Let $A$ be a sheaf of commutative rings on a site $X$. (In my applications, $X$ comes from one of the standard Grothendieck topologies on algebraic varieties.) It should be true that $R\Gamma (X, A)$ has the structure of an $E_{\infty}$-algebra.

In my experience this fact is "obvious" to homotopy theorists, but I've never heard mention of a written reference that explains this fact. What is a suitable reference, if I want to use this in a formal writeup?

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    $\begingroup$ Lots of ways to see this. Here's one: Any homotopy limit of $E_\infty$ algebras (or any algebraic gadget) has the canonical structure of an $E_\infty$ algebra, and $R\Gamma(X,A)$ is the homotopy limit over $X^{op}$ of the values of the sheaf $A$ as objects in, say, the derived category $D(\mathbb{Z})$ (enhanced in some way so you can take homotopy limits). (In other words, view the sheaf $A$ as taking values in $E_{\infty}$-algebras in $D(\mathbb{Z})$ and take the homotopy limit there; the forgetful functor preserves holims so the underlying complex is $R\Gamma(X,A)$.) $\endgroup$ Dec 4, 2017 at 20:30
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    $\begingroup$ This sounds great. Any reference where these facts are explained suitably "from scratch", e.g. to an audience that is not familiar beforehand with the notion of $E_{\infty}$-algebra? I'm sure the answers are all somewhere in "Higher Algebra", but it's a big book to sift through... $\endgroup$
    – user84144
    Dec 5, 2017 at 2:11
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    $\begingroup$ @user84144 The problem is that I don't know of any reasonable explanation "from scratch" of E_∞-algebras. Would you be interested in an answer that presents the argument with references to the literature for standard facts about E_∞-algebras? $\endgroup$ Dec 5, 2017 at 8:22
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    $\begingroup$ Yes, I would be interested in an answer like Dylan's but which gives a precise reference for each fact used, even if "standard" or "obvious" to those within the field. $\endgroup$
    – user84144
    Dec 5, 2017 at 13:20
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    $\begingroup$ Ah, here is a paper of May where he gives an explicit construction of this structure (at least for sheaves on a space) using the 'Eilenberg-Zilber operad', if that's more to your liking: pdfs.semanticscholar.org/bd38/… $\endgroup$ Dec 5, 2017 at 17:47


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