In the paper Axiomatic Homotopy Theory for Operads Berger and Moerdijk construct a cooperad out of a commutative bimonoid (I have no idea, why they call the latter a Hopf-object, since in the whole paper no antipode is mentioned). It should be possible to produce an operad out of a cocommutative bimonoid as well, hence both an operad and a cooperad out of a bicommutative bimonoid.
Let me briefly sketch their construction: Given a bimonoid $B$, the cooperad is given by the symmetric sequence $(B^{\otimes n})_n$ with unit map given by the unit $\Bbb{1}\rightarrow B$ of $B$ and structure maps given by the composite $$\begin{array}{c} B^{\otimes n_1+...+n_k}\\ \downarrow_\Delta\\ (B^{\otimes n_1}\otimes ... \otimes B^{\otimes n_k}) \otimes (B^{\otimes n_1}\otimes ... \otimes B^{\otimes n_k})\\ \downarrow_{(m\otimes...\otimes m)\otimes id}\\ B^{k}\otimes B^{\otimes n_1}\otimes ... \otimes B^{\otimes n_k} \end{array}$$ where $\Delta$ is the diagonal/comultiplication of the comonoid $B^{\otimes n_1+...+n_k}$ and $m$ denotes the respective multiplications $B^{\otimes n_i} \rightarrow B$.
I would like to know, whether this is some kind of standard construction. I was unable to find anything about it on the internet and feel like I don't really have a good motivation for the construction.
Without thinking about it, I would have guessed that monoids give rise to monads, hence to operads, while comonoids give rise to comonads, hence cooperads. But Berger&Moerdijk's construction crucially uses both the multiplication and comultiplication of the bimonoid to define the cooperad, using the commutativity of the multiplication to verify the equivariance of the cocomposition...
Why does it make sense to consider this specific cooperad associated to a commutative bimonoid? What does it encode?
If anyone has an opinion or reference about it, please let me know. Regardless, thank you for your time.