This is a very naive question. Suppose we have a site $\mathcal C$ (we can just think of a topological space with it's open sets) and a cover in it $\cup_\alpha X_\alpha = X$, Then, given a presheaf of abelian groups $\mathscr F$ on $\mathcal C$, we can form the Cech complex with respect to the cover: $$0 \to \mathscr F(X) \to \prod_\alpha \mathscr F(X_\alpha) \to \prod_{\alpha,\beta} \mathscr F(X_\alpha \times_X X_\beta) \to \dots $$ where the maps are the alternating restriction maps.
We say $\mathscr F$ is a sheaf if the first homology of the above complex is $\mathscr F(X)$ for all covers. Is there a terminology for those presheaves $\mathscr F$ for which the entire complex is exact? What about if we only require the first $n$ terms to be exact? Are there interesting examples of such things?
In the case of an affine scheme with each open in the cover being affine (or a contractible nice space with a good open cover (all intersections are contractible), every quasi coherent sheaf automatically makes the complex exact. But how about other, more interesting examples?