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\begin{document}

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\section*{site}

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

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

[[!include topos 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{morita_equivalent_sites}{Morita equivalent sites}\dotfill \pageref*{morita_equivalent_sites} \linebreak
\noindent\hyperlink{subcanonical_sites}{Subcanonical sites}\dotfill \pageref*{subcanonical_sites} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{classes_of_sites}{Classes of sites}\dotfill \pageref*{classes_of_sites} \linebreak
\noindent\hyperlink{specific_sites}{Specific sites}\dotfill \pageref*{specific_sites} \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}

A \emph{site} is a [[locally presentable category|presentation]] of a [[sheaf topos]] as a structure [[free cocompletion|freely generated under colimits]] from a category, subject to the relation that certain [[covering]] colimits are preserved.

As such, sites generalise [[topological spaces]] and [[locales]], which present [[localic topos|localic]] sheaf toposes. More precisely, sites generalise and [[categorify]] [[posites]], which present localic toposes but also present locales themselves in a decategorified manner.

In technical terms, a \emph{site} is a [[small category]] equipped with a [[coverage]] or [[Grothendieck topology]]. The [[category of sheaves]] over a site is a [[sheaf topos]] and the site is a \emph{site of definition} for this topos.

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

\begin{defn}
\label{Site}\hypertarget{Site}{}
A \emph{site} $(C,J)$ is a [[category]] $C$ equipped with a [[coverage]] $J$.

For $\mathcal{E}$ a topos equipped with an [[equivalence of categories]]

\begin{displaymath}
\mathcal{E} \simeq Sh(C,J)
\end{displaymath}
to the [[sheaf topos]] over a site, one says that $(C,J)$ is a \textbf{site of definition} for $\mathcal{E}$.

\end{defn}
Some classes of sites have their special names

\begin{defn}
\label{TypesOfSites}\hypertarget{TypesOfSites}{}
A site is called

\begin{itemize}%
\item a [[small site]], [[large site]], \emph{essentially small site} if the underlying category is a [[small category]], [[large category]], [[essentially small category]], respectively;


\item a \textbf{[[cartesian site]]} if the underlying category is [[finitely complete category|finitely complete]] (which the [[Elephant]] calls a [[cartesian category]]);


\item a \textbf{[[standard site]]} if it is a cartesian site equipped with a [[subcanonical coverage]].



\end{itemize}
\end{defn}
The term [[standard site]] appears in (\hyperlink{Johnstone}{Johnstone, example A2.1.11}).

\begin{remark}
\label{SmallAndLarge}\hypertarget{SmallAndLarge}{}
Often a site is required to be a [[small category]]. But also [[large site]]s play a role.

\end{remark}
\begin{remark}
\label{CoveragesAndTopologies}\hypertarget{CoveragesAndTopologies}{}
Every [[coverage]] induces a [[Grothendieck topology]]. Often sites are defined to be categories equipped with a Grothendieck topology. Some constructions and properties are more elegantly handled with coverages, some with Grothendieck topologies.

Notice that there are many equivalent ways to define a [[Grothendieck topology]], for instance in terms of a system of [[local isomorphisms]], or in terms of a system of [[dense monomorphisms]] in the [[category of presheaves]] $PSh(S)$.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
For $(C,J)$ a site, we write $Sh_J(C)$ for the [[category of sheaves]] on $C$ with respect to the [[coverage]] $J$.

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

\hypertarget{morita_equivalent_sites}{}\subsubsection*{{Morita equivalent sites}}\label{morita_equivalent_sites}

Many inequivalent sites may have equivalent [[sheaf topos]]es.

\begin{prop}
\label{}\hypertarget{}{}
Every [[sheaf topos]] has a \hyperlink{TypesOfSites}{standard site} of definition.

\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, theorem C2.2.8 (iii)}).

\begin{remark}
\label{}\hypertarget{}{}
By \href{classifying+topos#SheafToposesAreClassifyingForTheirTheoryOfLocalAlgegras}{this corollary} at \emph{[[classifying topos]]} this means that every sheaf topos is the [[classifying topos]] for some [[theory]] of [[local algebras]].

\end{remark}
\hypertarget{subcanonical_sites}{}\subsubsection*{{Subcanonical sites}}\label{subcanonical_sites}

\begin{prop}
\label{CharacterizationOfSubcanonicalSites}\hypertarget{CharacterizationOfSubcanonicalSites}{}
For $\mathcal{E}$ a [[sheaf topos]], the [[essentially small category|essentially small]] sites of definition $(\mathcal{C}, J)$ of $\mathcal{E}$ such that $J$ is a [[subcanonical coverage]] are precisely the [[full subcategories]] on [[generating families]] of objects equipped with the coverages induced from the [[canonical coverage]] of $\mathcal{E}$.

\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, prop. C2.2.16}).

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

\hypertarget{classes_of_sites}{}\subsubsection*{{Classes of sites}}\label{classes_of_sites}

\begin{itemize}%
\item Every [[frame]] is canonically a site, where $U$ is covered by $\{U_i\}$ precisely if it is their [[join]].

A subclass of examples is the [[category of open subsets]] of a [[topological space]].

This are examples of [[posite]]s/[[(0,1)-site]].


\item Various categories come with canonical structures of sites on them:

\begin{itemize}%
\item For every category $C$ there is its [[canonical coverage]].


\item On every [[regular category]] there is its [[regular coverage]].


\item On every [[coherent category]] there is its [[coherent coverage]].


\item Generalizing the previous two examples, on an [[∞-ary regular category]] there is a $\kappa$-canonical coverage.



\end{itemize}
If the category in question is the [[syntactic category]] of a [[theory]], the corresponding site is the [[syntactic site]].


\item For every site there is the corresponding [[double negation topology]] that forces the sheaf topos to a [[Boolean topos]].



\end{itemize}
Other classes of sites are listed in the following.

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


\item [[dense sub-site]]


\item [[large site]], [[essentially small site]]


\item [[concrete site]]


\item [[locally connected site]] / [[locally ∞-connected site]]

\begin{itemize}%
\item [[connected site]] / [[∞-connected site]]


\item [[strongly connected site]] / [[strongly ∞-connected site]]


\item [[totally connected site]] / [[totally ∞-connected site]]



\end{itemize}

\item [[local site]] / [[∞-local site]]


\item [[cohesive site]], [[∞-cohesive site]]


\item [[atomic site]]



\end{itemize}
\hypertarget{specific_sites}{}\subsubsection*{{Specific sites}}\label{specific_sites}

\begin{itemize}%
\item Sites for [[big topos]]es defining notions of [[higher geometry|geometry]] are:

\begin{itemize}%
\item The sites that define the [[higher geometry|geometry]] called [[differential geometry]] are [[CartSp]]${}_{smooth}$, [[SmoothMfd]], etc, equipped with the [[open cover]] [[coverage]]. Or more generally [[smooth loci]] etc.


\item The sites that induce [[topological geometry]] are small versions of [[Top]] equipped with the [[open cover]] [[coverage]].


\item The sites that induce the [[higher geometry]] modeled on [[Euclidean topology]] are the large site of [[paracompact manifold]]s and its [[dense sub-site]] [[CartSp]]${}_{top}$.



\end{itemize}

\item The sites that define the [[higher geometry|geometry]] called [[algebraic geometry]] are site structures on categories of formal duals of [[commutative ring]]s or commutative [[associative algebra]]s

\begin{itemize}%
\item [[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]

[[crystalline site]]



\end{itemize}


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

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


\item \textbf{site}

\begin{itemize}%
\item [[internal site]]
\item [[∞-ary site]]

\end{itemize}

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


\item [[(∞,1)-site]]

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

\end{itemize}

\item [[morphism of sites]], [[covering lifting property]]



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

In

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Elephant|Sketches of an Elephant]]}

\end{itemize}
sites are discussed in section C2.1.

[[!redirects sites]]

[[!redirects 1-site]] [[!redirects (1,1)-site]] [[!redirects 1-sites]] [[!redirects (1,1)-sites]]



\end{document}