Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative dg-algebras over $X$. Let $\mathfrak{U}$ be a fixed open covering of $X$ and $C^\bullet(\mathfrak{U},\mathcal{F})$ be the Cech cochains on associated to this open cover. This has a natural cup product. Does this define an $E_{\infty}$ algebra? I am quite a novice on homotopy algebra, so a gentle reference would be great.
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1$\begingroup$ It does define an E_∞-algebras (because it's a homotopy limit of cdgas, in particular of E_∞-algebras). I don't know of a "gentle" introduction, but the fact that the forgetful functor from E_∞-algebras detects limits is, for example, Higher Algebra, Corollary 3.2.2.5 (in the case where O is the commutative operad). There must be more elementary presentations somewhere. $\endgroup$– Denis NardinCommented Jan 26, 2018 at 19:49
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$\begingroup$ Also, if you need a reference to understand what the heck homotopy limits have to do with Čech cochains, let me just drop this here... $\endgroup$– Denis NardinCommented Jan 26, 2018 at 20:28
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$\begingroup$ @DenisNardin Thanks for your comment. That does sound like an interesting perspective. I'll have to study your link to understand in what sense Cech cochains give rise to homotopy limits. $\endgroup$– algebrachallengedCommented Jan 26, 2018 at 20:57
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