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\begin{document}

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\section*{internal 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{externalization}{Externalization}\dotfill \pageref*{externalization} \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 [[site]] may be [[internalization|internalized]] in any [[topos]] to yield a notion of \emph{internal site}.

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

The definition of internal site is obvious and straightforward.

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{E}$ a [[topos]], an \textbf{internal site} in $\mathcal{E}$ is an [[internal category]] $\mathbb{C} = C_1 \rightrightarrows C_0$ equipped with an internal [[coverage]].

\end{defn}
Spelled out in components, this means the following (as in (\hyperlink{Johnstone}{Johnstone}), we shall only define \emph{sifted} coverages). First, we define the [[subobject]] $Sv(\mathbb{C}) \hookrightarrow PC_1$ of \emph{[[sieves]]}, where a subobject $S \hookrightarrow C_1$ is a sieve if the composite

\begin{displaymath}
S\times_{C_0} C_1 \to C_1\times_{C_0} C_1 \to C_1
\end{displaymath}
factors through $S$. Also recall the usual membership relation $\in_{C_1} \stackrel{(n,e)}{\to} PC_1 \times C_1$.

\begin{defn}
\label{}\hypertarget{}{}
An \emph{internal sifted coverage} is given by a span $C_0 \stackrel{b}{\leftarrow} T \stackrel{c}{\to} Sv(\mathbb{C})$ subject to:

\begin{itemize}%
\item The square

\begin{displaymath}
\itexarray{
T \times_{PC_1} \in_{C_1} & \stackrel{e pr_2}{\to} & C_1 \\
{}^{pr_1}\downarrow   & {}                & \downarrow^{s} \\
T                  & \stackrel{b}{\to} & C_0
}
\end{displaymath}
commutes, where the pullback in the top left corner is of the map $\in_{C_1} \to PC_1$ along $T \to Sv(\mathbb{C}) \hookrightarrow PC_1$.


\item If we define the subobject $Q\hookrightarrow T\times_{C_0} C_1 \times_{C_0} T$ as

\begin{displaymath}
Q := \{(t',a,t) | aa' \in t \forall a'\in t'\}
\end{displaymath}
(in the internal language), the composite $Q \hookrightarrow T\times_{C_0} C_1 \times_{C_0} T \stackrel{pr_{23}}{\to} C_1 \times_{C_0} T$ is required to be an epimorphism.



\end{itemize}
\end{defn}
We can additionally ask that more saturation conditions (as discussed at [[coverage]]) hold.

(\ldots{})

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

\hypertarget{externalization}{}\subsubsection*{{Externalization}}\label{externalization}

We discuss how to every internal site there is a corresponding external site such that the [[internal sheaf topos]] on the former agrees with the external sheaf topos on the latter.

\begin{defn}
\label{TheExternalizedSite}\hypertarget{TheExternalizedSite}{}
Let $\mathcal{C}$ be a [[small category]] and let $\mathcal{E} := [\mathcal{C}^{op}, Set]$ be its [[presheaf topos]]. Let $\mathbb{D} \in \mathcal{E}$ be an internal site. Regarded, by the [[Yoneda lemma]], as a functor $\mathbb{D} : \mathcal{C}^{op} \to Cat$, this induces via the [[Grothendieck construction]] a [[fibered category]] which we denote

\begin{displaymath}
\mathcal{C} \rtimes \mathbb{D} \to \mathcal{C}
  \,.
\end{displaymath}
\end{defn}
This is reviewed for instance in (\hyperlink{Johnstone}{Johnstone, p. 596}).

The notation is motivated from the following example.

\begin{example}
\label{}\hypertarget{}{}
Let $G$ be a [[group]] (in [[Set]], hence a [[discrete group]]) and let $\mathcal{C} := \mathbf{B}G$ be its [[delooping]] [[groupoid]]. Then

\begin{displaymath}
\mathcal{E} \simeq [\mathbf{B}G , Set]
\end{displaymath}
is the topos of [[permutation representation]]s of $G$. Let $H \in \mathcal{E}$ be a [[group object]]. This is equivalently a group in $Set$ equipped with a $G$-[[action]]. Its internal delooping gives the [[internal groupoid]] $\mathbb{D} := \mathcal{B}H$ in $\mathcal{E}$.

In this case we have that

\begin{displaymath}
\mathcal{C} \rtimes \mathbb{D} \simeq \mathbf{B}(G \rtimes H)
\end{displaymath}
is the delooping groupoid of the [[semidirect product]] group of the $G$-action on $H$.

\end{example}
Generally we have

\begin{remark}
\label{}\hypertarget{}{}
The category $\mathcal{C} \rtimes \mathbb{D}$ from def. \ref{TheExternalizedSite} is described as follows:

\begin{itemize}%
\item [[object]]s are pairs $(U,V)$ with $U \in Ob \mathcal{C}$ and $V \in Ob \mathbb{D}(U)$;


\item [[morphism]]s $(U',V') \to (U,V)$ are pairs $(a,b)$ where $a : U' \to U$ is in $\mathcal{C}$ and $b : V' \to \mathbb{D}(a)(V)$ in $\mathbb{D}(U')$.



\end{itemize}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
We have an [[equivalence of categories]]

\begin{displaymath}
[\mathbb{D}^{op}, [\mathcal{C}^{op}, Set]]
   \simeq
  [(\mathcal{C} \rtimes \mathbb{D})^{op}, Set]
\end{displaymath}
between the [[category of internal presheaves]] in $\mathcal{E}$ over the internal category $\mathbb{D}$, and external presheaves over the semidirect product site $\mathcal{C} \rtimes \mathbb{D}$.

\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, lemma C2.5.3}).

This result generalizes straightforwardly to an analogous statement for internal sheaves.

\begin{defn}
\label{CoverageOnTheExternalizedSite}\hypertarget{CoverageOnTheExternalizedSite}{}
If $\mathcal{C}$ is equipped with a [[coverage]] $J$ and $\mathbb{D}$ is equipped with an internal coverage $K$ , define a coverage $J \rtimes K$ on $\mathcal{C} \rtimes \mathbb{D}$ by declaring that a [[sieve]] on an object $(U,V)$ is $(J \times K)$-covering if there exists an element $S \in K(U)$ with $b(S) = V$, \ldots{}

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{E} = Sh_J(\mathcal{C})$ be a [[sheaf topos]] and $(\mathbb{D}, K)$ an internal site in $\mathcal{E}$. Then with def. \ref{CoverageOnTheExternalizedSite} we have an [[equivalence of categories]]

\begin{displaymath}
Sh_{K}(\mathbb{D})
  \simeq
  Sh_{J \rtimes K}(\mathcal{C} \rtimes \mathbb{D})
\end{displaymath}
between [[internal sheaves]] in $\mathcal{E}$ on $\mathbb{D}$ and [[sheaf|external sheaves]] on the semidirect product site $J \rtimes K$.

Moreover, the [[projection]] [[functor]] $P : \mathcal{C} \rtimes \mathbb{D}$ is [[cover-reflecting]] and induces a [[geometric morphism]]

\begin{displaymath}
\Gamma 
   \colon
  Sh_K(\mathbb{D})
  \stackrel{}{\to}
  \mathcal{E}
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, prop. C2.5.4}).

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

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


\item [[internal locale]], [[internal sheaf]]


\item [[internalization]], [[internal logic]]



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

Section C2.4 and C2.5 of

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

\end{itemize}
The semidirect product externalization of internal sites is due to

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Continuous fibrations and inverse limits of toposes}, Composition Math. 68 (1986) (\href{http://www.numdam.org/item?id=CM_1986__58_1_45_0}{NUMDAM})

\end{itemize}
[[!redirects internal sites]]



\end{document}