Condition on a Hopf operad for tensor product in the base category to be a (categorical) coproduct for algebras

Question

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. "Hopf" may not be the best terminology, as Hopf structure usually refers to one with counit and coinverse, but the notion here is not what is usually meant by "cooperad"). It allows one to define a $P$-algebra structure on the tensor product of two $P$-algebras (as for tensor product of modules over a coalgebra).

For instance, there is such a structure on the associative or commutative operads, as a tensor product (in the base category, usually sets or modules) of associative (resp. commutative) algebras is endowed with a structure of associative (resp. commutative) algebra as well. The first one is defined by diagonal morphisms $\Sigma_n \longrightarrow \Sigma_n \times \Sigma_n$ (that results on the Hopf coproduct on $k \Sigma_n$ after linearization). The second is obvious.

We can observe that tensor product of algebras does provide a (categorical) coproduct for commutative algebras, although it is not the case for associative ones. My question is the following : Are there known (and simple) conditions on a Hopf operad that makes tensor product in the base category into a (categorical) coproduct for algebras, like in the commutative case ?

I would like to apply it to $E_\infty$-operads, obtained by taking $\Sigma_*$-projective resolutions of the commutative operad (considered in a category of vector spaces over a field).

Answer 1

It is more a remark than an answer, if $P$ is an operad and if $F(P,V)$ is the free $P$-algebra on $V$ we have the formula: $$F(P,V\oplus W)\cong F(P,V)\coprod F(P,W)$$ where $\coprod$ is the coproduct of $P$-algebras. Now imagine conditions on a hopf operad $P$ in order to have $$F(P,V\oplus W)\cong F(P,V)\otimes F(P,W)$$ then you will deduce conditions on each $P(n)$. For $E_{\infty}$-algebras we only have a quasi-isomorphism $$A\coprod B\rightarrow A\otimes B$$ at least when $A$ and $B$ are cofibrant $E_{\infty}$-algebras. See for example section 3 (thm 3.4) Mandell, Michael A. $E_\infty$ algebras and $p$-adic homotopy theory. Topology 40 (2001), no. 1, 43–94. The morphism from the coproduct to the tensor product is not used in this section, but you can apply the techniques of this section to get the result above.

Edit: I chose to work with the Barratt-Eccles operad (it is a nice Hopf operad, algebras over this operad come equipped with a closed model category structure), the morphism $$A\coprod B\rightarrow A\otimes B$$ is a weak-equivalence of $E_{\infty}$-algebras, the source is cofibrant because it is a coproduct of cofibrant guys the target is fibrant because any guy is fibrant, thus this map is an homotopy equivalence of algebras over the Barratt-Eccles operad, in particular when forgetting the multiplicative structure a chain homotopy equivalence.