# Cohomology theories and sheaf cohomology

Let $$X$$ be a paracompact Hausdorff topological space, $$\mathcal U$$ an open covering of $$X$$ and $$\mathcal N(\mathcal U)$$ the nerve of the covering (https://en.wikipedia.org/wiki/Nerve_of_a_covering). It is known that, if the space $$X$$ is good enough, and the covering is sufficiently fine, we have the following isomorphisms $$H_{\Delta}^{p}(\mathcal N(\mathcal U)) \simeq \check{H}^p(X, \mathbb{Z})\simeq H^p(X, \mathbb Z).$$ Here I denote with $$H_{\Delta}^{p}$$ the symplicial cohomology, with $$\check{H}^p(X, \mathbb{Z})$$ the $$\check Cech$$ cohomology and with $$H^p(X, \mathbb Z)$$ the singular cohomology.

So it turns out that, if the space $$X$$ is sufficiently nice, the singular cohomology is just a "special case" of the sheaf cohomology. My question is, does this situation hold in general? Formally, let $$h$$ be a cohomology theory in the category of the paracompact Hausdorff topological spaces, that is a sequence of functors $$h^n$$ with values in the category of the abelian groups that satisfy the usual axioms of a cohomology theory (excission, exact sequence of the couple and so on); does there exist a sheaf $$\mathcal F$$ such that $$h^p(X)=\check H^p (X; \mathcal F)$$ for all $$p\in \mathbb Z?$$

• You might be able to construct a presheaf via $\mathcal{F}(U) = h^0(U)$ with restriction maps given by the structure of $h^0$ being a contravariant functor. I'm not at all sure if that would be a sheaf in general, though, given that things like Mayer-Vietoris would probably only give gluing conditions for finite covers of an open set. – Daniel Schepler Mar 13 at 23:09