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Let $X$ be a topological space, $\mathcal F$ a sheaf of abelian groups on $X$. Let $\breve{H}^n(X, \mathcal F)$ denote the $n$-th Čech cohomology of $X$ with coefficients in $\mathcal F$. Thus $\breve{H}^n(X, \mathcal F) = \varinjlim \breve{H}^n(\mathcal U, \mathcal F)$, the limit being taken over open coverings partially ordered under refinement.

The construction of Čech cohomology makes sense for a presheaf as well. If $\mathcal G$ is a presheaf, the sheafification map $\mathcal G \to \mathcal G^+$ induces maps $\breve{H}^n(X, \mathcal G) \to \breve{H}^n(X, \mathcal G^+)$. For $n=0$, this map is an isomorphism, on the one hand because $\breve{H}^0(X, \mathcal G^+) = H^0(X, \mathcal G^+)$, and on the other hand because the limit which defines $\breve{H}^0(X, \mathcal G)$ identifies this group with $H^0(X, \mathcal G^+)$ by the usual explicit description of sheafification.

What can one say about the maps $\breve{H}^n(X, \mathcal G) \to \breve{H}^n(X, \mathcal G^+)$ for $n>0$? Are they also isomorphisms? are they injective? surjective? Is there a spectral sequence hiding somewhere?...

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    $\begingroup$ I think they are isomorphisms. Here is a thing I think is true although I haven't checked the details: define a weak equivalence between two presheaves to be a map which is an isomorphism when restricted to some open cover (a "local isomorphism"). I think the localization of the category of presheaves at these weak equivalences is the category of sheaves. And Cech cohomology respects this notion of weak equivalence. (Again, I haven't checked any of these details.) $\endgroup$ – Qiaochu Yuan May 21 '14 at 4:21
  • $\begingroup$ @Qiaochu Hmm. Interesting idea! I will give it some thought. Thanks! $\endgroup$ – Bruno Joyal May 21 '14 at 4:23
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    $\begingroup$ I am more confident that the category of presheaves localized at stalkwise isomorphisms is the category of sheaves but correspondingly less confident that Cech cohomology respects this notion of weak equivalence. I haven't checked this either. $\endgroup$ – Qiaochu Yuan May 21 '14 at 4:25

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