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\begin{document}

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\section*{(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{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{of_sieves}{Of sieves}\dotfill \pageref*{of_sieves} \linebreak
\noindent\hyperlink{of_coverages}{Of coverages}\dotfill \pageref*{of_coverages} \linebreak
\noindent\hyperlink{of_sites}{Of sites}\dotfill \pageref*{of_sites} \linebreak
\noindent\hyperlink{incarnations_and_models}{Incarnations and models}\dotfill \pageref*{incarnations_and_models} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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 structure of an $(\infty,1)$-site on an [[(∞,1)-category]] $C$ is precisely the data encoding an [[(∞,1)-category of (∞,1)-sheaves]]

\begin{displaymath}
Sh(C) \hookrightarrow PSh(C)
\end{displaymath}
inside the [[(∞,1)-category of (∞,1)-presheaves]] on $C$.

The notion is the analog in [[(∞,1)-category]] theory of the notion of a [[site]] in 1-[[category theory]].

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

The definition of $(\infty,1)$-sites parallels that of 1-categorical [[site]]s closely. In fact the structure of an $(\infty,1)$-site on an $(\infty,1)$-category is equivalent to that of a 1-categorical site on its [[homotopy category of an (infinity,1)-category|homotopy category]] (see below).

\begin{udefn}
\textbf{($(\infty,1)$-Grothendieck topology)}

A \textbf{[[sieve]] in} an [[(∞,1)-category]] $C$ is a full [[sub-(∞,1)-category]] $D \subset C$ which is closed under precomposition with morphisms in $C$.

A \textbf{sieve on} an [[object]] $c \in C$ is a sieve in the [[over quasi-category|overcategory]] $C_{/c}$.

Equivalently, a sieve on $c$ is an [[equivalence class]] of [[monomorphism in an (infinity,1)-category|monomorphisms]] $U \to j(c)$ in the [[(∞,1)-category of (∞,1)-presheaves]] $PSh(C)$, with $j : C \to PSh(C)$ the [[(∞,1)-Yoneda embedding]]. (See below for the proof of this equivalence).

For $S$ a sieve on $c$ and $f : d \to c$ a [[morphism]] into $c$, we take the \textbf{pullback sieve} $f^* S$ on $d$ to be that spanned by all those morphisms into $d$ that become equivalent to a morphism in $S$ after postcomposition with $f$.

A \textbf{[[Grothendieck topology]]} on the $(\infty,1)$-category $C$ is the specification of a collection of sieves on each object of $C$ -- called the \textbf{covering sieves} , subject to the following conditions:

\begin{enumerate}%
\item \emph{the trivial sieve covers} -- For each object $c \in C$ the overcategory $C_{/c}$ regarded as a maximal subcategory of itself is a covering sieve on $c$. Equivalently: the monomorphism $Id : j(c) \to j(c)$ covers.


\item \emph{the pullback of a sieve covers} -- If $S$ is a covering sieve on $c$ and $f : d \to c$ a morphism, then the pullback sieve $f^* S$ is a covering sieve on $d$. Equivalently, the [[pullback]]

\begin{displaymath}
\itexarray{  
     f^* U &\to& U 
     \\
     \downarrow && \downarrow
     \\
     d &\stackrel{f}{\to}& c
  }
\end{displaymath}
in $PSh(C)$ is covering.


\item \emph{a sieve covers if its pullbacks cover} -- For $S$ a covering sieve on $c$ and $T$ any sieve on $c$, if the pullback sieve $f^* T$ for every $f \in S$ is covering, then $T$ itself is covering.



\end{enumerate}
An $(\infty,1)$-category equipped with a Grothendieck topology is an \textbf{$(\infty,1)$-site}.

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

\hypertarget{of_sieves}{}\subsubsection*{{Of sieves}}\label{of_sieves}

\begin{ulemma}
A sieve $S'$ on $c$ that contains a covering sieve $S \subset S'$ is itself covering.

\end{ulemma}
\begin{proof}
For every $f : d \to c$ an object of $S \subset C_{/c}$, the pullback sieve $f^* S'$ equals the pullback sieve $f^* S$. So it covers $d$ by the second axiom on sieves. So by the third axiom $S'$ itself is covering.

\end{proof}
\begin{uproposition}
There is a natural bijection between sieves on $c$ in $C$ and equivalence class of [[monomorphism in an (infinity,1)-category|monomorphisms]] $U \to j(C)$ in $PSh(C)$.

\end{uproposition}
This is [[Higher Topos Theory|HTT, prop. 6.2.2.5]].

\begin{proof}
First observe that equivalence classes of $(-1)$-[[truncated]] object of $PSh(C_{/c})$ are in bijection with sieves on $c$:

An $(\infty,1)$-presheaf $F$ is $(-1)$-truncated if its value on any object is either the empty [[∞-groupoid]] $\emptyset$ or a [[contractible]] $\infty$-groupoid. The full subcategory of $C_{/c}$ on those objects on which $F$ takes a contractible value is evidently a sieve (because there is no morphism from a contractible to the empty $\infty$-groupoid). Conversely, given a sieve $S$ on $c$ we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value $* = \Delta[0] \in \infty Grpd$ on those objects that are in $S$, and $\emptyset$ otherwise.

Now, as described at  we have an equivalence

\begin{displaymath}
PSh(C_{/c})
  \simeq
  PSh(C)_{/j(c)}
  \,.
\end{displaymath}
Under this equivalence our bijection above maps to the statement that there is a bijection between sieves on $c$ and equivalence class of $(-1)$-[[truncated]] objects in $PSh(C)_{/j (c)}$. But such a (-1)-truncated object is precisely a [[monomorphism in an (infinity,1)-category|monomorphism]] $U \to j(c)$.

\end{proof}
\hypertarget{of_coverages}{}\subsubsection*{{Of coverages}}\label{of_coverages}

\begin{ulemma}
The set of Grothendieck topologies on an $(\infty,1)$-category $C$ is in natural bijection with the set of Grothendieck topologies on its [[homotopy category of an (infinity,1)-category|homotopy category]].

\end{ulemma}
This is [[Higher Topos Theory|HTT, remark 6.2.2.3]].

\begin{proof}
Because picking full sub-1-categories as well as full sub-$(\infty,1)$-categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.

\end{proof}
\begin{ucorollary}
If the $(\infty,1)$-category $C$ happens to be an ordinary [[category]] (for instance in its incarnation as a [[quasi-category]] it is the [[nerve]] of an ordinary [[category]]), then the structure of an $(\infty,1)$-site on it is the same as the 1-categorical structure of a [[site]] on it.

\end{ucorollary}
\hypertarget{of_sites}{}\subsubsection*{{Of sites}}\label{of_sites}

\begin{uprop}
Structures of $(\infty,1)$-sites on an [[(∞,1)-category]] $C$ correspond bijectively to [[topological localization]]s of the [[(∞,1)-category of (∞,1)-presheaves]] to a [[(∞,1)-category of (∞,1)-sheaves]]. See there for more details.

\end{uprop}
\hypertarget{incarnations_and_models}{}\subsection*{{Incarnations and models}}\label{incarnations_and_models}

If [[(∞,1)-categories]] are incarnated as [[simplicially enriched categories]], then an $(\infty,1)$-site appears as an

\begin{itemize}%
\item [[sSet-site]]

\end{itemize}
If $(\infty,1)$-categories are [[presentable (∞,1)-category|presented]] by [[model categories]], then the notion of $(\infty,1)$-site appears as that of

\begin{itemize}%
\item [[model site]].

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The \textbf{trivial Grothendieck-topology} on an $(\infty,1)$-category is that where the only covering sieve on each object $c$ is $C_{/c}$ itself. Equivalently, where the only covering monomorphisms $U \to j(c)$ in $PSh(C)$ are the equivalences.

The [[(∞,1)-category of (∞,1)-sheaves]] on this site is just the [[(∞,1)-category of (∞,1)-presheaves]] itself. The localization is an equivalence.


\item [[étale (∞,1)-site]]



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

\begin{itemize}%
\item [[site]]


\item [[2-site]], [[(2,1)-site]]


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

\begin{itemize}%
\item [[model site]], [[simplicial site]]

\end{itemize}
\emph{[[infinity-cohesive site]]}


\item [[internal site]] / [[internal (infinity,1)-site]]



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

Section 6.2.2 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

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

[[!redirects (infinity,1)-sites]] [[!redirects (∞,1)-sites]] [[!redirects (∞,1)-Grothendieck topology]] [[!redirects (∞,1)-Grothendieck topologies]]



\end{document}