internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
The notion of site may be internalized in any topos to yield a notion of internal site.
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
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. 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
Last revised on March 9, 2023 at 02:31:40. See the history of this page for a list of all contributions to it.