I am familiar with the usual nerve of a small category. Is there any way to construct the nerve of an open covering $\{U_i\}$ as the nerve of some small category?
How does the homology/cohomology of this simplicial set relate to the concept of Cech cohomology?
Call the nerve of the open covering $\{U_i\} \space$ X. Then define a chain complex $C_i = $ Free abelian group generated by $NU_i$ with $\partial=\sum(-1)^jd_j$, $d_j$ being the degeneracy maps of NU. Then we may take the homology/cohomology by applying Hom$(-,\mathbb Z)$.
The question I have now is if the Cech cohomology of the constant presheaf (not sheaf) $\mathbb Z$ is equal to the cohomology of $C^i = Hom(C_i, \mathbb Z)$. It seems obvious to me but don't want to go through all of the symbol pushing to prove it.