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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 $p\in \mathbb Z?$

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  • $\begingroup$ 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. $\endgroup$ – Daniel Schepler Mar 13 at 23:09

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