Definition of E-infinity operad

Question

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ (Definition 2.24). Sorry but I'm confused with how does the version in $\mathbb{Z}/p\mathbb{Z}$ must looks like. Thanks!

Answer 1

Operads $\mathcal{C}$ can be defined in any symmetric monoidal category, and then $E_{\infty}$ operads are specified in accordance with the (or a) notion of equivalence relevant to that category. In any category, I prefer to insist that they be $\Sigma$-free, in the sense that the symmetric group $\Sigma_n$ acts freely on $\mathcal{C(n)}$.

In spaces, it is required that each $\mathcal{C}(n)$ be contractible, so that it is a universal cover of the orbit space $\mathcal{C}(n)/\Sigma_n$, which is a $K(\Sigma_n,1)$. (Some might prefer weakly contractible, which is the same when the spaces of the operad are CW homotopy types, as is true in all of the examples I know).
There is not and should not be a canonical example: that would lose the whole force of the examples, where very different $E_{\infty}$ operads act on different naturally occurring spaces.

In algebra, we can take our symmetric monoidal category to be the category of chain complexes of modules over any commutative ring $R$. In that case, it is reasonable to require $\mathcal{C}(n)$ to be free (or at least projective) over the group ring $R[\Sigma_n]$. While grading is somewhat negotiable, in homological grading I would insist that $\mathcal{C}(n)$ is zero in negative degrees and that it be an $R[\Sigma_n]$-projective resolution of the trivial $\Sigma_n$-module $R$. Again there are many examples, none thought of as canonical.

This gives the pretty picture that the chain complex with coefficients in $R$ of an $E_{\infty}$ operad of spaces is an $E_{\infty}$ operad of $R$-chain complexes, automatically giving many different examples. There are many other examples that do not arise from spaces.

The $\Sigma$-freeness forces one not to confuse things with $\mathcal{Com}$, which is not homologically correct or interesting; see Are $E_n$-operads not formal in characteristic not equal to zero? which is especially relevant to the case $\mathbf{Z}/p\mathbf{Z}$ in the question.

Answer 2

There exists several models of $E_{\infty}$-operads. One model which works in chain complexes over any ring is the Barratt-Eccles operad, defined by applying aritywise the normalized chain complex to a simplicial operad introduced by Barratt and Eccles in On $\Gamma_+$-structures. I. A free group functor for stable homotopy theory in the study of infinite loop spaces. In each arity $r$, this operad is given by the normalized bar construction on the symmetric group $\Sigma_r$. A nice and detailed exposition of the structure of the Barratt-Eccles operad is written for instance in Combinatorial operad actions on cochains by Berger-Fresse and works in particular over fields of positive characteristic.