Is the category of sheaves on a site always abelian?

Question

Let $\mathcal{C}$ be a site and $\mathcal{A}$ be an abelian category.

Suppose that the category of presheaves $$ Psh(\mathcal{C},\mathcal{A}) = \operatorname{Fun}(\mathcal{C}^{op},\mathcal{A}) $$ is abelian (this is always true if $\mathcal{C}$ is small). Does this always imply that the category of sheaves on $\mathcal{C}$ is abelian too?

I have seen a proof where $\mathcal{C}$ is the category of opens of a topological space, and am currently studying the proof where $\mathcal{C}$ is the étale category of a variety. I was wondering if this will work in general.

(This is for an essay for a course on homological algebra i am currently following, which comes down to studying a part of Milne's notes and writing down what i understood)

Answer 1

The case $\mathcal{A} = \textbf{Ab}$ is proved as Theorem 8.11 in [Johnstone, 1977, Topos theory]. In fact, the theorem says a lot more:

• If $\mathcal{E}$ is an elementary topos, then the category $\textbf{Ab}(\mathcal{E})$ of internal abelian groups in $\mathcal{E}$ is an abelian category.
• If $\mathcal{E}$ has a natural numbers object, then the forgetful functor $\textbf{Ab}(\mathcal{E}) \to \mathcal{E}$ is monadic.
• If $\mathcal{E}$ is a Grothendieck topos, then $\textbf{Ab}(\mathcal{E})$ is Grothendieck abelian category.

Similar statements can be made about the category of modules over an internal ring in a topos. The point, however, is not to think of $\textbf{Ab}(\mathcal{E})$ as being the category of $\textbf{Ab}$-valued sheaves on a site but rather to internalise the standard proof that $\textbf{Ab}$ itself is an abelian category using only the properties of (elementary) toposes.