# Is the category of sheaves on a site always abelian?

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)

-
It's certainly true in the special case where $\mathcal{A} = \textbf{Ab}$, but why do you want it in this generality? –  Zhen Lin Feb 4 '13 at 17:46
@ZhenLin, i am perfectly happy with $\mathcal{A}$ is abelian groups! Could you give a reference to a proof? –  Joachim Feb 4 '13 at 17:48
Is it as easy to do it with $\mathcal{A} = R$-Mod for a commutative ring $R$? –  Joachim Feb 4 '13 at 17:49
Both can be proven equally straightforwardly, I think. –  Zhen Lin Feb 4 '13 at 17:50

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.

-
Thanks! Is there in general an adjoint to the forgetful functor from sheaves on a topos to presheaves, like the sheafification functor for sheaves on topological spaces? –  Joachim Feb 4 '13 at 18:11
Zhen, this is a great answer! –  Martin Brandenburg Feb 4 '13 at 18:24
@Joachim Your question is not quite well-formed. A Grothendieck topos is a category of sheaves. It is true that there is a left exact sheafification functor $[\mathbb{C}^\textrm{op}, \textbf{Set}] \to \textbf{Sh}(\mathbb{C}, J)$ for any Grothendieck site $(\mathbb{C}, J)$, so by general nonsense this carries presheaves of abelian groups to sheaves of abelian groups. This is left adjoint to the forgetful functor by general nonsense regarding 2-functors. –  Zhen Lin Feb 4 '13 at 19:05
Aiai sorry i did not write at all what i meant i'm sorry, but you did answer what i meant, so thanks! –  Joachim Feb 4 '13 at 21:10