Timeline for Which triangulated categories are subcategories of compact objects "somewhere"?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 8, 2017 at 21:08 | vote | accept | Mikhail Bondarko | ||
| Nov 23, 2016 at 23:52 | history | edited | Denis Nardin | CC BY-SA 3.0 |
Added a disclaimer that the answer only covers a special case.
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| Nov 23, 2016 at 20:24 | comment | added | Mikhail Bondarko | Yes, $T$ is small in my question. Also, I would be satisfied with any $B$ satisfying my conditions; no "enhancements" for it are necessary. | |
| Nov 23, 2016 at 20:08 | comment | added | Denis Nardin | @DavidWhite The topological triangulated categories are precisely those that can be enhanced to a stable ∞-category, so the categories in Muro-Schwede-Strickland indeed do not come from a stable ∞-category. Also, unless I am mistaken in the OP there's the hypothesis that $T$ is small. Otherwise you have to change universes, as you say. This creates some problems because it changes which triangulated categories are cocomplete, and I believe it makes the question itself ill-posed. | |
| Nov 23, 2016 at 20:07 | comment | added | David White | @DenisNardin: Just had a look at 5.4.2.4, and it says C needs to be small. Does that cause any issue? I know Lurie sometimes increases Grothendieck universe to pretend large things are small, but wouldn't that also change the meaning of "compact" and of "small coproducts" in Mikhail's question? How would you argue that 5.4.2.4 still applies even if the word "small" is dropped? | |
| Nov 23, 2016 at 20:04 | comment | added | David White | @MikhailBondarko I very much doubt that a given triangulated category has a stable $\infty$ category associated to it. The example of Muro-Schwede-Strickland "Triangulated categories without models" should be a counterexample. I believe the obstructions they find are present even on the $\infty$-category level, not just the model category level. | |
| Nov 23, 2016 at 18:51 | comment | added | Mikhail Bondarko | Thank you! So, you essentially confirm my guess. I wonder whether there exist arguments that are quite distinct from "take a enhancement for $T$ and consider ind-objects for it". | |
| Nov 23, 2016 at 18:46 | comment | added | Denis Nardin | @MikhailBondarko Another reference that might be useful is Construction 1.3.1.6 in Higher Algebra, that for every dg-category constructs an ∞-category with the same homotopy category. I believe that when the original dg-category was triangulated this ∞-category will be automatically stable. | |
| Nov 23, 2016 at 18:38 | comment | added | Denis Nardin | @MikhailBondarko I cannot vouch for all sort of "enhancements", but a stable ∞-category is, roughly speaking, a triangulated category enriched in spectra; so all dg-enhancements should be special cases of it (e.g. arxiv.org/abs/1308.2587). | |
| Nov 23, 2016 at 18:36 | comment | added | Mikhail Bondarko | Thank you! May I ask you (sorry for my ignorance): can all triangulated categories that admit some sort of an "enhancement" be presented as stable $ \infty$ categories? | |
| Nov 23, 2016 at 18:31 | history | answered | Denis Nardin | CC BY-SA 3.0 |