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Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$ assigned to it in such a way that $\Gamma_i(\varphi_{ij}(U))=\Gamma_j(U)$ whenever $i\leq j$. Then $X:=\varprojlim X_i$ has a presheaf $\Gamma$ defined on it by $\Gamma(U):=\Gamma_i(\varphi_i(U))$ where $\varphi_i:X\to X_i$ is the map from $X$ as an inverse limit; this is well-defined since $\varphi_{ij}(\varphi_j(U))=\varphi_i(U)$ whenever $i\leq j$, which makes $\Gamma_i(\varphi_i(U))=\Gamma_i(\varphi_{ij}(\varphi_j(U)))=\Gamma_j(\varphi_j(U))$.

In this situation, does Čech cohomology satisfy a continuity property? That is, is it true that $\breve{H}^*(X,\Gamma)=\varinjlim \breve{H}^*(X_i,\Gamma_i)$? I've seen it claimed in some places, such as this question or even wikipedia's talk page for Čech cohomology, that Čech cohomology satisfies some kind of continuity property for "nice enough" spaces, but I can't seem to find any clear reference for this fact. The paper that question refers to seems to be concerned with a more general situation involving triangulable pairs, and I can't fully make sense of it.

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  • $\begingroup$ Did you look at Watanabe's paper that Matthias Wendt cites in your link? It seemed to say that the answer to your question is $i)$ no, in general, $ii)$ yes, if you are smart. Of course Watanabe works with constant coefficients. $\endgroup$
    – Tyrone
    Commented Oct 27, 2020 at 15:30
  • $\begingroup$ I wasn't sure I had access to that before, but I was able to get it. It looks like the negative answer is for spaces that aren't compact, but the paper also seems to say the continuity property for compact hausdorff spaces is proved in Eilenberg and Steenrod's Foundations of algebraic topology. I tried reading that before without much success, but I suppose I can try again. $\endgroup$
    – Xindaris
    Commented Oct 27, 2020 at 16:03
  • $\begingroup$ Okay, from what I can tell Eilenberg and Steenrod define Čech cohomology purely in terms of coefficients in a module rather than a presheaf; possibly their book predates sheaf theory in general. I wonder whether I should refine my question to ask whether sheaf cohomology satisfies a continuity property, since some sources appear to identify Čech cohomology as an "approximation" to that which is equal on nice enough spaces. $\endgroup$
    – Xindaris
    Commented Oct 27, 2020 at 19:52

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