Timeline for What are the reflective subcategories of the category of presentable categories?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2018 at 20:56 | comment | added | Tim Campion | @IvanDiLiberti :). I think the reason for the other convention is to agree with the conventions of Lawvere theories. But even with Lawvere theories, I usually find it less confusing to think about it this way! | |
| Jul 30, 2018 at 16:14 | vote | accept | KotelKanim | ||
| Jul 29, 2018 at 22:23 | comment | added | Ivan Di Liberti | Tim, I prefer your version of GU, but everybody else prefer the one I chose. We should keep using the one that we like to make the others change! ahah | |
| Jul 29, 2018 at 21:49 | comment | added | Tim Campion | @KotelKanim Lol, well hello again! You're right, I shouldn't have the "op" there. I'm not sure where I wrote "colimits" -- but maybe the confusion is that I used a dual description of Gabriel-Ulmer duality to Ivan di Liberti's: there is an equivalence between $\kappa-Cts$ and $\kappa-Cocts$ given by taking opposite categories (at the level of 2-categories, you also have to reverse 2-cells, but I've thrown away the noninvertibles). If $C$ is $\kappa$-complete and small, then $Ind_\kappa(C) = Fun_\kappa(C^{op},S)$, so both perspectives are natural in their own way. | |
| Jul 29, 2018 at 21:40 | answer | added | Tim Campion | timeline score: 9 | |
| Jul 29, 2018 at 16:41 | comment | added | KotelKanim | Hey Tim! We met at YTM (I'm Lior). So... Tomer and his students are having some trouble with this point ;) Btw, did you mean limits when you wrote colimits? And isn't $Pr_{\kappa}^L$ equivalent to $\kappa-Cocts$ (without the op) just by taking $\kappa$-compact objects? Sorry if I am being dumb, I am not very strong at the part of category theory which has $\kappa$ in it... | |
| Jul 29, 2018 at 16:00 | comment | added | Tim Campion | @KotelKanim A good way to see that $Pres^L_\kappa$ is presentable is, as Ivan di Liberti points out in his answer, to observe that it is equivalent to $\kappa-Cocts^{op}$, the opposite of the category of small, $\kappa$-cocomplete categories and $\kappa$-cocontinuous functors, and these are the algebras for an accessible monad on $Cat$. Maybe I'll mention that Tomer Schlank and his students have done some work studying smashing localizations of $Pr^L$. They call them "modes". | |
| Jul 29, 2018 at 14:21 | answer | added | Ivan Di Liberti | timeline score: 2 | |
| Jul 29, 2018 at 14:00 | comment | added | KotelKanim | Well, I think it's time to come clean :) In the end, I am indeed interested in "smashing localizations" of $Pr^L$. Namely, those which a realized by tensoring with an idempotent object with respect to the symmetric monoidal structure of $Pr^L$ in the sense of HA.4.8.2.1. But I do think that the question as stated it is also interesting by itself. The condition that the localization respects the monoidal structure is equivalent to the condition that the subcategory is closed under exponentiation by an arbitrary presentable $\infty$-category. But perhaps characterizing those directly is easier. | |
| Jul 29, 2018 at 13:49 | comment | added | Tim Campion | I could have sworn there was a question about smashing localizations of $Pr^L$ here not too long ago, but I can't find it now. Anyway, I think it makes sense to take into account the monoidal structure on $Pr^L$ and think about reflective subcategories which respect the monoidal structure. | |
| Jul 29, 2018 at 12:41 | comment | added | KotelKanim | Actually, I can't immediately see why it is presentable. It is equivalent to the $\infty$-category of small $\infty$-categories admitting $\alpha$-small colimits and functors which preserve them, right? It seems plausible to me that it is presentable, but I don't have a proof from the top of my head. | |
| Jul 29, 2018 at 12:27 | comment | added | KotelKanim | Right. I have considered this approach but was hoping I could avoid it. But now that you mention it, I wonder if there is a way to encapsulate this for all $\alpha$ together as a property of $Pr^L$ itself, which can be then used as a black box. | |
| Jul 29, 2018 at 9:06 | comment | added | Simon Henry | Maybe an interesting point of view: $Pr^L$ is not presentable, but for any regular cardinal $\alpha$ the category of $\alpha$-presentable categories and Left adjoint functor preserving $\alpha$-presented objects is a presentable category. And for large $\alpha$ this is good approximation of the category of all presentable categories. | |
| Jul 29, 2018 at 8:17 | history | edited | KotelKanim | CC BY-SA 4.0 |
added 95 characters in body
|
| Jul 29, 2018 at 8:06 | history | asked | KotelKanim | CC BY-SA 4.0 |