Suppose we have a site $\mathcal{C}$ and $S$ is an object of $\mathcal{C}$. Let $F$ be a sheaf of abelian groups on $\mathcal{C}$ and let $S_\bullet$ be a simplicial hypercover of $S$.
My question is: when is the induced complex
$$0 \to F(S) \to F(S_1) \to F(S_2) \to \dots$$
exact? (here the complex is taken to be the Moore complex of the induced cosimplicial abelian group, in the same way as Dold-Kan correspondence).
Is there an easy criterion for this?
It is not hard to show in the first case, since $F(S)$ is just the equaliser of the two maps $F(S_1) \to F(S_2)$, but I can't seem to prove it beyond that.
Would it help if $\mathcal{C}$ is the site of compact Hausdorff spaces with finite jointly surjective families of maps as covers