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Some background. Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of endomorphisms and the operad of coendomorphisms $$\mathtt{End}_X(r) = \operatorname{Mor}(X^{\otimes r}, X), \; \mathtt{coEnd}_X(r) = \operatorname{Mor}(X, X^{\otimes r}).$$

An algebra over an operad $\mathtt{P}$ is then a morphism $\mathtt{P} \to \mathtt{End}_X$, and a coalgebra over $\mathtt{P}$ is a morphism $\mathtt{P} \to \mathtt{coEnd}_X$. If one likes, everything can be done in the enriched setting, e.g. dg-modules over a field, and a coalgebra structure on $X$ is equivalent to coactions: $$\mathtt{P}(r) \otimes X \to X^{\otimes r}$$ that satisfy associativity, equivariance, unitality... relations.

Often, one hears that "coalgebras are encoded by cooperads". The definition of cooperads is formally dual to the definition of operads, and a coalgebra over a cooperad $\mathtt{C}$ is given by morphisms $$X \to \mathtt{C}(r) \otimes^{\Sigma_r} X^{\otimes r}$$ satisfying relations formally dual to the relations defining an algebra over an operad.

My problem. But it seems to me that, in algebra at least, most notions of coalgebras encountered in nature are more naturally seen as coalgebras over operads, rather than coalgebras over cooperads. For example, a coassociative coalgebra is more naturally seen as an operator $\Delta : C \to C \otimes C$ (so, something like $\mathtt{Ass}(2) \otimes C \to C \otimes C$), rather than a map $C \to \mathtt{Ass}^*(2) \otimes^{\Sigma_2} (C \otimes C)$.

I think it's even more evident with e.g. Poisson coalgebras. IMO it's natural to think there's a cobracket and a coproduct both acting on $C$. If $\mu$ and $\lambda$ are the two generators of $\mathtt{Poiss}(2)$, this is something like $\mathtt{Poiss}(2) \otimes C \to C \otimes C$ mapping $\mu \otimes x$ to the coproduct $\Delta(x)$ of $x$ and $\lambda \otimes x$ to the cobracket $\delta(x)$). But it's not natural at all (IMO) to think of a coaction that maps $$x \in C \mapsto \mu^* \otimes \Delta(x) + \lambda^* \otimes \delta(x) \in \mathtt{Poiss}^*(2) \otimes^{\Sigma_2} (C \otimes C);$$ everything is jumbled together.

Of course, when one deals with finite-type dg-modules over a field (of characteristic zero to deal with invariants vs. coinvariants), everything can be dualized and there's no difference (AFAIK) between coalgebras over $\mathtt{P}$ and coalgebras over $\mathtt{P}^*$.

Question. Are there examples of coalgebra types that are more naturally given by cooperads (without requiring to look at the dual notion first)? And what about topological operads, say, where things cannot be dualized as easily?

(Asked in February on math.SE)

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This is an example of something that is natural to do from the dual point of view, but not of something that is easier to do with cooperads than operads. It is a little too long for a comment.

The dual point of view is likely to be useful for defining the notion of a "divided power coalgebra". Suppose C is a cooperad. Then you may define a divided power coalgebra over C to be an object X equipped with morphisms $X \to C(n)\otimes_{\Sigma_n} X^{\otimes n}$, subject to compatibility conditions. You can equally well define divided power coalgebras over an operad $O$ in this way, via structure maps $X \to \hom(O(n), X^{\otimes n})_{\Sigma_n}$.

One reason to be interested in divided power coalgebras is that the Bar construction on an algebra over an operad $O$ is not just a coalgebra over the bar construction of $O$, but a divided power coalgebra. This is discussed in the paper of Francis and Gaitsgory. It is conjectured that under some additional hypothesis there is an equivalence of homotopy categories of algebras over $O$ and divided power coalgebras over the Koszul dual of $O$. Some version of this conjecture is proved in this paper of Ching and Harper.

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