The notion of site may be internalized in any topos to yield a notion of internal site.
The definition of internal site is obvious and straightforward.
For $\mathcal{E}$ a topos, an internal site in $\mathcal{E}$ is an internal category $\mathbb{C} = C_1 \rightrightarrows C_0$ equipped with an internal coverage.
Spelled out in components, this means the following (as in (Johnstone), we shall only define sifted coverages). First, we define the subobject $Sv(\mathbb{C}) \hookrightarrow PC_1$ of sieves, where a subobject $S \hookrightarrow C_1$ is a sieve if the composite
factors through $S$. Also recall the usual membership relation $\in_{C_1} \stackrel{(n,e)}{\to} PC_1 \times C_1$.
An internal sifted coverage is given by a span $C_0 \stackrel{b}{\leftarrow} T \stackrel{c}{\to} Sv(\mathbb{C})$ subject to:
The square
commutes, where the pullback in the top left corner is of the map $\in_{C_1} \to PC_1$ along $T \to Sv(\mathbb{C}) \hookrightarrow PC_1$.
If we define the subobject $Q\hookrightarrow T\times_{C_0} C_1 \times_{C_0} T$ as
(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.
We can additionally ask that more saturation conditions (as discussed at coverage) hold.
(…)
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.
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
This is reviewed for instance in (Johnstone, p. 596).
The notation is motivated from the following example.
Let $G$ be a group (in Set, hence a discrete group) and let $\mathcal{C} := \mathbf{B}G$ be its delooping groupoid. Then
is the topos of permutation representations 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
is the delooping groupoid of the semidirect product group of the $G$-action on $H$.
Generally we have
The category $\mathcal{C} \rtimes \mathbb{D}$ from def. 3 is described as follows:
We have an equivalence of categories
between the category of internal presheaves in $\mathcal{E}$ over the internal category $\mathbb{D}$, and external presheaves over the semidirect product site $\mathcal{C} \rtimes \mathbb{D}$.
This appears as (Johnstone, lemma C2.5.3).
This result generalizes straightforwardly to an analogous statement for internal sheaves.
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 $b(S) = V$, …
Let $\mathcal{E} = Sh_J(\mathcal{C})$ be a sheaf topos and $(\mathbb{D}, K)$ an internal site in $\mathcal{E}$. Then with def. 4 we have an equivalence of categories
between internal sheaves in $\mathcal{E}$ on $\mathbb{D}$ and 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
This appears as (Johnstone, prop. C2.5.4).
Section C2.4 and C2.5 of
The semidirect product externalization of internal sites is due to