Is there a description of of sheaf cohomology for the sheaf of sections of a continuous function in terms of common constructions in Algebraic Topology?
In more detail: Any sheaf on a space X can be described as the sheaf of sections of some continuous map from the étale space Y to X. In fact, the category of sheaves (of sets) on X is equivalent to the category of maps to X which are local homeomorphisms. A sheaf of Abelian groups is the same as an Abelian group object in the category of sheaves of sets, so instead of talking about cohomology of sheaves, we could talk about cohomology of an Abelian group object in the category of local homeomorphisms to X, that is, a local homeomorphism from some space Y to X such that, roughly, every fibre has an Abelian group structure where all the multiplications (of the fibres) put together form a continuous map from Y × X Y to Y.
It seems like there should be a simple description of cohomology of X with coefficient in a sheaf of Abelian groups in terms of the corresponding map Y → X that uses only usual constructions in Algebraic Topology and the (fibrewise) group structure of Y. Is there one?
Best Answer
In the case that $Y = A \times X$ is an untwisted sheaf, then there is an easy description (for reasonable spaces X and top. abelian groups A (say Hausdorff, compactly generated, locally contractible)) which is proven in G. Segal "Cohomology of Topological Groups" Sym. Math. Vol IV 1970 pg. 377. From the results of that paper it follows that for $i \geq 1$,
$$H^i(X, \mathcal{O}_A) \cong [X, B^i A]$$
where this is sheaf cohomology and $[-, -]$ denotes homotopy classes of maps, and $B^iA$ is the $i^{\text{th}}$ iterated classifying space. (Note that when A is abelian, BA is again an abelian topological group).
For twisted coefficients (i.e. arbitrary Y), there is a similar description, but you must work in the over category of spaces over X.