# Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg coalgebra is $$\mathbb{Z}$$-graded?.

Thanks.

• The result the OP is pointing to in Getzler-Goerss is Corollary 1.6. – David White Apr 29 at 18:02
• @david Indeed, and I also expect to extend this proposition to differential $\mathbb{Z}$-graded coalgebras by using the previous lemmas of the article. I hope an artifice might suffice. – Victor TC Apr 30 at 9:33

Yes. The category of $$\mathbb{Z}$$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model categories by Ching and Riehl. See also Lemma 5.2 of Model Structures for Coalgebras by Drummond-Cole and Hirsh. This paper of Adamek and Porst might also be helpful.

By the way, the main result of the Getzler-Goerss paper you cite is generalized in Corollary 6.3.5 of A necessary and sufficient condition for induced model structures by Hess, Kedziorek, Riehl, and Shipley. It works for any $$\mathbb{Z}$$-graded coalgebras over any commutative ring $$R$$.

• Thank you very much, I will check the references. – Victor TC Apr 29 at 19:08
• @VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou. – Daniel Robert-Nicoud Apr 29 at 21:36
• N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you. – Gabriel C. Drummond-Cole Apr 30 at 0:14
• @DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below? – Gabriel C. Drummond-Cole Apr 30 at 0:15
• @Gabriel sorry, I forgot to mention that in Brice's article the coalgebras are supposed to be conilpotent (as can be gathered from the start of the proof of Lemma 5). I suppose Leonid's example is non-conilpotent. – Daniel Robert-Nicoud Apr 30 at 6:37

For coassociative dg-coalgebras over any field $$k$$ the answer is positive, because:

1. Let $$C$$ be a $$\mathbb Z$$-graded coalgebra and $$D\subset C$$ a finite-dimensional ungraded subcoalgebra (of the underlying ungraded coalgebra) of $$C$$. Let $$D^{gr}\subset C$$ denote the graded vector subspace spanned by all the grading components of the elements of $$D$$. Then $$D\subset D^{gr}$$ and $$D^{gr}$$ is a finite-dimensional graded subcoalgebra of $$C$$.

2. Let $$(C,d)$$ be a dg-coalgebra and $$D\subset C$$ be a finite-dimensional graded subcoalgebra of $$C$$. Set $$D^{dg}=D+d(D)\subset C$$. Then $$D\subset D^{dg}$$ and $$D^{dg}$$ is a finite-dimensional dg-subcoalgebra of $$C$$.

Using the observations 1. and 2. and the fact that any ungraded coassociative coalgebra is the union of its finite-dimensional subcoalgebras, one deduces the assertion that any $$\mathbb Z$$-graded dg-coalgebra is the union of its finite-dimensional dg-subcoalgebras.

Possible generalizations: One can replace a field $$k$$ by a Noetherian commutative ring $$k$$ and speak about subcoalgebras that are finitely generated as $$k$$-modules (instead of "finite-dimensional"). All the assertions remain true.

EDIT: I've realized that the preceding paragraph is problematic for the following reason: given a $$k$$-submodule $$D$$ is a $$k$$-module $$C$$, the tensor product $$D\otimes_k D$$ is not a submodule of the tensor product $$C\otimes_k C$$, generally speaking. So the very notion of a $$k$$-subcoalgebra for a commutative ring $$k$$ is problematic, or at least requires extra care with nonexact tensor products. So, I retract the preceding paragraph.

One cannot drop the coassociativity condition. Indeed, even for ungraded coalgebras over a field of characteristic $$0$$, there is an example of infinite-dimensional Lie coalgebra $$L$$ having no nonzero finite-dimensional subcoalgebras. The Lie coalgebra $$L$$ is simplest described in terms of its dual topological Lie algebra structure (on a pro-finite-dimensional topological vector space): $$L^*=\mathfrak g=k[[z]]\,d/dz$$, the Lie algebra of vector fields on the formal disk.

• Thank you!. Apparently you did not need $C$ to be counital, did you?. – Victor TC Apr 30 at 11:34
• No, counitality is not necessary. However, there is a problem with the next-to-last paragraph of my answer. I am now adding an edit. – Leonid Positselski Apr 30 at 11:40
• Concerning counitality, it is of course not needed in 1. and 2., but the key question is extending to noncounital coassociative ungraded coalgebras $C$ the standard result that coassociative coalgebras are unions of their finite-dimensional subcoalgebras. The standard arguments that I know are using counitality, but it can be avoided. Alternatively, you can adjoint a counit to $C$ formally, passing from $C$ to the counital coalgebra $C'=k\oplus C$, represent $C'$ as the union of its finite-dimensional subcoalgebras $D'$, and conclude that $C=C'/k$ is the union of $D=D'\bmod k$. – Leonid Positselski Apr 30 at 12:02
• Thank you for the clarification. I apologize for my delay in aswering, I tried to better understand your edit. With this result in hand, is it reasonable to conclude that the category of dg coalgebras has all (small) limits?. I mean, by applying dualization on the finite dimensional dg subcoalgebras and assuming the existence of all small colimits in the category of dg algebras. – Victor TC May 2 at 15:10
• The assertion is correct: all small limits exist in the category of dg-coalgebras. I do not immediately see how this follows from all dg-coalgebras being unions of their finite-dimensional subcoalgebras. Maybe one can deduce the former from the latter by proving that the category of dg-coalgebras is locally presentable (locally finitely presentable, in fact). – Leonid Positselski May 2 at 18:40