Timeline for Are accessible $\infty$-categories closed under accessible localizations?
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15 events
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| Jul 3, 2016 at 17:25 | comment | added | Dylan Wilson | Yes. I don't think there's anything different between the 1-categorical and infty-categorical setting here. Adameck and Rosicky is a good reference for 1-categorical results about accessible and presentable categories. I think they treat localizations in that book as well. | |
| Jul 3, 2016 at 16:41 | comment | added | Tim Campion | Oh. Lurie constructs localization functors at strongly saturated classes $\mathcal{W}$. The strong saturation condition 5.5.4.5. is much stronger than just being saturated -- it requires being closed under co-base change and all colimits in the arrow category. At which point it becomes quite plausible that a theorem like this could be proven in the 1-categorical setting. So I think I'm back to thinking it's not that localization is intrinsically better-behaved $\infty$-categorically, but rather that limits and colimits are better in categories of interest. | |
| Jul 3, 2016 at 16:35 | comment | added | Tim Campion | @DylanWilson I don't think there's anything surprising going on in 5.5.4.2., since part of the hypotheses include having a localization functor, which Lurie defines to be a left adjoint to a fully faithful functor. So I think 5.5.4.2. should go through just fine in the 1-categorical context. But yeah -- I probably should have read 5.5.4. carefully before asking this question! | |
| Jul 3, 2016 at 16:28 | comment | added | Dylan Wilson | @TimCampion see Proposition 5.5.4.2 for the saturated case at least. At this point maybe you should just read the whole of section 5.5.4 :) | |
| Jul 3, 2016 at 16:17 | comment | added | Tim Campion | Beautiful. So I guess I'm just left wondering about the case where the ambient $\infty$category is merely accessible. Since there aren't that many accessible-but-non-presentable $\infty$categories that anybody presently cares about, maybe I shouldn't be too worried. | |
| Jul 3, 2016 at 16:03 | comment | added | Dylan Wilson | @TimCampion "accessible classes" as you call them, are always generated by a set. See Lemma 5.5.4.14 in HTT. | |
| Jul 3, 2016 at 15:48 | comment | added | Kaya Arro | Yes, I agree! One feels that ``localization'' is much more naturally an $\infty$-categorical concept than a 1-categorical concept. Unfortunately, I don't know a reference for accessible rather than small classes of morphisms. Usually when I encounter a size issue, I try to see if I can move into a larger Grothendieck universe, so that whatever object I want to be small actually is small. But I don't think that solves all the problems that might arise here, since perhaps in that case the $(\infty,1)$-category in question would no longer be presentable? Size issues clearly aren't my specialty. | |
| Jul 3, 2016 at 15:47 | comment | added | Tim Campion | where can I read about this? In HTT 5.5.4 he constructs the localization I'm talking about for a small class of morphisms. Is the generalization to an accessible class written up somwhere? | |
| Jul 3, 2016 at 15:43 | comment | added | Tim Campion | Wow, this is amazing! Given how badly it fails in the ordinary context. | |
| Jul 3, 2016 at 15:40 | comment | added | Kaya Arro | Yes, it is true that I have not fully answered your question because my response only applies in the context of presentable $(\infty,1)$-categories; however, what I have written holds for presentable $(\infty,1)$-categories: localization in the broad sense coincides with localization in the narrow sense in the presence of those hypotheses, and localization at $W$ really is the same as localization at its saturation, so questions of accessibility are unimportant in this context. I do not know the answer for non-presentable $(\infty,1)$-categories. | |
| Jul 3, 2016 at 15:39 | comment | added | Tim Campion | Or perhaps there is a sense that, even if what I'm asking for is not true, is it the case that every localization which is actually interesting turns out to be reflective in the $\infty$-setting? So that in practice the answer is yes? | |
| Jul 3, 2016 at 15:31 | comment | added | Tim Campion | Regarding your second paragraph: In ordinary category theory the word "localization" is often taken to mean taking a subcategory orthogonal to some objects, analogous to the sort of localization you describe. But I mean a localization in the broadest possible sense of universally inverting a class of morphisms, accessibly embedded in the arrow category. E.g. the construction of the homotopy category is a localization in the broad sense but not the narrow sense. Do you say that every localization in this broad sense is a reflective localization in the context of presentable $\infty$-categories? | |
| Jul 3, 2016 at 15:30 | comment | added | Tim Campion | Regarding your first paragraph: I'm worried that the saturation of an accessible, accessibly embedded subcategory of the arrow category might fail to be accessible / accessibly embedded, so I think it potentially does make a difference. | |
| Jul 3, 2016 at 15:24 | history | edited | Kaya Arro | CC BY-SA 3.0 |
added 145 characters in body
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| Jul 3, 2016 at 15:17 | history | answered | Kaya Arro | CC BY-SA 3.0 |