Timeline for Are dg-modules over a cofibrant dg-category cofibrant?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2020 at 2:27 | vote | accept | Stahl | ||
| Dec 13, 2020 at 2:25 | comment | added | Stahl | I see, I misunderstood the statement you made. Thanks for the clarification! | |
| Dec 13, 2020 at 2:24 | comment | added | Dmitri Pavlov | @Stahl: All generating cofibrations are functors between small (in fact, finite) categories, but not all functors between small (or finite) categories are generating cofibrations. The inclusion is in one direction only. | |
| Dec 13, 2020 at 2:22 | comment | added | Stahl | Sorry, I mixed up the variance. In either case, if the generating cofibrations are functors between small categories, then working inside $\mathsf{dgCat}_{\Bbb{V}}$ (choosing a universe $\Bbb{V}$ in which the objects and morphisms of $\mathsf{dgMod}_{C,\Bbb{U}}$ form sets) we should have that $\emptyset\to \mathsf{dgMod}_{C,\Bbb{U}}$ would be one of the generating cofibrations in $\mathsf{dgCat}_{\Bbb{V}},$ as in $\Bbb{V}$ both $\mathsf{dgMod}_{C,\Bbb{U}}$ and the initial dg-category $\emptyset$ are small. | |
| Dec 13, 2020 at 2:15 | comment | added | Dmitri Pavlov | @Stahl: I do not understand your last sentence: "so the functor to the terminal object should be a cofibration by the results you've cited": for cofibrancy, one needs the functor from the initial object to be a cofibration, and how does it follow from the cited results? | |
| Dec 12, 2020 at 23:07 | comment | added | Stahl | However, it then seems like if I choose an appropriate universe $\Bbb{V}\ni\Bbb{U},$ and consider $\mathsf{dgMod}_{C,\Bbb{U}}$ inside $\mathsf{dgCat}_{\Bbb{V}},$ it should now be cofibrant in $\mathsf{dgCat}_{\Bbb{V}}$: it is now small, and so the functor to the terminal object should be a cofibration by the results you've cited. What am I missing? | |
| Dec 12, 2020 at 23:07 | comment | added | Stahl | Thanks for the reference! I'm a little confused by your last comment, though: suppose I fix a Grothendieck universe $\Bbb{U},$ and fix $C\in\mathsf{dgCat}_{\Bbb{U}}$ (cofibrant, although this now seems irrelevant). Then I can form the category $\mathsf{dgMod}_{C,\Bbb{U}}$ of dg-modules over $C$ in $\mathsf{dgCat}_{\Bbb{U}}$ which as you've stated is not cofibrant in $\mathsf{dgCat}_{\Bbb{U}}.$ | |
| Dec 12, 2020 at 18:45 | comment | added | Dmitri Pavlov | @Stahl: There are other problems apart from size issues, e.g., cofibrant dg-categories are (roughly) retracts of free dg-categories, and this imposes an obstruction that one can show is violated for dg-categories of modules over simplest dg-categories, e.g., dg-modules over an ordinary ring. | |
| Dec 12, 2020 at 18:41 | comment | added | Dmitri Pavlov | @Stahl: Generating cofibrations are functors between categories with 0, 1, or 2 objects, I added a reference to the answer. | |
| Dec 12, 2020 at 18:38 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Dec 12, 2020 at 18:33 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Dec 12, 2020 at 5:53 | comment | added | Stahl | Hi Dmitri, thanks for your answer! I didn't expect set-theoretic issues to get in the way, but perhaps I should have given that Toën is intentionally making use of universe-changing in the paper. It sounds like I might be able to get what I want if I play with changing universes like Toën does, but I'll have to think about this a bit more. By the way, do you know of a reference where I might find the proof that generating cofibrations of $\mathsf{dgCat}$ are functors between small categories? | |
| Dec 7, 2020 at 15:52 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |