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.
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?