20 August 2019, Pages 56-132 ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))

* [[David Carchedi]], _Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi_, Memoirs of the American Mathematical Society 2020; 120 ([arXiv:1312.2204](https://arxiv.org/abs/1312.2204), [ISBN:978-1-4704-5810-2](https://bookstore.ams.org/memo-264-1282))

and aspects of their [[geometric realization]]/[[shape modality]] are discussed in

* {#Carchedi15} [[David Carchedi]], _On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces_, Advances of Mathematics, Volume 294, 2016, Pages 756-818 ([arXiv:1504.02394](http://arxiv.org/abs/1504.02394))

* [[David Carchedi]], _On the étale homotopy type of higher stacks_ ([arXiv:1511.07830](https://arxiv.org/abs/1511.07830))

A formalization as [[V-manifolds]] in terms of [[differential cohesion]] is discussed at _[differential cohesion -- Structures -- cohesive étale ∞-groupoids](differential+cohesive+%28infinity%2C1%29-topos#EtaleObjects)_.

The formalization of this in [[homotopy type theory]] is discussed in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

[[!redirects étale infinity-groupoids]]

[[!redirects étale ∞-groupoid]]
[[!redirects étale ∞-groupoids]]

[[!redirects étale ∞-stack]]
[[!redirects étale ∞-stacks]]

[[!redirects etale infinity-groupoid]]
[[!redirects etale infinity-groupoids]]

[[!redirects étale infinity-stack]]
[[!redirects étale infinity-stacks]]

[[!redirects etale infinity-stack]]
[[!redirects etale infinity-stacks]]

[[!redirects etale ∞-groupoid]]
[[!redirects etale ∞-groupoids]]


étale map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Étale morphisms
+-- {: .hide}
[[!include etale morphisms - contents]]
=--
#### Geometry
+-- {: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The notion of _étale map_ is an abstraction of that of [[local homeomorphism]] in [[topology]].   The concept is usually found in places with a geometric or topological flavour.

## Examples

### Between topological spaces

An étale map between [[topological spaces]] is a  [[local homeomorphism]]; see [[étalé space]] (which is the total space of such a map viewed as a [[bundle]]).

### Between smooth manifolds

An étale map between [[smooth spaces]] is a [[local diffeomorphisms|local diffeomorphism]], which is in particular a [[local homeomorphism]] on the underlying [[topological spaces]].

### Between schemes (affine schemes / rings)

For an étale map between [[schemes]] see _[[étale morphism of schemes]]_.

Restricted to [[affine schemes]], this yields, [[Isbell duality|dually]], a notion of étale morphisms between [[rings]]. Étale maps between noncommutative [[rings]] have also been considered.

### Between $E_\infty$-rings

* [[étale morphism of E-∞ rings]]

### Between analytic spaces

*  Étale maps between [[analytic spaces]] are closely related to étale maps between schemes, while also (when the spaces are smooth) a special case of an étale map between smooth spaces.

+-- {: .query}
Zoran: I do not understand this statement. Analytic spaces have a different structure sheaf; in general nilpotent elements are allowed. This is additional structure not present in theory of smooth spaces.

Toby: Is it correct now?
=--

### Between toposes

* [[étale geometric morphism]]

### Between anabelioids

* [[finite étale morphism of anabelioids]]

## Axiomatization

The idea of _étale morphisms_ can be axiomatized in any [[topos]]. This idea goes back to lectures by [[André Joyal]] in the 1970s. See ([Joyal-Moerdijk 1994](#JoyalMoerdijk)) and ([Dubuc 2000](#Dubuc)).

## Related concepts

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

* [[separated morphism]]

* [[formally étale morphism]]

* [[log-étale morphism]]

## References

Axiomatizations of the notion of étale maps in general [[toposes]] are discussed in

* [[André Joyal]], [[Ieke Moerdijk]], _A completeness theorem for open maps_, Annals of Pure and Applied Logic __70__, Issue 1, 18 November 1994, p. 51-86, [MR95j:03104](http://www.ams.org/mathscinet-getitem?mr=1303663), doi
{#JoyalMoerdijk}

* [[Eduardo Dubuc]], _Axiomatic etal maps and a theory of spectrum_, Journal of pure and applied algebra __149__ (2000) 15--45
 {#Dubuc}

[[!redirects étale map]]
[[!redirects étale maps]]
[[!redirects etale map]]
[[!redirects etale maps]]
[[!redirects etalé map]]
[[!redirects etalé maps]]
[[!redirects étalé map]]
[[!redirects étalé maps]]

[[!redirects étale morphism]]
[[!redirects étale morphisms]]
[[!redirects etale morphism]]
[[!redirects etale morphisms]]


étale morphism of E-∞ rings

#Contents#
* table of contents
{:toc}

## Definition

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

A morphism $f \colon \mathbf{A} \to \mathbf{B}$ of [[E-∞ rings]] is an _étale morphism_ if the underlying homomorphism of [[commutative rings]] $\pi_0 f\colon \pi_0 \mathbf{A}\to \pi_0 \mathbf{B}$ is an [[étale morphism]], and the map $\pi_n(A) \otimes_{\pi_0(A)} \pi_0(B) \to \pi_n(B)$ is an isomorphism of abelian groups for every integer $n$.

=--

([Lurie, def.7.5.1.4](#Lurie))

## Properties

étale morphisms of underlying rings lift essentially uniquely to étale morphisms of [[E-∞ rings]]:

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

For $\mathbf{A}$ an [[E-∞ ring]] and $\pi_0 \mathbf{A} \to B$ a homomorphism to an ordinary ring $B$, then there is an essentially unique $E_\infty$-ring $\mathbf{B}$ with $\pi_0 \mathbf{B} \simeq B$ and étale morphism $\mathbf{A}\to \mathbf{B}$.

=--

([Lurie, theorem 7.5.0.6](#Lurie))

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

Proposition \ref{EssentiallyUniqueLiftsFromOrdinaryRings} is a central ingredient in the characterization of the moduli stack of [[derived elliptic curves]] as having underlying it the ordinaty [[moduli stack of elliptic curves]].

=--

+-- {: .num_remark #LocalizationOfRings}
###### Remark
**(localization of $E_\infty$-rings)**

Proposition \ref{EssentiallyUniqueLiftsFromOrdinaryRings} serves to lift [[localization of rings]] from rings to $E_\infty$-rings: for $\mathbf{A}$ an [[E-∞ ring]] and $a\in \pi_0 A$ an element, then the map $\pi_0 \mathbf{A} \to (\pi_0 \mathbf{A})[a^{-1}]$ of [[localization of a ring]] away from $a$ lifts to yield an [[E-∞ ring]] $\mathbf{A}[a^{-1}]$ with étale morphism $\mathbf{A} \to \mathbf{A}[a^{-1}]$. See also at _[[localization of a module]]_ for more on this.

=--

## References

* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_

[[!redirects étale morphisms of E-∞ rings]]

[[!redirects etale morphism of E-∞ rings]]


étale site

[[!redirects etale site]]

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

#Contents#
* table of contents
{:toc}

## Idea

The _étale site_ of a [[scheme]] is an analog of the [[category of open subsets]] of a [[topological space]]. The corresponding [[cohomology]] is [[étale cohomology]].

The étale topology has similar cohomological properties as the [[complex analytic geometry|complex analytic topology]], and in particular it is much finer for cohomological purposes than the [[Zariski topology]].

## Definition

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

Let $X$ be a [[scheme]].

The **[[big site|big]] étale [[site]]** $Sch_{/X,et}$ of $X$ is the [[over category]] $Sch_{/X}$ of [[scheme]]s over $X$ equipped with the [[coverage]] given by [[étale covers]] (after forgetting the maps to $X$).

The **[[small site|small]] étale [[site]]** $i : X_{et} \hookrightarrow Sch/{X,et}$ is the full [[subcategory]] of $Sch_{/X}$on the [[étale morphism]]s $U \to X$.

=--

The [[abelian sheaf cohomology]] of the étale site is called [[étale cohomology]].

## Properties

### Cofinal affine covers
  {#CofinalAffineCovers}

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

For $X = Spec(A)$ an [[affine scheme]] and $\{Y_i  \to X\}$
an étale cover, then there exists a refinement to an étale cover
$\{U_i  \to X\}$ such that each $U_i$ is an [[affine scheme]].

=--

### Cohomology

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

The [[inverse image]] restriction functor $i^* Sh(Sch_{/X,et}, Ab) \to Sh(X_{et}, Ab)$ on the [[categories of sheaves]] with values in [[Ab]]

* is an [[exact functor]]

* maps [[injective object]]s to injective objects.

=--

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

For $X$ a [[scheme]] and $F \in Sh(Sch_{/{X,et}}, Ab)$ an [[abelian sheaf]] on its big site, then the [[etale cohomology]] of $X$ with [[coefficients]] in $F$ may equivalently be computed on the small site:

$$
  H^p(X_{et}, F|_{et}) \simeq H^p(X,F)
  \,.
$$

=--

This appears for instance in ([deJong, prop. 3.4](#deJong)).

### Derived geometry

The [[derived geometry]] of the étale site is the [[étale (∞,1)-site]]. The precise statement is at derived étale geometry.

## Related concepts

* [[étale morphism]], **étale site**, [[étale topos]], [[étale cohomology]], [[étale homotopy]]

* The [[lisse-étale site]] of some $X$ consists of [[smooth morphisms]] $U \to X$.

[[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ **étale site** $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]

* [[pro-étale site]]

* [[Weil-étale topology for arithmetic schemes]]

* [[étale (∞,1)-site]]

* [[constructible sheaf]]

* [[étale homotopy]]

## References

The classical references are

* [[Pierre Deligne]] et al., _Cohomologie étale_ , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.
{#Deligne}

Textbooks include

* [[Günter Tamme]], _[[Introduction to Étale Cohomology]]_

* [[James Milne]], _Etale cohomology_, Princeton Mathematical Series __33__, 1980. xiii+323 pp.

A detailed survey is in chapter 34 of

* [[Aise Johan de Jong]], _[[The Stacks Project]]_

Lecture notes include

* [[James Milne]], _Lectures on etale cohomology_ ([pdf](http://www.jmilne.org/math/CourseNotes/LEC.pdf))

* {#deJong} [[Aise Johan de Jong]], _Étale cohomology_ ([pdf](http://math.columbia.edu/~pugin/Teaching/Etale_files/EtaleCohomology.pdf))

A variant, the _[[pro-étale site]]_ (locally contractible in some sense) is discussed in

* [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))

[[!redirects étale site]]

[[!redirects etale sites]]
[[!redirects étale sites]]

[[!redirects small étale site]]
[[!redirects small etale sites]]

[[!redirects small étale sites]]
[[!redirects small etale sites]]

[[!redirects big étale site]]
[[!redirects big etale sites]]

[[!redirects big étale sites]]
[[!redirects big etale sites]]

[[!redirects étale topology]]
[[!redirects etale topology]]

[[!redirects étale topologies]]
[[!redirects etale topologies]]


étale space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### étale morphisms
+-- {: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Let [[Top]] be a category of [[topological space]]s and $B$ an object in $Top$ (the 'base' space). The [[over category|slice category]] $Top/B$ is called the category of (topological) spaces over $B$ (or sometimes simply [[bundles]]).

An __étale space__ (or _[[étale map]]_, sometimes *étalé space*) over $B$ is an object $p:E\to B$ in $Top/B$ such that $p$ is a [[local homeomorphism]]: that is, for every $e\in E$, there is an open set $U \ni e$ such that the [[image]] $p(U)$ is open in $B$ and the restriction of $p$ to $U$ is a [[homeomorphism]] $p|_U: U \to p(U)$.

The set $E_x = p^{-1}(x)$ where $x\in B$ is called the __[[stalk]]__ of $p$ over $x$.

The underlying set of the _total space_ $E$ is the union of its stalks (notice that we do not say fiber!).  $p$ is sometimes refered to as the _projection_.

## Definition for arbitrary toposes

The following is adapted from a [MathOverflow answer](https://mathoverflow.net/questions/78681/etal%c3%a9-space-construction-for-presheaves-on-a-grothendieck-site/111293#111293) by [[David Carchedi]]:

> As long as $C$ has a [[small set]] of topological generators (i.e., as long as $Sh(C,J)$ isn't too large to be a [[topos]]), there always exists a certain version of an étale space: If $F$ is a [[sheaf]] on $(C,J)$, the [[slice topos]] $Sh(C,J)/F$ has a canonical étale projection  $\pi_F:Sh(C,J)/F \to Sh(C,J)$. This map is a [[local homeomorphism of topoi]]. This topos with this local homeomorphism is the étale space of $F$. Indeed, we may make this construction for each object $c \in C$, call it $U(c) \coloneqq Sh(C,J)/y(c)$, where $y(c)$ is the (possibly sheafified) [[Yoneda embedding|Yoneda embedded]] object. Then, "[[sections]] of $\pi_F$ over $U(c) \to Sh(C,J)$" are in [[bijection]] with [[elements]] of $F(c)$. If the Grothendieck [[site]] $(C,J)$ happened to be the [[canonical site]] of a [[topological space]], then each [[slice]] $Sh(C,J)/F$ is equivalent to [[category of sheaves|sheaves]] on the étale space of that [[sheaf]], and the projection corresponds to the usual one. In particular, $U(c) \to Sh(C,J)$ corresponds to the inclusion of an [[open subset]]. So, this is reduces to the usual construction for spaces. Another example is, if $(C,J)$ were the [[small étale site]] of some [[scheme]] $S$, then each $Sh(C,J)/F$ is the small étale site of some algebraic space (with no separation conditions) étale over $S$, with $\pi_F$ corresponding to the étale map from this algebraic space to $S$.

## Properties

### Relation to sheaves
 {#RelationToSheaves}

Let $p:E\to B$ be in $Top/B$.  The (local) __[[sections]]__ of $p$ over an open set $U\subseteq B$ are the [[continuous maps]] $s:U\to E$ such that $p\circ s = \mathrm{id}_U$. It is an elementary but central fact that for an étale map $p$, _the images of local sections form a base for the topology_ of the total space $E$. The topology of $E$ is then typically non-[[Hausdorff space|Hausdorff]].

The set of sections of $p$ over $U$ is denoted by $\Gamma_U p = (\Gamma p)(U) = \Gamma_U E = (\Gamma E)(U)$ and may be shown to extend to a [[functor]] $\Gamma : Top/B\to PShv_B$ where $PShv_B$ is the [[category of presheaves]] over $B$.  The functor $\Gamma$ has a [[left adjoint]] $L : PShv_B\to Top/B$, whose [[essential image]] is the [[full subcategory]] $Et/B$ of étale spaces over $B$.  The [[essential image]] of the functor $\Gamma$ is the [[category of sheaves]] $Shv_B$  over $B$, and this [[adjunction]] restricts to an [[equivalence of categories]] between $Et/B$ and $Shv_B$ (that is, it is an [[idempotent adjunction]]).

If $P:Open(B)^{op}\to Set$ is a [[sheaf]], then one sometimes calls the total space $E(P)$ of the étale space $L(P) = (E(P)\to B)$ the __space of the sheaf__ $P$, having in mind the adjoint equivalence above.  (This is also called the __sheaf space__ or the __display space__ (alias __étale space__, or in French: __espace étalé__); compare also a [[display morphism]] of [[contexts]].)  The associated sheaf functor $a:PShv_B\to Shv_B\hookrightarrow PShv_B$ decomposes as $a = \Gamma\circ L$, and $a$ may be considered as an endofunctor part of an [[idempotent monad]] in $PShv_B$ whose corresponding [[reflective subcategory]] is $Shv_B$.

(e.g. [MacLane-Moerdijk, section II.5, II.6](#MacLaneMoerdijk))

### Relation to covering spaces

Every [[covering space]] (even in the more general sense not requiring any connectedness axiom) is étale but not vice versa:

* for a covering space the [[inverse image]] of some [[open subset]] in the base $B$ needs to be, by the definition, a [[disjoint union]] of homeomorphic open sets in $E$; however the 'size' of the [[open neighborhoods]] over various $e$ in the same [[stalk]] required in the definition of étale space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.

* even if the the stalks of the étale space are finite, it need not be locally trivial. For instance the [[disjoint union]] $\coprod_i U_i$ of a collection of [[open subsets]] of a topological space $X$ with the obvious projection $(\coprod_i U_i) \to X$ is étale, but does not have a typical fiber: the fiber over a given point has [[cardinality]] the number of open sets $U_i$ that contain this particular point.

## Grammar/pronunciation note
{#grammar}

### Grammar and spelling

In French, the verb 'étaler' means, roughly, to spread out; '-er' becomes '-é' to make a past participle.  So an 'espace étalé' is a space that has been spread out over $B$.  On the other hand, 'étale' is a (relatively obscure, distantly related) nautical adjective that can be translated as 'calm' or 'slack'.

To quote from the [Wiktionnaire français](https://fr.wiktionary.org/wiki/%C3%A9tale):

> 'étale' _qualifie la mer qui ne monte ni ne descend à la fin du flot ou du jusant_
('flot' = 'flow' and 'jusant' = 'ebb').

* [Further reference](https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive11#french_spelling)

### Pronunciation

The French pronunciation of "étalé" is three syllables; that of "étale" is two syllables, with the final "e" silent. Many English-speaking mathematicians follow the respective French pronunciations; see for example [this nForum thread](https://nforum.ncatlab.org/discussion/11283/etale-space/?Focus=84522#Comment_84522).

Others use the two-syllable pronunciation (corresponding to French "étale") for both written forms "étale" and "étalé". This group included [[Grothendieck]] himself, when speaking English. (See [[Colin McLarty#McLarty18|McLarty18]], from 4:45 to 6:05 and particularly [at 5:26](https://www.youtube.com/watch?v=5AR55ZsHmKI&t=326s).)

### Poem

There is an interesting stanza from a song of Léo Ferré:

> Et que les globules figurent

> Une mathématique bleue,

> Sur cette mer jamais étale

> D'où me remonte peu à peu

> Cette mémoire des étoiles.

> — (Léo Ferré, La mémoire et la mer)

He also mentions geometry and 'théorème' elsewhere in the song.

## Related concepts

* [[étale groupoid]], [[étale infinity-groupoid]]

## References

* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], sections II.5 and II.6 of _[[Sheaves in Geometry and Logic]]_

For generalizations to étale spaces of stacks in groupoids see

* [[David Carchedi]], _An étalé space construction for stacks_,  Algebr. Geom. Topol. 13 (2013), no. 2, 831–903.

[[!redirects etale space]]
[[!redirects etale spaces]]
[[!redirects étalé space]]
[[!redirects étalé spaces]]
[[!redirects étale space]]
[[!redirects étale spaces]]
[[!redirects etalé space]]
[[!redirects etalé spaces]]
[[!redirects espace étalé]]
[[!redirects espaces étalés]]
[[!redirects espace etale]]
[[!redirects espaces etales]]

[[!redirects sheaf space]]
[[!redirects sheaf spaces]]

[[!redirects display space]]
[[!redirects display spaces]]


étale theta function

A concept in [[arithmetic geometry]] introduced by [[Shinichi Mochizuki]] in the context of his [[inter-universal Teichmüller theory]].

###Related concepts

* [[Frobenioid]]

## References

* [[Shinichi Mochizuki]],  (2009), 'The étale theta function and its Frobenioid-theoretic manifestations', Kyoto University. Research Institute for Mathematical Sciences. Publications 45 (1): 227–349. ([pdf](http://www.kurims.kyoto-u.ac.jp/~prims/pdf/45-1/45-1-7.pdf))

* [[Minhyong Kim]], _[What is an étale theta function?](http://mathoverflow.net/q/195841/447)_

[[!redirects étale theta functions]]


étale topos

[[!redirects etale topos]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In the context of the [[geometry]] of [[schemes]] there is a traditional notion of [[étale morphism of schemes]] and an _étale topos_ is a [[category of sheaves]] on the [[étale site]] of a [[scheme]], consisting of [[covers]] by such [[étale morphism of schemes|étale morphisms]]. This traditional notion we discuss in

* [Étale topos of a scheme](#EtaleToposOfAScheme).

More abstractly, given that [[étale morphisms of schemes]] may be characterized as modal morphisms with respect to an [[infinitesimal shape modality]], one can consider étale toposes in every context of [[differential cohesion]]. This we discuss in

* [Étale topos of a differentially cohesive object](#GeneralAbstract)

### Étale topos of a scheme
 {#EtaleToposOfAScheme}

An **étale topos** is the [[sheaf topos]] over an [[étale site]], hence over a site whose "open subsets" are [[étale morphisms]] into the base [[space]]. The intrinsic [[cohomology]] of an étale [[(∞,1)-topos]] is _[[étale cohomology]]_.

More generally there is the pro-étale topos over a [[pro-étale site]], which is a bit better behaved. In particular the intrinsic [[cohomology]] of a pro-étale [[(∞,1)-topos]] includes the [[Weil cohomology theory]] [[ℓ-adic cohomology]].

Generally, given that an [[étale morphism of schemes]] is a [[formally étale morphism]] subject to a size constraint on its [[fibers]] -- for an actual [[étale morphism of schemes]] the fibers are [[finite sets]] in the suitable sense (formal duals to [[étale algebras]]) while for a [[pro-étale morphism of schemes]] they are [[pro-objects]] of such fibers --
in a suitable ambient context ("[[differential cohesion]]") one can drop all finiteness conditions and consider just opens given by [[formally étale morphisms]] as encoded by an [[infinitesimal shape modality]]. This we discuss [below](#GeneralAbstract).

### Étale topos of a differentially cohesive object
 {#GeneralAbstract}

We discuss how in [[differential cohesion]] $\mathbf{H}_{th}$ every object $X$ canonically induces its étale topos $Sh_{\mathbf{H}_{th}}(X)$.

For $X \in \mathbf{H}_{th}$ any object in a [[differential cohesion|differential cohesive]] $\infty$-topos, we formulate

1. the [[(∞,1)-topos]] denoted $\mathcal{X}$ or $Sh_\infty(X)$ of [[(∞,1)-sheaves]] over $X$, or rather of formally étale maps into $X$;

1. the [[structure (∞,1)-sheaf]] $\mathcal{O}_{X}$ of $X$.

The resulting structure is essentially that discussed ([Lurie, Structured Spaces](#Lurie)) if we regard $\mathbf{H}_{th}$ equipped with its formally étale morphisms, ([def.](differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh)), as a ([[large category|large]]) [[geometry for structured (∞,1)-toposes]].

One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let $G \in Grp(\mathbf{H}_{th})$ a differential cohesive [[∞-group]] with [de Rham coefficient object](cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology) $\flat_{dR}\mathbf{B}G$ and for $X \in \mathbf{H}_{th}$ any differential homotopy type, the product projection

$$
  X \times \flat_{dR} \mathbf{B}G \to X
$$

regarded as an object of the [[slice (∞,1)-topos]] $(\mathbf{H}_{th})_{/X}$ _almost_ qualifies as a "bundle of flat $\mathfrak{g}$-valued differential forms" over $X$: for $U \to X$ an cover (a [[1-epimorphism]]) regarded in $(\mathbf{H}_{th})_{/X}$, a $U$-plot of this product projection is a $U$-plot of $X$ together with a flat $\mathfrak{g}$-valued de Rham cocycle on $X$.

This is indeed what the sections of a corresponding bundle of differential forms over $X$ are supposed to look like -- but only _if_ $U \to X$ is sufficiently _spread out_ over $X$, hence sufficiently [[étale map|étale]]. Because, on the extreme, if $X$ is the point (the [[terminal object]]), then there should be no non-trivial section of differential forms relative to $U$ over $X$, but the above product projection instead reproduces all the sections of $\flat_{dR} \mathbf{B}G$.

In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be _restricted_ to plots out of suficiently étale maps into $X$. In order to correctly test differential form data, "suitable" here should be "formally", namely infinitesimally. Hence the restriction should be along the full inclusion

$$
  (\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}
$$

of the formally étale maps (see [def.](differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh)) into $X$. Since on formally étale covers the sections should be those given by $\flat_{dR}\mathbf{B}G$, one finds that the corresponding "cotangent bundle" must be the [[coreflective subcategory|coreflection]] along this inclusion. The following proposition establishes that this coreflection indeed exists.

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

For $X \in \mathbf{H}_{th}$ any object, write

$$
  (\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}
$$

for the full [[sub-(∞,1)-category]] of the [[slice (∞,1)-topos]] over $X$ on those maps into $X$ which are formally étale, (see [def.](differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh)).

We also write $FEt_{\mathbf{X}}$ or $Sh_{\mathbf{H}}(X)$ for $(\mathbf{H}_{th})_{/X}^{fet}$.

=--

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

The inclusion $\iota$ of def. \ref{EtaleSlice} is both [[reflective sub-(∞,1)-category|reflective]] as well as [[coreflective subcategory|coreflective]], hence it fits into an [[adjoint triple]] of the form

$$
  (\mathbf{H}_{th})_{/X}^{fet}
  \stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{\iota}{\hookrightarrow}}{\underset{Et}{\leftarrow}}}
  (\mathbf{H}_{th})_{/X}
  \,.
$$

=--

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

By the general discussion at _[[reflective factorization system]]_, the reflection is given by sending a morphism $f \colon Y \to X$ to
$X \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y$ and the reflection unit is the left horizontal morphism in

$$
  \array{
    Y &\to& X \times_{\mathbf{\Pi}_{inf}(Y)} \mathbf{\Pi}_{inf}(Y) &\to& \mathbf{\Pi}_{inf}(Y)
   \\
    & \searrow & \downarrow^{} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}}
   \\
   && X &\to& \mathbf{\Pi}_{inf}(X)
  }
  \,.
$$

Therefore $(\mathbf{H}_{th})_{/X}^{fet}$, being a reflective subcategory of a [[locally presentable (∞,1)-category]], is (as discussed there) itself locally presentable. Hence by the [[adjoint (∞,1)-functor theorem]] it is now sufficient to show that the inclusion preserves all small [[(∞,1)-colimits]] in order to conclude that it also has a right [[adjoint (∞,1)-functor]].

So consider any [[diagram]] [[(∞,1)-functor]] $I \to (\mathbf{H}_{th})_{/X}^{fet}$ out of a [[small (∞,1)-category]]. Since the inclusion of $(\mathbf{H}_{th})_{/X}^{fet}$ is full, it is sufficient to show that the $(\infty,1)$-colimit over this diagram taken in $(\mathbf{H}_{th})_{/X}$ lands again in $(\mathbf{H}_{th})_{/X}^{fet}$ in order to have that $(\infty,1)$-colimits are preserved by the inclusion. Moreover, colimits in a slice of $\mathbf{H}_{th}$ are computed in $\mathbf{H}_{th}$ itself (this is discussed at _[slice category - Colimits](overcategory#LimitsAndColimits)_).

Therefore we are reduced to showing that the square

$$
  \array{
    \underset{\to_i}{\lim} Y_i &\to& \mathbf{\Pi}_{inf} \underset{\to_i}{\lim} Y_i
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_{inf}(X)
  }
$$

is an [[(∞,1)-pullback]] square. But since $\mathbf{\Pi}_{inf}$ is a [[left adjoint]] it commutes with the $(\infty,1)$-colimit on objects and hence this diagram is equivalent to

$$
  \array{
    \underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \mathbf{\Pi}_{inf} Y_i
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_{inf}(X)
  }
  \,.
$$

This diagram is now indeed an [[(∞,1)-pullback]] by the fact that we have [[universal colimits]] in the [[(∞,1)-topos]] $\mathbf{H}_{th}$, hence that on the left the component $Y_i$ for each $i \in I$ is the [[(∞,1)-pullback]] of $\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally étale morphisms.

=--

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

The $\infty$-category $(\mathbf{H}_{th})_{/X}^{fet}$ is an [[(∞,1)-topos]] and the canonical inclusion into $(\mathbf{H}_{th})_{/X}$ is a [[geometric embedding]].

=--

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

By prop. \ref{EtalificationIsCoreflection} the inclusion $(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ is [[reflective sub-(infinity,1)-category|reflective]] with reflector given by the $(\mathbf{\Pi}_{inf}-equivalences , \mathbf{\Pi}_{inf}-closed)$ factorization system. Since $\mathbf{\Pi}_{inf}$ is a [[right adjoint]] and hence in particular preserves [[(∞,1)-pullbacks]], the $\mathbf{\Pi}_{inf}$-equivalences are stable under pullbacks. By the discussion at _[[stable factorization system]]_ this is the case precisely if the corresponding reflector preserves [[finite (∞,1)-limits]]. Hence the embedding is a [[geometric embedding]] which exhibits a [[sub-(∞,1)-topos]] inclusion.

=--

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

For $X \in \mathbf{H}_{th}$ we speak of

$$
  \mathcal{X} \coloneqq Sh_{\mathbf{H}_{th}}(X)
  \coloneqq
  (\mathbf{H}_{th})_{/X}^{fet}
$$

also as the ([[petit (∞,1)-topos|petit]]) [[(∞,1)-topos]] of $X$, or the **étale topos** of $X$.

Write

$$
  \mathcal{O}_X
    \colon
  \mathbf{H}_{th}
    \stackrel{(-) \times X}{\to}
  (\mathbf{H}_{th})_{/X}
    \stackrel{Et}{\to}
  (\mathbf{H}_{th})_{/X}^{fet}
$$

for the composite [[(∞,1)-functor]] that sends any $A \in \mathbf{H}_{th}$ to the etalification, prop. \ref{EtalificationIsCoreflection}, of the projection $A \times X \to X$.

We call $\mathcal{O}_X$ the **[[structure sheaf]]** of $X$.

=--

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

For $X, A \in \mathbf{H}_{th}$ and for $U \to X$ a [[formally étale morphism]] in $\mathbf{H}_{th}$ (hence like an [[open subset]] of $X$), we have that

$$
  \begin{aligned}
    \mathcal{O}_{X}(A)(U)
    & \coloneqq
    Sh_{\mathbf{H}_{th}}(X)( U , \mathcal{O}_{X}(A) )
    \\
    & \coloneqq
    Sh_{\mathbf{H}_{th}}(X)( U , Et(X \times A) )
    \\
    & \simeq
    (\mathbf{H}_{th})_{/X}(U, X \times A)
    \\
    & \simeq
    \mathbf{H}_{th}(U,A)
    \\
    & \simeq
    A(U)
  \end{aligned}
  \,,
$$

where we used the [[adjoint (∞,1)-functor|∞-adjunction]] $(\iota \dashv Et)$ of prop. \ref{EtalificationIsCoreflection} and the [[(∞,1)-Yoneda lemma]].

This means that $\mathcal{O}_{X}(A)$ behaves as the _sheaf of $A$-valued functions over $X$_.

Since $\mathcal{O}_{X}$ is [[right adjoint]] to the forgetful functor

$$
  Sh_{\mathbf{H}}(X)
    \simeq
  (\mathbf{H}_{th})_{/X}^{fet}
    \hookrightarrow
  (\mathbf{H}_{th})_{/X}
    \stackrel{\underset{X}{\sum}}{\to}
  \mathbf{H}_{th}
$$

it preserves [[(∞,1)-limits]]. Therefore this is a [[structure sheaf]] which exhibits $Sh_{\mathbf{H}_{th}}(X)$ as a [[structured (∞,1)-topos]] over $\mathbf{H}_{th}$ regarded as a (large) [[geometry (for structured (∞,1)-toposes)]], with the formally étale morphisms being the "admissible morphisms".

=--

+-- {: .num_example #CotangentBundle}
###### Example

Let $G \in Grp(\mathbf{H}_{th})$ be an [[∞-group]] and write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object.

Then

$$
  \mathcal{O}_X(\flat_{dR}\mathbf{B}G)
  \in
  Sh_{\mathbf{H}}(X)
$$

we may call the **$G$-valued flat cotangent sheaf** of $X$.

=--

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

For $U \in \mathbf{H}_{th}$ a test object (say an object in a [[(∞,1)-site]] of definition, under the [[Yoneda embedding]]) a formally étale morphism $U \to X$ is like an [[open map]]/open embedding. Regarded as an object in $(\mathbf{H}_{th})_{/X}^{fet}$ we may consider the sections over $U$ of the cotangent bundle as defined above, which in $\mathbf{H}_{th}$ are diagrams

$$
  \array{
    U &&\to&& \mathcal{O}_X(\flat_{dR}\mathbf{B}G)
    \\
    & \searrow && \swarrow
    \\
    && X
  }
  \,.
$$

By the fact that $Et(-)$ is [[right adjoint]], such diagrams are in bijection to diagrams

$$
  \array{
    U &&\to&& X \times \flat_{dR} \mathbf{B}G
    \\
    & \searrow && \swarrow
    \\
    && X
  }
$$

where we are now simply including on the left the formally étale map $(U \to X)$ along $(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$.

In other words, the sections of the $G$-valued flat cotangent sheaf $\mathcal{O}_X(\flat_{dR}\mathbf{B}G)$ are just the sections of $X \times \flat_{dR}\mathbf{B}G \to X$ itself, only that the _domain_ of the section is constrained to be a formally é patch of $X$.

But then by the very nature of $\flat_{dR}\mathbf{B}G$ it follows that the flat sections of the $G$-valued cotangent bundle of $X$ are indeed nothing but the flat $G$-valued differential forms on $X$.

=--

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

For $X \in \mathbf{H}_{th}$ an object in a differentially cohesive
$\infty$-topos, then its petit structured $\infty$-topos
$Sh_{\mathbf{H}_{th}}(X)$, according to def. \ref{TheStructureSheafOfX},
is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]].

=--

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

We need to check that the composite

$$
  \infty Grpd \stackrel{Disc}{\longrightarrow} \mathbf{H}_{th}
   \stackrel{(-) \times X}{\longrightarrow}
   (\mathbf{H}_{th})_{/X}
  \stackrel{L}{\longrightarrow}
  Sh_{\mathbf{H}}(X)
$$

preserves [[(∞,1)-limits]], so that it has a further
[[left adjoint]]. Here $L$ is the
reflector from prop. \ref{EtalificationIsCoreflection}.
Inspection shows that this composite sends an object $A \in \infty Grpd$ to
$\mathbf{\Pi}_{inf}(Disc(A)) \times X \to X$:

$$
  \array{
    \mathbf{\Pi}_{inf}(Disc(A)) \times X
     &\longrightarrow&
    \mathbf{\Pi}_{inf}(Disc(A) \times X) & \simeq \mathbf{\Pi}_{inf}(Disc(A)) \times \mathbf{\Pi}_{inf}(X)
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    X &\longrightarrow& \mathbf{\Pi}_{inf}(X)
  }
  \,.
$$

By the discussion at [slice (∞,1)-category -- Limits and colimits](slice+infinity-category#LimitsAndColimits) an [[(∞,1)-limit]] in the slice $(\mathbf{H}_{th})_{/X}$ is computed as an [[(∞,1)-limit]] in $\mathbf{H}$ of the [[diagram]] with the slice [[cocone]] adjoined. By [[right adjoint|right adjointness]] of the inclusion $Sh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X}$ the same is then true for $Sh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}$.

Now for $A \colon J \to \infty Grpd$ a [[diagram]], it is taken to the diagram $j \mapsto \mathbf{\Pi}_{inf}(Disc(A_j)) \times X \to X$ in $Sh_{\mathbf{H}}(X)$ and so its $\infty$-limit is computed in $\mathbf{H}$ over the diagram locally of the form

$$
  \array{
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j}))
     &&\longrightarrow&&
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j'}))
     \\
     & \searrow && \swarrow
     \\
     && X
  }
  \simeq
  \array{
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j}))
     &&\longrightarrow&&
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j'}))
     \\
     & \searrow && \swarrow
     \\
     && X \times \ast
  }
  \,.
$$

Since $\infty$-limits commute with each other this limit is the product of

1.  $\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))$

1. $\underset{\leftarrow}{\lim}_{J \star \Delta^0} X$ (over the co-coned diagram constant on $X$).

For the first of these, since the [[infinitesimal shape modality]] $\mathbf{\Pi}_{inf}$
is in particular a [[right adjoint]] (with [[left adjoint]] the
[[reduction modality]]), and since $Disc$ is also [[right adjoint]] by [[cohesion]], we have a [[natural equivalence]]

$$
  \underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))
  \simeq
   \mathbf{\Pi}_{inf}(Disc(\underset{\leftarrow}{\lim}_j(A_j)))
  \,.
$$

For the second, the $\infty$-limit over an $\infty$-category $J \star \Delta^0$ of a functor constant on $X$ is

$$
  \begin{aligned}
     \underset{\leftarrow}{\lim}_{J \star \Delta^0} X
     & \simeq
     \underset{\leftarrow}{\lim}_{J \star \Delta^0} [\ast, X]
     \\
     & \simeq
     [\underset{\rightarrow}{\lim}_{J \star \Delta^0} \ast, X]
     \\
     & \simeq
     [{\vert {J \star \Delta^0}\vert}, X]
     \\
     & \simeq
     [\ast, X] \simeq X
  \end{aligned}
  \,,
$$

where the last line follows since ${J \star \Delta^0}$
has a terminal object and hence contractible geometric realization.

In conclusion this shows that $\infty$-limits are preserved by $L \circ (-)\times X\circ Disc$.

=--

## Properties

### Sheaf condition and examples of étale sheaves
 {#SheafConditionAndExamples}

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

For $X$ a [[scheme]], and $A \in PSh(X_{et})$ a [[presheaf]],
for checking the [[sheaf]] condition it is sufficient to
check [[descent]] on the following two kinds of [[covers]]
in the [[étale site]]

1. jointly surjective collections of [[open immersions of schemes]];

1. single surjective/[[étale morphism of schemes|étale]] morphisms between [[affine schemes]]

(all over $X$).

=--

([Tamme, II Lemma (3.1.1)](#Tamme), [Milne, prop. 6.6](#Milne))

+-- {: .proof}
###### Proof (sketch)

Since [[covers]] by standard [[open immersions of schemes]] in the [[Zariski topology]] are also [[étale morphisms of schemes]] and étale covers, we may take any étale cover $\{Y_i \to Y\}$ over $X$, find an Zariski cover $\{U_i \to X\}$ of $X$, pull back the original cover to that and in turn cover the pullbacks themselves by Zariski covers. The result is still a cover and is so by a collection of [[open immersions of schemes]].
Now using compactness assumptions we find finite subcovers of all these covers. This makes their [[disjoint union]] be a single morphisms of affines.

=--

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

For $Z \to X$ any [[scheme]] over a [[scheme]] $X$, the induced [[presheaf]] on the [[étale site]]

$$
  (U_Y \to X) \mapsto Hom_X(U_Y, Z)
$$

is a [[sheaf]].

=--

This is due to ([[Grothendieck]], [[SGA]]1 exp. XIII 5.3) A review is in ([Tamme, II theorem (3.1.2)](#Tamme), [Milne, 6.2](#Milne)).

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

By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers} we are reduced to
showing that the represented presheaf satisfies [[descent]] along collections of open immersions and along surjective maps of affines. For the first this is clear (it is [[Zariski topology]]-descent). For the second case of a [[faithfully flat]] cover of affines $Spec(B) \to Spec(A)$ it follows with the exactness of the correspomnding [[Amitsur complex]]. See there for details.

=--

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

This map from $X$-schemes to sheaves on $X_{et}$ is not injective, different $X$-schemes may represent the same sheaf on $X_{et}$.
Unique representatives are given by [[étale morphism of schemes|étale schemess]] over $X$.

=--

(e.g. [Tamme, II theorem 3.1](#Tamme))

We consider some examples of [[sheaves of abelian groups]]
induced by prop. \ref{XSchemesRepresentSheaves} from  [[group schemes]]
over $X$.

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

The [[additive group]] over $X$ is the [[group scheme]]

$$
  \mathbb{G}_a  \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X
  \,.
$$

By the [[universal property]] of the [[pullback]], the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment

$$
  \begin{aligned}
     (\mathbb{G}_a)_X(U_X \to X) & =
     Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X)
     \\
     & =
     Hom(U_X, Spec(\mathbb{Z}[t]))
     \\
     & =
     Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X}))
     \\
     & = \Gamma(U_X, \mathcal{O}_{U_X})
  \end{aligned}
  \,.
$$

In other words, the sheaf represented by the [[additive group]] is the [[abelian sheaf]] underlying the [[structure sheaf]] of $X$.

=--

Similarly one finds

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

The [[multiplicative group]] over $X$

$$
  \mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X
$$

represents the sheaf $(\mathbb{G}_m)_X$ given by

$$
  (\mathbb{G}_m)_X(U_X)
  \mapsto
  \Gamma(U_X, \mathcal{O}_{U_X})^\times
  \,.
$$

=--

(e.g. [Tamme, II, 3](#Tamme))

### Base change and sheaf cohomology
 {#BaseChange}

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

For $f \colon X \longrightarrow Y$ a [[homomorphism]] of [[schemes]], there is induced a [[functor]] on the [[categories]] underlying the [[étale site]]

$$
  f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}
$$

given by sending an [[object]] $U_Y \to Y$ to the [[fiber product]]/[[pullback]] along $f$

$$
  f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X)
  \,.
$$

=--

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

The morphism in def. \ref{BaseChangeOnSites} is a [[morphism of sites]]
and hence induces a [[geometric morphism]] between the étale toposes

$$
  (f^\ast \dashv f_\ast)
  \;\colon\;
  Sh(X_{et})
  \stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}
  Sh(Y_{et})
  \,.
$$

Here the [[direct image]] is given on a [[sheaf]] $\mathcal{F} \in Sh(X_{et})$ by

$$
  f_\ast \mathcal{F} \;\colon\;  (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y)
$$

while the [[inverse image]] is given on a [[sheaf]] $\mathcal{F} \in Sh_(Y_{et})$ by

$$
  f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y)
  \,.
$$

=--

By the discussion at _[morphisms of sites -- Relation to geometric morphisms](morphism+of+sites#RelationToGeometricMorphisms)_.
See also for instance ([Tamme I 1.4](#Tamme)).

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

For $X_{et}$ an [[étale site]], write $\mathcal{D}(X_{et})$ for the [[derived category]] of the [[abelian category]] $Ab(Sh(X_{et}))$ of [[abelian sheaves]] on $X$.

=--

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

The $q$th [[derived functor]] $R^q f_\ast$ of the [[direct image]] functor of def. \ref{DirectAndInverseImageAlongMapOfBases} sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]]

$$
  (U_Y \to Y)
  \mapsto
  H^q(X \times_Y U_Y, \mathcal{F})
  \,,
$$

where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$ ([[étale cohomology]]).

=--

By the discussion at _[[direct image]]_ and at _[[abelian sheaf cohomology]]_. See e.g. ([Tamme, II (1.3.4)](#Tamme), [Milne prop. 12.1](#Milne)).

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

For $O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z$
two composable [[morphisms of sites]],
the [[Leray spectral sequence]] for the corresponding [[direct images]] exists and is of the form

$$
  E^{p,q}_2 = R^p f_\ast(R^q g_\ast(\mathcal{F}))
  \Rightarrow
  E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F})
  \,.
$$

For the special case that $S_Z = \ast$ and $g^{-1}$ includes an [[étale morphism of schemes|étale morphism]] $U_Y \to Y$ this yields

$$
  E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F})
  \Rightarrow
  E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F})
  \,.
$$

=--

### Quasi-coherent modules
 {#QuasiCoherentModules}

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

For $X$ a [[scheme]] and $N$ a [[quasicoherent module]] over its [[structure sheaf]] $\mathcal{O}_X$, then this induces an [[abelian sheaf]] on the [[étale site]] by

$$
  N_{et} \;\colon\; (U_X \to X)
   \mapsto
  \Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_X})
  \,.
$$

=--

(e.g. [Tamme, II 3.2.1](#Tamme))

### Relation to Zariski topos

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

A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every étale [[sheaf]] is a Zariski sheaf, but not necessarily conversely. Expressed in a different way, the étale topos is a [[subtopos]] of the Zariski topos.

=--

For more see at _[étale cohomology -- Properties -- Relation to Zariski cohomology](etale+cohomology#RelationZariskiEtaleCohomology)_.

### As a classifying topos

The étale topos over the big étale site of [[commutative rings]] is the [[classifying topos]] for [[strict local rings]].

## Related concepts

* [[basics of étale cohomology]]

* [[étale homotopy]]

* [[étale cohomology]]

## References

### Etale topos of a schemes

* [[Günter Tamme]], section II 1 of _[[Introduction to Étale Cohomology]]_
 {#Tamme}

* [[James Milne]], section 7 of _[[Lectures on Étale Cohomology]]_
 {#Milne}

### Etale topos of a differentially cohesive object

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

[[!redirects etale toposes]]
[[!redirects etale topoi]]

[[!redirects étale topos]]
[[!redirects étale toposes]]
[[!redirects étale topoi]]

[[!redirects étale (∞,1)-topos]]
[[!redirects etale (∞,1)-topos]]
[[!redirects étale (∞,1)-toposes]]
[[!redirects etale (∞,1)-toposes]]

[[!redirects étale (infinity,1)-topos]]

[[!redirects etale (infinity,1)-topos]]
[[!redirects étale (infinity,1)-toposes]]
[[!redirects etale (infinity,1)-toposes]]


étendue

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

# Contents
* automatic table of contents goes here
{:toc}

## Idea

An **étendue** (also 'etendue', 'étendu', or 'etendu'; from French 'étendue' (fem.)- _extent_) is a [[topos]] $\mathcal{Y}$ that locally looks like the category of sheaves on a space:
>Briefly, the slogan is that $\mathcal{Y}$ is locally a topological space. ([Lawvere 1976](#Lawvere76), p.129)

In some papers, [[topological spaces]] are replaced by [[locales]]
in the definition of [[étendues]].
For example, see Definition 1.1 in [Rosenthal](#Rosenthal) or Definition 1.12 in [Kock and Moerdijk](#KockMoerdijk).

Originally defined by [[A. Grothendieck]] in one of the famous 'exercises' of [SGA4](#SGA4) (ex. 9.8.2) as a [[Grothendieck topos]] $\mathcal{Y}$ that has a well-supported object $X$ such that the [[slice topos]] $\mathcal{Y}/X$ is equivalent to a sheaf topos on a topological space, the definition was generalized by [[Lawvere]] (1975,1976) by dropping the spatiality of the slice and require only that $\mathcal{Y}/X$ is a [[localic topos]].

Several characterizations of étendues are known and the _Ur_-example of an étendue, the presheaf topos $\mathcal{S}^G$ of group actions, exhibits one in terms of sites rather directly: it has a [[site]] where every morphism is monic. Other characterizations involve (local) [[equivalence relations]] and yield connections to [[orbifolds]], [[foliations]], and [[stacks]], which are instrumental for the generalization to _$\infinity$-étendues_ (cf. [Carchedi 2013](#Carchedi13)).

Étendues play an important role in Lawvere's approach to [[cohesion]] and the distinction between [[petit and gros toposes]] where they provide one of the classes of **petit** toposes (generalized spaces). In this context, Lawvere (1989,1991) interprets the cancellative property of the site as enabling an interpretation of étendues as _categories of processes_.

## Definition

A topos $\mathcal{Y}$ is called an _étendue_ if there is an object $X\in|\mathcal{Y}|$ such that the unique $X\rightarrow 1$ is epic and the [[slice topos]] $\mathcal{Y}/X$  is a [[localic topos]].[^fine]

[^fine]:  An epic $k:X\rightarrow Y$ induces a [[geometric morphism]] $k_\ast:\mathcal{Y}/X\rightarrow \mathcal{Y}/Y$ whose inverse image part, the change of base functor, $k^\ast:\mathcal {Y}/Y\rightarrow\mathcal{Y}/X$ is faithful, which says by definition that $k_\ast$ is a surjection, and in case $Y=1$, one says that $\mathcal{Y}/X$ _covers_ $\mathcal{Y}$. $k_\ast$ is an [[étale geometric morphism]].

## Examples

> The first example of an étendue seems to have been the [[space of moduli]] of algebraic curves, which is prevented from being globally a space due to the action of the [[Galois groups]] within each point. Yes, something vaguely reminiscent of particle spin is going on in such spaces, and the most naked form is that for any group G, the category $\mathcal{S}^G$ is an étendue with only one point! This is easily seen from the observations that $\mathcal{S}^G/G\cong\mathcal{S}^G$ and that $G\twoheadrightarrow 1$ where the last two $G$'s denote the regular representation.      ([Lawvere 1976](#Lawvere76), pp.129-130)

* The [[Sierpinski topos]] $\mathcal{S}^{\cdot\rightarrow\cdot}$, as the sheaf topos on the [[Sierpinski space]], is an étendue.

* The topos  $\mathcal{S}^{\cdot\rightrightarrows\cdot}$ of _directed [[graphs]]_ (aka [[quivers]]; Lawvere calls them *irreflexive graphs*) is an étendue, as it is locally equivalent to the sheaf topos on a three point space  (Lawvere 1986). The contrast between $\mathcal{S}^{\cdot\rightrightarrows\cdot}$ and the topos  $\mathcal{S}^{\Delta_1^{op}}$ of _reflexive graphs_ is a paradigmatic example of the distinction between a _petit_ and a _gros_ topos.

* The [[Jónsson-Tarski topos]] $\mathcal{J}_2$ is an étendue, as it is locally equivalent to the sheaf topos on the [[Cantor space]]. It is discussed as a petit topos for _labeled graphs_ in (Lawvere 1989).

## Properties

* **Proposition**. A Grothendieck topos $\mathcal{Y}$ is an étendue iff there exists a [[site]] $(\mathcal{C}, J)$ for $\mathcal{Y}$ such that every morphism of $\mathcal{C}$ is monic.

* In particular for small $\mathcal{C}$, the presheaf topos $\mathcal{S}^{\mathcal{C}^{op}}$ is an étendue iff all morphisms in $\mathcal{C}$ are monic. In particular for [[monoid]]s $\mathcal{C}$ this is sometimes called [[left cancellative category|left cancellative]] (e.g. each free monoid is left cancellative).

* [[Subtoposes]] of étendues are étendues. K. Rosenthal uses this together with the preceding remark on $\mathcal{S}^{\mathcal{C}^{op}}$ for all-monic $\mathcal{C}$ in order to construct further étendues $Sh_j(\mathcal{S}^{\mathcal{C}^{op}})$ via a topology $j$ from a suitable functor $H:\mathcal{C}\to\mathcal{S}$ (for further details see [[Jónsson-Tarski topos]] or Rosenthal(1981)).

* A Grothendieck topos is a _[[boolean topos|Boolean]] étendue_ precisely if it satisfies the _internal [[axiom of choice]]_ (Freyd&Scedrov 1990). An example of such a Boolean étendue is $\mathcal{S}^G$, for $G$ a group.

* Étendues are 'locally co-decidable' in the sense that for a small $\mathcal{C}$ the functor category $[\mathcal{C},Set]$ is a [[locally decidable topos]] precisely if $[\mathcal{C}^{op},Set]$ is an [[étendue]]. Also the all-monic-site property is dual to the all-epic-site property of locally decidable toposes. Both concepts are subsumed under the notion of having a (sub canonical) site representation with no (non-trivial) [[idempotents]] (McLarty 2006, Lawvere 2007).

## Slice toposes of étendues are étendues

One can use the site characterization to show that being an étendue topos is a _local property_.

+-- {: .num_prop #etendue_slices}
###### Proposition
Let $Sh(\mathcal{C},J)$ be an étendue topos with $(\mathcal{C}, J)$ being an all-monic-site presentation and $P:C^{op}\to Set$ be a presheaf on $\mathcal{C}$  that is a $J$-sheaf. Then $Sh(\mathcal{C},J)/P$ is an étendue topos.
=--

+-- {: .proof}
###### Proof
It suffices to show that $Sh(\mathcal{C},J)/P$ has an all-monic site presentation. By exercise III.8 in [Mac Lane-Moerdijk (1994, p.157)](#MacLaneMoerdijk) there exists a topology $J'$ on the [[category of elements]] $\int_\mathcal{C} P$ such that
$$Sh(\mathcal{C},J)/P\simeq Sh(\int_\mathcal{C} P,J')\; .$$
Whence it suffices to show that $\int_\mathcal{C} P$ is all-monic: Let $f:(C,x)\to (D,y)$ be a morphism in $\int_\mathcal{C} P$ and $g,h:(B,z)\rightrightarrows (C,x)$ such that $f\cdot g=f\cdot h$ then $g=h$ since composition in $\int_\mathcal{C} P$ is inherited from $\mathcal{C}$ and the latter was all-monic by assumption.

=--

+-- {: .num_prop #IAC_local}
###### Corollary
Let $\mathcal{E}$ be a Grothendieck topos satisfying the [[axiom of choice|internal axiom of choice]] (IAC). Then any [[slice topos]] $\mathcal{E}/X$ satisfies the internal axiom of choice as well.
=--
+-- {: .proof}
###### Proof
By Freyd-Scedrov ([1990](#FreydScedrov90), p.181) a Grothendieck topos $\mathcal{E}$ satisfies IAC iff $\mathcal{E}$ is a Boolean étendue. But being [[Boolean category|Boolean]] is a local property whence by the preceding proposition all slices $\mathcal{E}/X$ are Boolean étendues.
=--

## Related entries

* [[localic topos]]

* [[petit topos]]

* [[locally decidable topos]]

* [[Deligne-Mumford stack]]

* [[bicategory of fractions]]

## References

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, Springer LNM vol.269 (1972), pp.479-484.

* {#Carchedi13}[[David Carchedi]], _Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi_ , arXiv1312.2204 (2013). ([pdf](http://arxiv.org/pdf/1312.2204v1.pdf))

* Darian De Wolf, [[Dorette Pronk|D. Pronk]], _A Double-Categorical View on Representations of Etendues_, Cah. Top. Géom. Diff. Cat. **LXI** 1 (2020) 3-56 [, [pdf](http://cahierstgdc.com/wp-content/uploads/2020/01/DEWOLF-PRONK-61-1.pdf), [arXiv:1910.02540](https://arxiv.org/abs/1910.02540)]

* {#FreydScedrov90}[[Peter Freyd|P. J. Freyd]], A. Scedrov, _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990.

* [[Peter Johnstone]], _Sketches of an [[Elephant]] II_, Oxford UP 2002, pp.769-775.

* {#KockMoerdijk} [[A. Kock]], [[I. Moerdijk]], _Presentations of étendues_ , Cah. Top. Géom. Diff. Cat. **XXXII** 2 (1991) pp.145-164. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_2/CTGDC_1991__32_2_145_0/CTGDC_1991__32_2_145_0.pdf))

* [[A. Kock]], [[I. Moerdijk]], _Every étendue comes from a local equivalence relation_ , JPAA **82** (1992) pp.155-174.

* [[M. V. Lawson]], [[B. Steinberg]], _Ordered groupoids and étendues_ , Cah. Top. Géom. Diff. Cat. **XXXXV** 2 (2004) pp.82-108. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_2004__45_2/CTGDC_2004__45_2_82_0/CTGDC_2004__45_2_82_0.pdf))

* [[F. William Lawvere]], _Variable sets etendu and variable structure in topoi_ , Lecture notes University of Chicago 1975.

* {#Lawvere76}[[F. William Lawvere]], _Variable quantities and variable structures in topoi_ , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.

* [[F. William Lawvere]], _Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs_, Revista Colombiana de Matemáticas **XX** (1986) pp.179-186. Reprinted with commentary in TAC **9** (2005) pp.1-7. ([pdf](http://www.tac.mta.ca/tac/reprints/articles/9/tr9.pdf))

* [[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Cont. Math. **92** (1989) pp.261-299.

* [[F. William Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in Springer LNM vol. 1488 (1991).

* [[F. William Lawvere]], _Axiomatic Cohesion_ , TAC **19** no. 3 (2007) pp.41-49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

* [[F. William Lawvere]], _Cohesive Toposes: Combinatorial and Infinitesimal Cases_, Como Ms. 2008. ([pdf](http://comocategoryarchive.com/Archive/temporary_new_material/FWLawvere-Cohesive-Toposes-Como-January-2008.pdf))

* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994.

* [[Colin McLarty]], _Every Grothendieck Topos has a One-Way Site_ , TAC **16** no. 5 (2006) pp.123-126. ([pdf](http://www.tac.mta.ca/tac/volumes/16/5/16-05.pdf))

* [[Ieke Moerdijk]], _Foliations, groupoids and Grothendieck étendues_ , Rev. Acad. Cienc. Zaragoza (2) **48** (1993) pp.5-33.

* [[Dorette A. Pronk]], _Etendues and stacks as bicategory of fractions_ , Comp. Math. **102** 3 (1996) pp.243-303. ([pdf](http://archive.numdam.org/ARCHIVE/CM/CM_1996__102_3/CM_1996__102_3_243_0/CM_1996__102_3_243_0.pdf))

* [[Pedro Resende]], _Groupoid Sheaves as Quantale Modules_ , arXiv.0807.4848v3 (2011). ([pdf](http://arxiv.org/pdf/0807.4848v3.pdf))

* {#Rosenthal} [[Kimmo I. Rosenthal]], _Étendues and Categories with Monic Maps_ , JPAA **22** (1981) pp.193-212.

* [[Kimmo I. Rosenthal]], _Sheaves and Local Equivalence Relations_ , Cah. Top. Géom. Diff. Cat. **XXV** 2 (1984) pp.179-206. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1984__25_2/CTGDC_1984__25_2_179_0/CTGDC_1984__25_2_179_0.pdf))

* [[Kimmo I. Rosenthal]], _Covering Étendues and Freyd's Theorem_ , Proc. AMS **99** 2 (1987) pp.221-222. ([pdf](http://www.ams.org/journals/proc/1987-099-02/S0002-9939-1987-0870775-X/S0002-9939-1987-0870775-X.pdf))

[[!redirects etendue]]
[[!redirects etendues]]
[[!redirects étendues]]


Évariste Galois

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

* [MacTutorBiographies page](https://mathshistory.st-andrews.ac.uk/Biographies/Galois/)

## Selected writings

* [[Évariste Galois]], *[[Galois' last letter|letter to Auguste Chevallier]]* (29 May 1832) [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)]

## Related entries

* [[group]], [[group theory]]

* [[Galois theory]]

* [[Galois connection]]

category: people

[[!redirects Galois]]
[[!redirects Evariste Galois]]


Øystein Ore

__Øystein Ore__ was a Norwegian algebraist and graph theorist. His Norwegian name in his mathematical articles was however spelled as Oystein Ore. He introduced [[Ore extension]]s and Ore conditions for [[Ore set]]s (in the context of study of linear equations over noncommutative [[Ore domain]]s) of rings. Ore introduced name groupoid for what we now call [[magma]] (cf. [[historical notes on quasigroups]]).

* wikipedia: [Øystein Ore](http://en.wikipedia.org/wiki/%C3%98ystein_Ore), [Ore extension](http://en.wikipedia.org/wiki/Ore_extension), [Ore condition](http://en.wikipedia.org/wiki/Ore_condition), [Ore's theorem](http://en.wikipedia.org/wiki/Ore%27s_theorem)
* [biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Ore.html) at MacTutor History of Mathematics archive, University of St Andrews

category: people
[[!redirects Oystein Ore]]