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For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...

I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak. Question: Let $M$ be a ...