Over topos is a topos

Question

Suppose we have a topos $Shv(\mathcal{C})$, category of sheaves on a small site $\mathcal{C}$ and $S$ is an object of $Shv(\mathcal{C})$. Then whether the over category $Shv(\mathcal{C})/S$ is equivalent with a category of sheaves on some site? I was reading an article where the answer was given to be yes and it was written that $Shv(\mathcal{C})/S$ is equivalent to $Shv(\mathcal{C/S})$. The reference was given as SGA-1(Volume 1). But I don't know French. Please help with detailed arguement of equivalence. Thanks in advance.

Answer 1

This is indeed true; a reference is for instance Moerdijk/Mac Lane Sheaves in geometry and logic, where it is stated on page 157 as Exercise III.8.

Answer 2

In the case where $\mathcal{C}$ is a one-point category so that $Shv(\mathcal{C})$ is equivalent to $Set$, consider an object of $Set / S$, i.e. a set $X$ and a function $f : X \to S$. Then we can consider this as being equivalent to specifying the fibers $f^{-1}(\{ s \})$ for each $s \in S$. More generally, if $\mathcal{C}$ is a site, then we can generalize this thinking to consider an object of $Shv(\mathcal{C}) / S$ as being equivalent to specifying the fibers $f^{-1}(\{ s \})$ for each object $U$ of $\mathcal{C}$ and section $s \in S(U)$, in a "compatible" manner. That leads to the following definition of a site:

Let the objects be given by a pair of $U$ an object of $\mathcal{C}$, and a section $s \in S(U)$. The morphisms of the site from $(V, t)$ to $(U, s)$ are given by $\{ \phi \in \operatorname{Hom}_{\mathcal{C}}(V, U) \mid \phi^* s = t \}$. And given a family of morphisms $(V_i, t_i) \to (U, s)$, we consider this to be a covering family exactly when the underlying morphisms $\phi_i \in \operatorname{Hom}_{\mathcal{C}}(V_i, U)$ form a covering family of $U$.

It now remains to verify the conditions of a site, and to show the desired equivalence of categories. As a hint on the latter, given an object of $Shv(\mathcal{C}) / S$, i.e. an object $T$ of $Shv(\mathcal{C})$ and a morphism $\phi : T \to S$, we map to the sheaf on the new site whose sections on $(U, s)$ are $\{ t \in T(U) \mid \phi_U(t) = s \}$. Conversely, given a sheaf $\mathcal{F}$ on the new site, we map to the sheaf on $\mathcal{C}$ whose sections on $U$ are $\bigsqcup_{s\in S(U)} \mathcal{F}((U, s))$ along with the morphism to $S$ given on $U$ by projection of an element of the disjoint union to the value of $s \in S(U)$ specifying the component it's from.