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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.)

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We actually discussed this briefly over at this MO question where I added some comments below your question.

The main obstruction to the category of cocommutative comonoids being a topos is that the category generally fails to be regular. In particular, quotients fail to be preserved by pulling back. However, as mentioned in one of my comments, it has many of the other desirable features of a topos.

Some of these features are pertinent to part of your main question: first, as you point out, $Cog(dgVect)$ is locally finitely presentable. It is also an (infinitary) extensive category, so that arbitrary coproducts are disjoint and are preserved by pulling back, and even hyper-extensive; in this case every object is canonically a coproduct of connected (i.e., indecomposable) objects. I'm having some trouble finding a completely satisfactory reference for this fact, but see page 7 of 43 of this paper by Adamek, Milius, and Velebil, especially 2.9-2.11.

I won't be able to say much now about the other aspects of your question, but it's possible you'd be interested in the model category aspects of $Cog(dgVect)$, as discussed by Getzler and Goerss; look here under Preprints.

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Ah, yes, now I remember learning the facts I cite in my first paragraph from you even before I learned them from Alex. But I had forgotten them by the time Alex told me, so I was surprised a second time. I will probably continue to be surprised by facts like this. –  Theo Johnson-Freyd Mar 8 '11 at 16:35
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