Timeline for Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?
Current License: CC BY-SA 4.0
9 events
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| May 3, 2019 at 8:51 | comment | added | Victor TC | I see, this assertion is proposition 1.8 in the article by Getzler and Goerss and is proved by using a strange a strange category to me named the category of **profinite ** differential graded algebras, can you please tell me a more direct argument to deduce such a result?. | |
| May 2, 2019 at 18:40 | comment | added | Leonid Positselski | 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). | |
| May 2, 2019 at 15:10 | comment | added | Victor TC | 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. | |
| Apr 30, 2019 at 12:02 | comment | added | Leonid Positselski | 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 $D=D'\bmod k$. | |
| Apr 30, 2019 at 11:45 | history | edited | Leonid Positselski | CC BY-SA 4.0 |
Edit inserted (the next-to-last paragraph problematic)
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| Apr 30, 2019 at 11:40 | comment | added | Leonid Positselski | 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. | |
| Apr 30, 2019 at 11:34 | comment | added | Victor TC | Thank you!. Apparently you did not need $C$ to be counital, did you?. | |
| Apr 29, 2019 at 22:40 | history | edited | Leonid Positselski | CC BY-SA 4.0 |
two paragraphs on "possible generalizations" added at the end
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| Apr 29, 2019 at 22:04 | history | answered | Leonid Positselski | CC BY-SA 4.0 |