Reference for Cech cohomology on sites (not pre-topologies) I'm searching for a reference dealing Cech cohomology on sites (not pre-topologies). In general, when dealing with Cech cohomology on sites, one admits that the category has finite limits so you can always makes the nerves with the pullback of a covering, however I've never seen any reference in the context of categories without pullbacks. I know that I can use the Yoneda embedding and, then, take the pullbacks, but since I've never seen this approach before, I'm afraid that maybe some problems could occur.
Thanks in advance.
 A: Čech cohomology (with respect to a fixed covering) is essentially the derived functor cohomology of the topos of presheaves: the Čech chain complex is precisely a projective resolution of the free abelian presheaf generated by that covering – when the site has finite products. Then, taking the colimit over all coverings turns out to compute the derived functors of the functor that sends a presheaf to its associated separated presheaf. This is explained quite nicely in [Johnstone, Topos theory, §8.2]. 
Fix a small category $\mathcal{C}$ and consider presheaves on $\mathcal{C}$. Let $U$ be a presheaf (of sets) and let $V$ be the presheaf image of the unique morphism $U \to 1$. We can then construct the Čech chain complex $\check{\mathscr{C}}_{\bullet} (U)$ where $\check{\mathscr{C}}_n (U)$ is the free abelian presheaf generated by $U^{n+1}$ and the differentials are defined by the evident alternating sum of projections. It is not hard to check that $\check{\mathscr{C}}_{\bullet} (U)$ is a resolution of the free abelian presheaf $\mathbb{Z} V$ generated by $V$, but the real question is whether it is a projective resolution. When $\mathscr{C}$ has finite products and $U$ is a coproduct of representable presheaves, each $U^{n + 1}$ is then a coproduct of representable presheaves (by the distributivity of products and coproducts) and hence the abelian presheaf $\check{\mathscr{C}}_n (U)$ is projective. But in general there is no reason to believe $\check{\mathscr{C}}_{\bullet} (U)$ is a projective resolution of $V$, and so there is no reason to believe that the cohomology groups $\check{H}{}^n (U, \mathscr{F}) = H^n (\mathrm{Hom} (\check{\mathscr{C}}_{\bullet} (U), \mathscr{F}))$ coincide with $R^n \Gamma (V, \mathscr{F})$.
The solution is to consider projective resolutions of $\mathbb{Z} V$ in general and not just the Čech chain complex. But can this really be called Čech cohomology?
