I asked this question on MSE(https://math.stackexchange.com/q/1579026/239218). Let $A^{\bullet}$ be a cosimplicial commutative algebra over a field $\Bbbk$. Denote with $N(A)^{\bullet}$ the conormalized Moore complex. Since $A^{\bullet}$ is equipped with a product, the Alexander–Whitney map induces an associative product $$\Delta : N(A)^{\bullet}\otimes N(A)^{\bullet}\to N(A)^{\bullet}.$$ which is commutative in cohomology.

If $A$ is good enough, the product above may be interpreted as a piece of some complicated operads $P$ (Eilenberg–Zilber operad, or Barratt–Eccles operad). In particular they are a cofibrant replacement of the operad $\mathsf{Comm}$, and they induce an $E_{\infty}$ structure on $N(A)^{\bullet}$ (see cochain on a simplicial set). It seems to me that these two operads are defined in order to find an $E_{\infty}$ structure on $N(A)^{\bullet}$, when the characteristic of the field is positive (so the hard case).

My question is: what happens for characteristic zero? Are there some simpler $E_{\infty}$ structure on $N(A)^{\bullet}$?

Here a suggestion: $(N(A)^{\bullet}, \Delta)$ is a differential associative algebra, where the product is graded commutative in cohomology. Is that an $E_{\infty}$ algebra?