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 categoy 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).