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!



Definition of E-infinity operad
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?](https://mathoverflow.net/questions/218610/are-e-n-operads-not-formal-in-characteristic-not-equal-to-zero/218886#218886)
which is especially relevant to the case $\mathbf{Z}/p\mathbf{Z}$ in the question. 


