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\section*{2-site}

This entry is about the notion of [[site]] in [[2-category]] theory. For the notion ``bisite'' of a 1-categorical site equipped with two coverages see instead [[separated presheaf]].

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,2)$-Topos theory}}\label{topos_theory}

[[!include (infinity,2)-topos theory - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{saturation_conditions}{Saturation conditions}\dotfill \pageref*{saturation_conditions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \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 notion of [[2-site]] is the generalization of the notion of [[site]] to the [[higher category theory]] of [[2-categories]] ([[bicategories]]).

Over a 2-site one has a [[2-topos]] of [[2-sheaves]].

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

A \textbf{[[coverage]]} on a [[2-category]] $C$ consists of, for each object $U\in C$, a collection of families $(f_i: U_i\to U)_i$ of morphisms with codomain $U$, called \emph{covering families}, such that

\begin{itemize}%
\item If $(f_i:U_i\to U)_i$ is a covering family and $g:V\to U$ is a morphism, then there exists a covering family $(h_j:V_j\to V)_j$ such that each composite $g h_j$ factors through some $f_i$, up to isomorphism.

\end{itemize}
This is the 2-categorical analogue of the 1-categorical notion of [[nlab:coverage|coverage]] introduced in the [[nlab:Elephant|Elephant]].

A 2-category equipped with a coverage is called a \textbf{2-site}.

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

\begin{itemize}%
\item If $C$ is a [[michaelshulman:regular 2-category]], then the collection of all singleton families $(f:V\to U)$, where $f$ is eso, forms a coverage called the \emph{regular coverage}.


\item Likewise, if $C$ is a [[michaelshulman:coherent 2-category]], the collection of all finite jointly-eso families forms a coverage called the \emph{coherent coverage}.


\item On $Cat$, the \emph{canonical coverage} consists of all families that are jointly essentially surjective on objects.



\end{itemize}
\hypertarget{saturation_conditions}{}\subsection*{{Saturation conditions}}\label{saturation_conditions}

A \textbf{pre-Grothendieck coverage} on a 2-category is a coverage satisfying the following additional conditions:

\begin{itemize}%
\item If $f:V\to U$ is an equivalence, then the one-element family $(f:V\to U)$ is a covering family.


\item If $(f_i:U_i\to U)_{i\in I}$ is a covering family and for each $i$, so is $(h_{i j}:U_{i j} \to U_i)_{j\in J_i}$, then $(f_i h_{i j}:U_{i j}\to U)_{i\in I, j\in U_i}$ is also a covering family.



\end{itemize}
This is the 2-categorical version of a [[Grothendieck pretopology]] (minus the common condition of having actual [[pullbacks]]).

Now, a \textbf{sieve} on an object $U\in C$ is defined to be a functor $R:C^{op}\to Cat$ with a transformation $R\to C(-,U)$ which is objectwise fully faithful (equivalently, it is a [[fully faithful morphism]] in $[C^{op},Cat]$). Equivalently, it may be defined as a subcategory of the [[slice 2-category]] $C/U$ which is closed under precomposition with all morphisms of $C$.

Every family $(f_i\colon U_i\to U)_i$ generates a sieve by defining $R(V)$ to be the full subcategory of $C(V,U)$ on those $g:V\to U$ such that $g \cong f_i h$ for some $i$ and some $h:V\to U_i$. The following observation is due to \hyperlink{StreetCBS}{StreetCBS}.

\begin{lemma}
\label{}\hypertarget{}{}
A 2-presheaf $X:C^{op}\to Cat$ is a 2-sheaf for a covering family $(f_i:U_i\to U)_i$ if and only if

\begin{displaymath}
X(U) \simeq[C^{op},Cat](C(-,U),X) \to [C^{op},Cat](R,X)
\end{displaymath}
is an equivalence, where $R$ is the sieve on $U$ generated by $(f_i:U_i\to U)_i$.

\end{lemma}
Therefore, just as in the 1-categorical case, it is natural to restrict attention to covering \emph{sieves}. We define a \textbf{Grothendieck coverage} on a 2-category $C$ to consist of, for each object $U$, a collection of sieves on $U$ called covering sieves, such that

\begin{itemize}%
\item If $R$ is a covering sieve on $U$ and $g:V\to U$ is any morphism, then $g^*(R)$ is a covering sieve on $V$.


\item For each $U$ the sieve $M_U$ consisting of \emph{all} morphisms into $U$ (the sieve generated by the singleton family $(1_U)$) is a covering sieve.


\item If $R$ is a covering sieve on $U$ and $S$ is an arbitrary sieve on $U$ such that for each $f:V\to U$ in $R$, $f^*(S)$ is a covering sieve on $V$, then $S$ is also a covering sieve on $U$.



\end{itemize}
Here if $R$ is a sieve on $U$ and $g:V\to U$ is a morphism, $g^*(R)$ denotes the sieve on $V$ consisting of all morphisms $h$ into $V$ such that $g h$ factors, up to isomorphism, through some morphism in $R$.

As in the 1-categorical case, one can then show that every coverage generates a unique Grothendieck coverage having the same 2-sheaves.

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

\begin{itemize}%
\item The [[2-category]] of [[2-sheaf|2-sheaves]] on a 2-site is a [[Grothendieck 2-topos]].


\item If $C$ is a [[1-category]] regarded as a 2-category with only [[identity morphism|identity]] [[2-morphisms]], then a coverage (pretopology, topology) on $C$ reduces to the usual notion of [[coverage]], [[Grothendieck pretopology]], or [[Grothendieck topology]].



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

\begin{itemize}%
\item [[(n,r)-site]]


\item [[1-site]]


\item \textbf{2-site}, [[(2,1)-site]]


\item [[(∞,1)-site]]

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

\end{itemize}


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

Strict 2-sites were considered in

\begin{itemize}%
\item [[Ross Street]], \emph{Two-dimensional sheaf theory} J. Pure Appl. Algebra \textbf{23} (1982), no. 3, 251-270, \href{http://www.ams.org/mathscinet-getitem?mr=644277}{MR83d:18014}, 

\end{itemize}
Bicategorical 2-sites in

\begin{itemize}%
\item [[Ross Street]], [[zoranskoda:Characterization of Bicategories of Stacks]], \href{http://www.ams.org/mathscinet-getitem?mr=682967}{MR84d:18006}, p. 282-291 in: Category theory (Gummersbach 1981) Springer LNM \textbf{962}, 1982

\end{itemize}
See also \emph{[[StreetCBS]]}.

More discussion is in

\begin{itemize}%
\item [[Michael Shulman]], \emph{[[michaelshulman:2-site]]}

\end{itemize}
[[!redirects 2-sites]]

[[!redirects (2,2)-site]] [[!redirects (2,2)-sites]]

[[!redirects bisite]] [[!redirects bisites]] [[!redirects Grothendieck 2-topology]] [[!redirects 2-Grothendieck topology]] [[!redirects Grothendieck 2-topologies]] [[!redirects 2-Grothendieck topologies]] [[!redirects Grothendieck 2-pretopology]] [[!redirects 2-Grothendieck pretopology]] [[!redirects Grothendieck 2-pretopologies]] [[!redirects 2-Grothendieck pretopologies]] [[!redirects 2-coverage]] [[!redirects 2-coverages]] [[!redirects topology on a 2-category]] [[!redirects topologies on 2-categories]] [[!redirects Grothendieck topology on a 2-category]] [[!redirects Grothendieck topologies on 2-categories]] [[!redirects coverage on a 2-category]] [[!redirects coverages on 2-categories]] [[!redirects topology on a bicategory]] [[!redirects topologies on bicategories]] [[!redirects Grothendieck topology on a bicategory]] [[!redirects Grothendieck topologies on bicategories]] [[!redirects coverage on a bicategory]] [[!redirects coverages on bicategories]]



\end{document}