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\begin{document}

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\section*{étale (infinity,1)-site}

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

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

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

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

[[!include etale morphisms - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsDerivedGeometry}{Derived \'e{}tale geometry}\dotfill \pageref*{AsDerivedGeometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{\'e{}tale (∞,1)-site} is an [[(∞,1)-site]] whose [[(∞,1)-topos]] encodes the [[derived geometry]] version of the geometry encoded by the [[topos]] over the [[étale site]].

Its underlying [[(∞,1)-category]] is the [[opposite (∞,1)-category]] $sCAlg_k^{op}$ of commutative [[simplicial algebras]] over a commutative ring $k$, whose [[covering]] families are essentially those which under [[decategorification]] become coverings in the [[étale site]] of ordinary $k$-algebras.

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

Let $k$ be a commutative ring. Let $T$ be the [[Lawvere theory]] of commutative [[associative algebra]]s over $k$.

\begin{defn}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
\infty CAlg_k := T Alg_\infty
\end{displaymath}
be the [[(∞,1)-category]] of [[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]] over $T$ regarded as an [[(∞,1)-algebraic theory]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Let $sCAlg_k = (T Alg)^{\Delta^{op}}$ be the [[sSet]]-[[enriched category]] of [[simplicial algebra|simplicial commutative associative k-algebras]] equipped with the standard [[model structure on simplicial T-algebras]]. Write $sCAlg_k^\circ$ for the $(\infty,1)$-category [[presentable (∞,1)-category]]. Then we have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\infty CAlg_k \simeq (sCAlg_k)^\circ
  \,.
\end{displaymath}
\end{prop}
This is a special case of the general statement discussed at [[(∞,1)-algebraic theory]]. See also (\hyperlink{Lurie}{Lurie, remark 4.1.2}).

\begin{defn}
\label{}\hypertarget{}{}
For $X \in \infty CAlg_k^{op}$ we write $\mathcal{O}(X)$ for the corresponding object in $\infty CAlg_k$ and conversely for $A \in \infty CAlg_k$ we write $Spec A$ for the corresponding object in $\infty CAlg_k^{op}$.

So $Spec \mathcal{O} X = X$ and $\mathcal{O} Spec A = A$, by definition of notation.

\end{defn}
Notice from the discussion at [[model structure on simplicial algebras]] the [[homotopy group]] functor

\begin{displaymath}
\pi_* : sCAlg_k \to CAlg_k
  \,.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
A morphism $Spec A  \to Spec B$ in $\infty CAlg_k^{op}$ is an \textbf{\'e{}tale morphism} if

\begin{enumerate}%
\item The underlying morphism $Spec \pi_0(A) \to Spec \pi_0(B)$ is an [[étale morphism]] of [[schemes]];


\item for each $i \in \mathbb{N}$ the canonical morphism

\begin{displaymath}
\pi_i(A) \otimes_{\pi_0(A)} \pi_0(B) \to \pi_i(B)
\end{displaymath}
is an [[isomorphism]].



\end{enumerate}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The \textbf{\'e{}tale $(\infty,1)$-site} is the [[(∞,1)-site]] whose underlying $(\infty,1)$-category is the [[opposite (∞,1)-category]] $\infty CAlg_k^{op}$ and whose [[covering]] famlies $\{Spec A_i \to Spec B\}_{i \in I}$ are those collections of morphisms such that

\begin{enumerate}%
\item every $Spec A_i \to Spec B$ is an \'e{}tale morphism


\item there is a [[finite set|finite subset]] $J \subset I$ such that the underlying decategorified family $\{Spec \pi_0(A_j) \to Spec \pi_0(B)\}_{j \in J}$ is a covering family in the 1-[[étale site]].



\end{enumerate}
\end{defn}
This appears as (\hyperlink{ToenVezzosi}{To\"e{}nVezzosi, def. 2.2.2.12}) and as (\hyperlink{Lurie}{Lurie, def. 4.3.3; def. 4.3.13}).

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

\hypertarget{AsDerivedGeometry}{}\subsubsection*{{Derived \'e{}tale geometry}}\label{AsDerivedGeometry}

The following definition and theorem show how the \'e{}tale $(\infty,1)$-site arises naturally from the [[étale site|étale 1-site]], and naturally encodes the [[derived geometry]] induced by the \'e{}tale site.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(\'e{}tale pregeometry)}

Let $\mathcal{T}_{et}$ be the 1-[[étale site]] regarded as a [[pregeometry (for structured (∞,1)-toposes)]] as follows.

\begin{itemize}%
\item the underlying [[(∞,1)-category]] is the 1-[[category]]

\begin{displaymath}
(CAlg_k^{sm})^{op} \hookrightarrow CAlg_k ^{op}
  \,,
\end{displaymath}
which is the full [[subcategory]] of $CAlg_k$ on those objects $A \in CAlg_k$ for which there exists an [[étale morphism]] $k[x^1, \cdots, x^n] \to A$ from the [[polynomial]] algebra in $n$ generators for some $n \in \mathbb{N}$;


\item the \emph{admissible morphisms} in the pregeometry are the [[étale morphism]]s;


\item a collection of admissible morphisms is a [[covering]] family if it is so as a family of morphisms in the [[étale site]].



\end{itemize}
\end{defn}
This is (\hyperlink{Lurie}{Lurie, def. 4.3.1}).

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(\'e{}tale geometry)}

Let $\mathcal{G}_{et}$ be the [[geometry (for structured (∞,1)-toposes)]] given by

\begin{itemize}%
\item the underlying [[(∞,1)-site]] is the \'e{}tale $(\infty,1)$-site;


\item the admissible morphisms are the \'e{}tale morphisms.



\end{itemize}
\end{defn}
This is (\hyperlink{Lurie}{Lurie, def. 4.3.13}).

\begin{theorem}
\label{}\hypertarget{}{}
The [[geometry (for structured (∞,1)-toposes)|geometry]] generated by the \'e{}tale pregeometry $\mathcal{T}_{et}$ is the \'e{}tale geometry $\mathcal{G}_{et}$.

\end{theorem}
This is (\hyperlink{Lurie}{Lurie, prop. 4.3.15}).

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

\begin{itemize}%
\item [[étale morphism]], [[étale site]], [[étale cohomology]]


\item [[étale morphism of E-∞ rings]]


\item \textbf{\'e{}tale $(\infty,1)$-site}



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

In its [[presentable (∞,1)-category|presentation]] as a [[model site]] the \'e{}tale $(\infty,1)$-site is given in definition 2.2.2.12 of

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Homotopical Algebraic Geometry II: geometric stacks and applications} (\href{http://arxiv.org/abs/math/0404373}{arXiv}) .

\end{itemize}
A discussion in the context of [[structured (∞,1)-toposes]] is

\begin{itemize}%
\item [[Jacob Lurie]], section 4.3 of \emph{[[Structured Spaces]]}

\end{itemize}
\begin{itemize}%
\item [[Jacob Lurie]], section 3 and 4 of, \emph{Descent theorems} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item [[Benjamin Antieau]], [[David Gepner]], \emph{Brauer groups and \'e{}tale cohomology in derived algebraic geometry} (\href{http://arxiv.org/abs/1210.0290}{arXiv:1210.0290})

\end{itemize}
[[!redirects étale (∞,1)-site]] [[!redirects etale (∞,1)-site]]

[[!redirects etale (infinity,1)-site]]



\end{document}