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Let $\mathcal V$ be a reasonably nice category — I'm interested in the case when $\mathcal V$ is $\mathbb K$-linear for some field $\mathbb K$, abelian, and has all products and coproducts (hence all limits and colimits, as it is abelian), but I don't mind if you demand similar or weaker properties — with a reasonably nice monoidal structure $\otimes$. Recall that a (counital coassiciative) coalgebra in $\mathcal V$ is an object $A\in\mathcal V$ along with maps $A \to 1$ and $A \to A\otimes A$ that are coassiciative and counital in the sense that certain diagrams commute. The notion of homomorphism of coalgebras is obvious.

When $\mathcal V = \text{VECT}$ is the category of vector spaces, then coalgebras satisfy a particular fundamental property that makes them essentially easy. Namely, any coalgebra is $\text{VECT}$ is the (vector space) sum of its finite-dimensional subcoalgebras. On the other hand, the corresponding statement in $\text{VECT}^{\rm op}$ fails: it is not true that every algebra in $\text{VECT}$ is a pullback of its finite-dimensional quotient algebras. This is in spite of the fact that for many purposes $\text{VECT}$ and $\text{VECT}^{\rm op}$ are equally nice categories.

For a general sufficiently nice category $\mathcal V$, I should replace the word "sum" by "limit" and I should replace "finite-dimensional" by "dualizable". All together, my question is:

For which sufficiently nice monoidal categories is it true that every coalgebra object is a limit of its dualizable subcoalgebra objects?

This is, of course, an open ended question. The very best would be some necessary and sufficient conditions that are easier to check, but that's probably too hard: natural (and naturally occurring) easily-checked sufficient conditions would suffice.

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Let us replace "dualizable" with "Noetherian". Then, I believe, for any locally Noetherian Grothendieck (abelian) category with an exact, associative tensor product functor preserving direct limits (perhaps one should also require that the Noetherian objects form a tensor subcategory), any coassociative coalgebra is a filtered direct limit of its subcoalgebras that are Noetherian objects in the abelian category. The proof would proceed roughly as follows: given a coalgebra C and a Noetherian subobject V in it, consider the full preimage of C⊗V⊗C under the double comultiplication map C → C⊗C⊗C. This full preimage should be a subcoalgebra of C contained in V; and the whole of C should be the union of such subcoalgebras.

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There are some results in this direction in Paddy McCrudden's paper "Tannaka duality for Maschkean Categories", JPAA 168, 2002.

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