An interpretation of this construction giving an operad from a bialgebra?

Question

Let $A$ be a cocommutative bialgebra object (or even a Hopf algebra) in a symmetric monoidal category. Define an operad $\mathtt{P}_A$ by $\mathtt{P}_A(r) = A^{\otimes r}$ (so that $\mathtt{P}_A(0) = 1$ is the unit of $\otimes$), and the composition is given by the following formula: $$\gamma(a_1 \otimes \dots \otimes a_r; \underline{b}_1, \dots, \underline{b}_r) := (a_1 \cdot \underline{b}_1 \otimes \dots \otimes a_r \cdot \underline{b}_r)$$ where $a_i \in A$, $\underline{b}_j \in A^{\otimes k_j}$, and $A^{\otimes k}$ becomes an $A$-module using the cocommutative diagonal of $A$, and using the counit for $A^{\otimes 0} = 1$ (I'm writing this formula in the case of vector spaces, I don't want to draw a humongous commutative diagram; I hope the general definition is clear).

Is this construction an example of a more general phenomenon / can it be described differently? It appears when defining semi-direct products of operads (as in the paper of Salvatore and Wahl), for example. It feels like something rather simple that appears naturally (I don't know, maybe the free construction on something), but I can't write it as an application of a more general construction; I realize it's the semi-direct product $\mathtt{Com} \rtimes A$, but it feels a bit circular.

Answer 1

It might be worth to first consider the particular case of the symmetric monoidal category $({\rm Set},\times)$ of sets and Cartesian products. Let us mildly extend the setting to include possibly multi-colored operads. Then your construction becomes a particular case of the construction which associates to a category $\mathcal{C}$ the colored operad $\mathcal{C}^{\coprod}$ whose colors are the objects of $\mathcal{C}$ and such that for every finite set $I$, every $I$-tuple of objects $\{x_i\}_{i \in I}$ of $\mathcal{C}$ and every object $y \in \mathcal{C}$ the set of multi-operations $\{x_i\}_{i \in I} \to y$ is the set $\prod_i{\rm Hom}(x_i,y)$ (in other words, a multi-operation from $\{x_i\}_{i \in I}$ to $y$ is given by a collection of maps $f_i:x_i \to y$ for $i \in I$). The operad $\mathcal{C}^{\coprod}$ has several interesting features:

(1) If the category $\mathcal{C}$ has finite coproducts then $\mathcal{C}^{\coprod}$ is the underlying operad of the symmetric monoidal category $(\mathcal{C},\coprod)$.

(2) The operad $\mathcal{C}^{\coprod}$ is equivalent to the Boardman-Vogt tensor product of $\mathcal{C}$ (considered as a colored operad with no non-1-ary operations) and ${\rm Com}$. As a result, the notion of a $\mathcal{C}^{\coprod}$-algebra in a symmetric monoidal category $(\mathcal{D},\otimes)$ is equivalent to that of a $\mathcal{C}$-indexed diagram of commutative algebra objects in $\mathcal{D}$. More generally, if $\mathcal{O}$ is any operad then the category of operad maps $\mathcal{C}^{\coprod} \to \mathcal{O}$ is equivalent to the category of functors from $\mathcal{C}$ to the category of operad maps ${\rm Com} \to \mathcal{O}$.

(3) If $\mathcal{O}$ is a unital colored operad (that is, each color has a unique $0$-ary operation) then operad maps $\mathcal{O} \to \mathcal{C}^{\coprod}$ are the same as maps from the underlying category of $\mathcal{O}$ (obtained by taking only the $1$-ary operations) to $\mathcal{C}$.

In particular, the operad $\mathcal{C}^{\coprod}$ enjoys both a mapping property out of it and a mapping property into it (at least for unital operads), both of which characterize it up to a unique isomorphism. There is also a similar story in the setting of $\infty$-operads, where $\mathcal{C}$ is allowed to be any $\infty$-category, and the $\infty$-categorical analogue of all three properties above holds.

Returning to the single colored case where $\mathcal{C}$ is taken to be a monoid rather than a category we can still formulate (2) above as giving a characterization of the notion of an algebra object over $\mathcal{C}^{\coprod}$ in a general symmetric monoidal category. Your construction is a generalization of this case in another direction, where you replace your monoid (which can be considered as a bialgebra in the symmetric monoidal category $({\rm Set},\times)$) by a bialgebra $A$ in a general symmetric monoidal category $\mathcal{V}$. In this case, you can still characterize $\mathtt{P}_A$ as above by specifying what are $\mathtt{P}_A$-algebras in a given symmetric monoidal $\mathcal{V}$-enriched category $\mathcal{D}$: these will correspond to what one might call commutative $A$-algebra objects in $\mathcal{D}$, that is, objects equipped with a commutative algebra structure and an $A$-module structure which are compatible with each other. This compatibility can be formulated using the structure of $A$ as a bialgebra. For example, you can describe this structure as being a commutative algebra object in the category of $A$-modules, where the latter has a symmetric monoidal structure induced by the coproduct of $A$.

One might summarize this discussion as follows: the "reason" why there is a natural operad $\mathtt{P}_A$ is that for a bialgebra $A$ there is a natural notion of a commutative $A$-algebra, and so we can expect to have an associated operad which encodes this theory.