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\begin{document}

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\section*{étale topos}

[[!redirects etale topos]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{tale_morphisms}{}\paragraph*{{\'E{}tale morphisms}}\label{tale_morphisms}

[[!include etale morphisms - contents]]

\hypertarget{topos_theory_2}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory_2}

[[!include (infinity,1)-topos - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{EtaleToposOfAScheme}{\'E{}tale topos of a scheme}\dotfill \pageref*{EtaleToposOfAScheme} \linebreak
\noindent\hyperlink{GeneralAbstract}{\'E{}tale topos of a differentially cohesive object}\dotfill \pageref*{GeneralAbstract} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SheafConditionAndExamples}{Sheaf condition and examples of \'e{}tale sheaves}\dotfill \pageref*{SheafConditionAndExamples} \linebreak
\noindent\hyperlink{BaseChange}{Base change and sheaf cohomology}\dotfill \pageref*{BaseChange} \linebreak
\noindent\hyperlink{QuasiCoherentModules}{Quasi-coherent modules}\dotfill \pageref*{QuasiCoherentModules} \linebreak
\noindent\hyperlink{relation_to_zariski_topos}{Relation to Zariski topos}\dotfill \pageref*{relation_to_zariski_topos} \linebreak
\noindent\hyperlink{as_a_classifying_topos}{As a classifying topos}\dotfill \pageref*{as_a_classifying_topos} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{etale_topos_of_a_schemes}{Etale topos of a schemes}\dotfill \pageref*{etale_topos_of_a_schemes} \linebreak
\noindent\hyperlink{etale_topos_of_a_differentially_cohesive_object}{Etale topos of a differentially cohesive object}\dotfill \pageref*{etale_topos_of_a_differentially_cohesive_object} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In the context of the [[geometry]] of [[schemes]] there is a traditional notion of [[étale morphism of schemes]] and an \emph{\'e{}tale 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

\begin{itemize}%
\item \hyperlink{EtaleToposOfAScheme}{\'E{}tale topos of a scheme}.

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

\begin{itemize}%
\item \hyperlink{GeneralAbstract}{\'E{}tale topos of a differentially cohesive object}

\end{itemize}
\hypertarget{EtaleToposOfAScheme}{}\subsubsection*{{\'E{}tale topos of a scheme}}\label{EtaleToposOfAScheme}

An \textbf{\'e{}tale 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 \'e{}tale [[(∞,1)-topos]] is \emph{[[étale cohomology]]}.

More generally there is the pro-\'e{}tale topos over a [[pro-étale site]], which is a bit better behaved. In particular the intrinsic [[cohomology]] of a pro-\'e{}tale [[(∞,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 \hyperlink{GeneralAbstract}{below}.

\hypertarget{GeneralAbstract}{}\subsubsection*{{\'E{}tale topos of a differentially cohesive object}}\label{GeneralAbstract}

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

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

\begin{enumerate}%
\item the [[(∞,1)-topos]] denoted $\mathcal{X}$ or $Sh_\infty(X)$ of [[(∞,1)-sheaves]] over $X$, or rather of formally \'e{}tale maps into $X$;


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



\end{enumerate}
The resulting structure is essentially that discussed (\hyperlink{Lurie}{Lurie, Structured Spaces}) if we regard $\mathbf{H}_{th}$ equipped with its formally \'e{}tale morphisms, (\href{differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh}{def.}), 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 \href{cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology}{de Rham coefficient object} $\flat_{dR}\mathbf{B}G$ and for $X \in \mathbf{H}_{th}$ any differential homotopy type, the product projection

\begin{displaymath}
X \times \flat_{dR} \mathbf{B}G \to X
\end{displaymath}
regarded as an object of the [[slice (∞,1)-topos]] $(\mathbf{H}_{th})_{/X}$ \emph{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 \emph{if} $U \to X$ is sufficiently \emph{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 \emph{restricted} to plots out of suficiently \'e{}tale 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

\begin{displaymath}
(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}
\end{displaymath}
of the formally \'e{}tale maps (see \href{differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh}{def.}) into $X$. Since on formally \'e{}tale 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.

\begin{defn}
\label{EtaleSlice}\hypertarget{EtaleSlice}{}
For $X \in \mathbf{H}_{th}$ any object, write

\begin{displaymath}
(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}
\end{displaymath}
for the full [[sub-(∞,1)-category]] of the [[slice (∞,1)-topos]] over $X$ on those maps into $X$ which are formally \'e{}tale, (see \href{differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleInHTh}{def.}).

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

\end{defn}
\begin{prop}
\label{EtalificationIsCoreflection}\hypertarget{EtalificationIsCoreflection}{}
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

\begin{displaymath}
(\mathbf{H}_{th})_{/X}^{fet} 
  \stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{\iota}{\hookrightarrow}}{\underset{Et}{\leftarrow}}}
  (\mathbf{H}_{th})_{/X}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the general discussion at \emph{[[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

\begin{displaymath}
\itexarray{
    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)
  }
  \,.
\end{displaymath}
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 \emph{\href{overcategory#LimitsAndColimits}{slice category - Colimits}}).

Therefore we are reduced to showing that the square

\begin{displaymath}
\itexarray{
    \underset{\to_i}{\lim} Y_i &\to& \mathbf{\Pi}_{inf} \underset{\to_i}{\lim} Y_i 
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_{inf}(X)
  }
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    \underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \mathbf{\Pi}_{inf} Y_i 
    \\
    \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_{inf}(X)
  }
  \,.
\end{displaymath}
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 \'e{}tale morphisms.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
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]].

\end{prop}
\begin{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 \emph{[[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.

\end{proof}
\begin{defn}
\label{TheStructureSheafOfX}\hypertarget{TheStructureSheafOfX}{}
For $X \in \mathbf{H}_{th}$ we speak of

\begin{displaymath}
\mathcal{X} \coloneqq Sh_{\mathbf{H}_{th}}(X)
  \coloneqq
  (\mathbf{H}_{th})_{/X}^{fet}
\end{displaymath}
also as the ([[petit (∞,1)-topos|petit]]) [[(∞,1)-topos]] of $X$, or the \textbf{\'e{}tale topos} of $X$.

Write

\begin{displaymath}
\mathcal{O}_X 
    \colon
  \mathbf{H}_{th}
    \stackrel{(-) \times X}{\to}
  (\mathbf{H}_{th})_{/X}
    \stackrel{Et}{\to}
  (\mathbf{H}_{th})_{/X}^{fet}
\end{displaymath}
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 \textbf{[[structure sheaf]]} of $X$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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{displaymath}
\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}
  \,,
\end{displaymath}
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 \emph{sheaf of $A$-valued functions over $X$}.

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

\begin{displaymath}
Sh_{\mathbf{H}}(X) 
    \simeq 
  (\mathbf{H}_{th})_{/X}^{fet} 
    \hookrightarrow 
  (\mathbf{H}_{th})_{/X} 
    \stackrel{\underset{X}{\sum}}{\to}
  \mathbf{H}_{th}
\end{displaymath}
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 \'e{}tale morphisms being the ``admissible morphisms''.

\end{remark}
\begin{example}
\label{CotangentBundle}\hypertarget{CotangentBundle}{}
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

\begin{displaymath}
\mathcal{O}_X(\flat_{dR}\mathbf{B}G)
  \in 
  Sh_{\mathbf{H}}(X)
\end{displaymath}
we may call the \textbf{$G$-valued flat cotangent sheaf} of $X$.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
For $U \in \mathbf{H}_{th}$ a test object (say an object in a [[(∞,1)-site]] of definition, under the [[Yoneda embedding]]) a formally \'e{}tale 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

\begin{displaymath}
\itexarray{
    U &&\to&& \mathcal{O}_X(\flat_{dR}\mathbf{B}G)
    \\
    & \searrow && \swarrow
    \\
    && X
  }
  \,.
\end{displaymath}
By the fact that $Et(-)$ is [[right adjoint]], such diagrams are in bijection to diagrams

\begin{displaymath}
\itexarray{
    U &&\to&& X \times \flat_{dR} \mathbf{B}G
    \\
    & \searrow && \swarrow
    \\
    && X
  }
\end{displaymath}
where we are now simply including on the left the formally \'e{}tale 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 \emph{domain} of the section is constrained to be a formally \'e{} 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 $X$.

\end{remark}
\begin{prop}
\label{StructuredPetitToposesAreLocallyContractible}\hypertarget{StructuredPetitToposesAreLocallyContractible}{}
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]].

\end{prop}
\begin{proof}
We need to check that the composite

\begin{displaymath}
\infty Grpd \stackrel{Disc}{\longrightarrow} \mathbf{H}_{th}
   \stackrel{(-) \times X}{\longrightarrow}
   (\mathbf{H}_{th})_{/X}
  \stackrel{L}{\longrightarrow}
  Sh_{\mathbf{H}}(X)
\end{displaymath}
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 $\mathbf{\Pi}_{inf}(Disc(A)) \times X \to X$:

\begin{displaymath}
\itexarray{
    \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)
  }
  \,.
\end{displaymath}
By the discussion at \href{slice+infinity-category#LimitsAndColimits}{slice (∞,1)-category -- Limits and colimits} 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

\begin{displaymath}
\itexarray{
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j}))
     &&\longrightarrow&&
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j'}))
     \\
     & \searrow && \swarrow
     \\
     && X
  }
  \simeq
  \itexarray{
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j}))
     &&\longrightarrow&&
     X \times \mathbf{\Pi}_{inf}(Disc(A_{j'}))
     \\
     & \searrow && \swarrow
     \\
     && X \times \ast
  }
  \,.
\end{displaymath}
Since $\infty$-limits commute with each other this limit is the product of

\begin{enumerate}%
\item $\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))$


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



\end{enumerate}
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]]

\begin{displaymath}
\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))
  \simeq
   \mathbf{\Pi}_{inf}(Disc(\underset{\leftarrow}{\lim}_j(A_j)))
  \,.
\end{displaymath}
For the second, the $\infty$-limit over an $\infty$-category $J \star \Delta^0$ of a functor constant on $X$ is

\begin{displaymath}
\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}
  \,,
\end{displaymath}
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 $L \circ (-)\times X\circ Disc$.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SheafConditionAndExamples}{}\subsubsection*{{Sheaf condition and examples of \'e{}tale sheaves}}\label{SheafConditionAndExamples}

\begin{prop}
\label{EtaleDescentDetectedOnOpenImmersionCovers}\hypertarget{EtaleDescentDetectedOnOpenImmersionCovers}{}
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]]

\begin{enumerate}%
\item jointly surjective collections of [[open immersions of schemes]];


\item single surjective/[[étale morphism of schemes|étale]] morphisms between [[affine schemes]]



\end{enumerate}
(all over $X$).

\end{prop}
(\hyperlink{Tamme}{Tamme, II Lemma (3.1.1)}, \hyperlink{Milne}{Milne, prop. 6.6})

\begin{proof}
Since [[covers]] by standard [[open immersions of schemes]] in the [[Zariski topology]] are also [[étale morphisms of schemes]] and \'e{}tale covers, we may take any \'e{}tale 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.

\end{proof}
\begin{prop}
\label{XSchemesRepresentSheaves}\hypertarget{XSchemesRepresentSheaves}{}
For $Z \to X$ any [[scheme]] over a [[scheme]] $X$, the induced [[presheaf]] on the [[étale site]]

\begin{displaymath}
(U_Y \to X) \mapsto Hom_X(U_Y, Z)
\end{displaymath}
is a [[sheaf]].

\end{prop}
This is due to ([[Grothendieck]], [[SGA]]1 exp. XIII 5.3) A review is in (\hyperlink{Tamme}{Tamme, II theorem (3.1.2)}, \hyperlink{Milne}{Milne, 6.2}).

\begin{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.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
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 $X$.

\end{remark}
(e.g. \hyperlink{Tamme}{Tamme, II theorem 3.1})

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

\begin{example}
\label{}\hypertarget{}{}
The [[additive group]] over $X$ is the [[group scheme]]

\begin{displaymath}
\mathbb{G}_a  \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X
  \,.
\end{displaymath}
By the [[universal property]] of the [[pullback]], the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment

\begin{displaymath}
\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}
  \,.
\end{displaymath}
In other words, the sheaf represented by the [[additive group]] is the [[abelian sheaf]] underlying the [[structure sheaf]] of $X$.

\end{example}
Similarly one finds

\begin{example}
\label{}\hypertarget{}{}
The [[multiplicative group]] over $X$

\begin{displaymath}
\mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X
\end{displaymath}
represents the sheaf $(\mathbb{G}_m)_X$ given by

\begin{displaymath}
(\mathbb{G}_m)_X(U_X)
  \mapsto
  \Gamma(U_X, \mathcal{O}_{U_X})^\times
  \,.
\end{displaymath}
\end{example}
(e.g. \hyperlink{Tamme}{Tamme, II, 3})

\hypertarget{BaseChange}{}\subsubsection*{{Base change and sheaf cohomology}}\label{BaseChange}

\begin{defn}
\label{BaseChangeOnSites}\hypertarget{BaseChangeOnSites}{}
For $f \colon X \longrightarrow Y$ a [[homomorphism]] of [[schemes]], there is induced a [[functor]] on the [[categories]] underlying the [[étale site]]

\begin{displaymath}
f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}
\end{displaymath}
given by sending an [[object]] $U_Y \to Y$ to the [[fiber product]]/[[pullback]] along $f$

\begin{displaymath}
f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{DirectAndInverseImageAlongMapOfBases}\hypertarget{DirectAndInverseImageAlongMapOfBases}{}
The morphism in def. \ref{BaseChangeOnSites} is a [[morphism of sites]] and hence induces a [[geometric morphism]] between the \'e{}tale toposes

\begin{displaymath}
(f^\ast \dashv f_\ast)
  \;\colon\;
  Sh(X_{et})
  \stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}
  Sh(Y_{et})
  \,.
\end{displaymath}
Here the [[direct image]] is given on a [[sheaf]] $\mathcal{F} \in Sh(X_{et})$ by

\begin{displaymath}
f_\ast \mathcal{F} \;\colon\;  (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y)
\end{displaymath}
while the [[inverse image]] is given on a [[sheaf]] $\mathcal{F} \in Sh_(Y_{et})$ by

\begin{displaymath}
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)
  \,.
\end{displaymath}
\end{prop}
By the discussion at \emph{\href{morphism+of+sites#RelationToGeometricMorphisms}{morphisms of sites -- Relation to geometric morphisms}}. See also for instance (\hyperlink{Tamme}{Tamme I 1.4}).

\begin{defn}
\label{}\hypertarget{}{}
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 $X$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
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]]

\begin{displaymath}
(U_Y \to Y)
  \mapsto
  H^q(X \times_Y U_Y, \mathcal{F})
  \,,
\end{displaymath}
where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$ ([[étale cohomology]]).

\end{prop}
By the discussion at \emph{[[direct image]]} and at \emph{[[abelian sheaf cohomology]]}. See e.g. (\hyperlink{Tamme}{Tamme, II (1.3.4)}, \hyperlink{Milne}{Milne prop. 12.1}).

\begin{remark}
\label{}\hypertarget{}{}
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

\begin{displaymath}
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})
  \,.
\end{displaymath}
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

\begin{displaymath}
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})
  \,.
\end{displaymath}
\end{remark}
\hypertarget{QuasiCoherentModules}{}\subsubsection*{{Quasi-coherent modules}}\label{QuasiCoherentModules}

\begin{prop}
\label{}\hypertarget{}{}
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

\begin{displaymath}
N_{et} \;\colon\; (U_X \to X) 
   \mapsto 
  \Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_X})
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Tamme}{Tamme, II 3.2.1})

\hypertarget{relation_to_zariski_topos}{}\subsubsection*{{Relation to Zariski topos}}\label{relation_to_zariski_topos}

\begin{remark}
\label{}\hypertarget{}{}
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 \'e{}tale [[sheaf]] is a Zariski sheaf, but not necessarily conversely. Expressed in a different way, the \'e{}tale topos is a [[subtopos]] of the Zariski topos.

\end{remark}
For more see at \emph{\href{etale+cohomology#RelationZariskiEtaleCohomology}{\'e{}tale cohomology -- Properties -- Relation to Zariski cohomology}}.

\hypertarget{as_a_classifying_topos}{}\subsubsection*{{As a classifying topos}}\label{as_a_classifying_topos}

The \'e{}tale topos over the big \'e{}tale site of [[commutative rings]] is the [[classifying topos]] for [[strict local rings]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[basics of étale cohomology]]


\item [[étale homotopy]]


\item [[étale cohomology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{etale_topos_of_a_schemes}{}\subsubsection*{{Etale topos of a schemes}}\label{etale_topos_of_a_schemes}

\begin{itemize}%
\item [[Günter Tamme]], section II 1 of \emph{[[Introduction to Étale Cohomology]]}

\end{itemize}
\begin{itemize}%
\item [[James Milne]], section 7 of \emph{[[Lectures on Étale Cohomology]]}

\end{itemize}
\hypertarget{etale_topos_of_a_differentially_cohesive_object}{}\subsubsection*{{Etale topos of a differentially cohesive object}}\label{etale_topos_of_a_differentially_cohesive_object}

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://arxiv.org/abs/1310.7930}{arXiv:1310.7930})

\end{itemize}
[[!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]]



\end{document}