Let $\mathcal C$ be a *closed* symmetric monoidal category. By $\operatorname{Cog}(\mathcal C)$ I denote the category of cocommutative (counital coassociative) coalgebras in $\mathcal C$. Suppose that $\mathcal C$ is presentable; then it is a nontrivial fact that $\operatorname{Cog}(\mathcal C)$ is also presentable (in particular, complete and cocomplete). The forgetful functor $\operatorname{Cog}(\mathcal C) \to \mathcal C$ is a left adjoint (in particular, cocontinuous; its right adjoint is $\operatorname{Sym}:\mathcal C \to \operatorname{Cog}(\mathcal C)$ computing the cofree cocommutative coalgebra on an object). The tensor product of cocommutative coalgebras is their "Cartesian" product in $\operatorname{Cog}(\mathcal C)$. Indeed, $\operatorname{Cog}(\mathcal C) \to \mathcal C$ is the universal symmetric-monoidal left adjoint to $\mathcal C$ from a Cartesian closed presentable category. I learned these facts from Alex Chirvasitua.

Thus $\operatorname{Cog}(\mathcal C)$ feels very much like a category of sets — it's not far from being a topos.

Side Question:How far is $\operatorname{Cog}(\mathcal C)$ from a topos?

Here's a very hands-on example. Fix a field $k$, and let $\mathcal C$ be the category of vector spaces over $k$, with the usual symmetric monoidal structure. Then every object in $\operatorname{Cog}(\mathcal C)$ breaks up *canonically* as a disjoint union of "fuzzy heavy points". A "heavy point" is (the cococommutative coalgebra dual to) a finite-dimensional field extension of $k$. A "fuzzy point" is a point with some "nilpotent" fuzz. There can be a lot of fuzz, but any fuzzy point is a union (over a point) of $n$-jets of $N$-dimensional spaces, where $n = 0,1,\dots,\infty$, and $N$ is any cardinal. A typical example is $\mathcal U\mathfrak g$, the universal enveloping algebra of a Lie algebra $\mathfrak g$ (in $\operatorname{char} k = 0$, say). As a coalgebra, $\mathcal U\mathfrak g$ is a fuzzy (but non-heavy) point, and the fuzz is an $\infty$-jet of $\mathfrak g$ near $0$ — as a coalgebra, $\mathcal U\mathfrak g = \operatorname{Sym}(\mathfrak g)$. The algebra structure on $\mathcal U\mathfrak g$ makes this point into a "formal group", in particular a group object in $\operatorname{Cog}(k\text{-Vect})$.

I have been unable to compute examples beyond this, although I haven't tried super hard.

Side Question:Is there a similarly straightforward description of $\operatorname{Cog}(\text{AbGp})$?

Main Question:Is there a similarly straightforward description of $\operatorname{Cog}(\text{dgVect})$ (at least in characteristic $0$)? E.g. does every object break up canonically into a disjoint union of indecomposables? What are said indecomposables?

I'm not optimistic about the Main Question. For example, the data of a Lie algebra structure on $\mathfrak g\in \text{Vect}$ is the same as the data of a dg structure on $\operatorname{Sym}(g[1])$, at least up to a sign convention, so an answer to the Main Question will probably include something like "... is a Lie algebra" (and classification of Lie algebras is probably hopeless).

(This question deserves more tags. If you have suggestions, feel free to add them.)