Timeline for What's the stabilization of the $\infty$-category of $\infty$-categories?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2016 at 22:04 | vote | accept | Yuri Sulyma | ||
| Aug 25, 2016 at 20:18 | comment | added | Dylan Wilson | @CharlesRezk you got me | |
| Aug 25, 2016 at 19:29 | answer | added | Yonatan Harpaz | timeline score: 21 | |
| Aug 25, 2016 at 17:39 | comment | added | Charles Rezk | It might be more interesting to think about stabilizations of slices $\infty\mathrm{Cat}_{/C}$. I'd guess that the stabilization of $\infty\mathrm{Cat}_{/\Delta^1}$ is something like spans of spectra. | |
| Aug 25, 2016 at 17:05 | comment | added | Charles Rezk | @DylanWilson That's what I thought. BTW, assuming J is a small category, every localization of Fun(J,Spaces) wrt a set of maps is accessible, and conversely all accessible localizations arise in this way. And all presentible infty-cats are equivalent to ones obtained this way. So I don't think I have to insert "accessible" or "presentable" anywhere :P | |
| Aug 25, 2016 at 16:56 | comment | added | Dylan Wilson | @CharlesRezk yes (after inserting 'presentable' and 'accessible' in a few places, at least). This follows from the universal property for maps out of a stabilization (that's where I use presentability), together with the universal property of maps out of a localization, right? Alternatively you could probably prove it by hand as in the proof of Higher Algebra 1.4.4.9 | |
| Aug 25, 2016 at 15:33 | comment | added | Elden Elmanto | @DenisNardin - ah alright good to know! | |
| Aug 25, 2016 at 15:28 | comment | added | Denis Nardin | If you are interested in things that are loosely speaking homology theories on categories, one interesting keyword is noncommutative motives. (@EldenElmanto the stabilization is always linear functors from $S^{fin}_*$, no presentability assumptions needed) | |
| Aug 25, 2016 at 15:19 | history | edited | Yuri Sulyma | CC BY-SA 3.0 |
clarified which question I care more about
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| Aug 25, 2016 at 14:29 | comment | added | Elden Elmanto | @CharlesRezk, are you thinking of describing stabilization of a presentable $C$ the localization of $Fun(S^{fin}_{\star}, C)$ at reduced and excisive functors? I think this also only works for presentable categories | |
| Aug 25, 2016 at 13:44 | comment | added | Charles Rezk | Isn't there some formula for computing stabilizations in terms of presentations? E.g., if C is a localization of Fun(J,Spaces) at some set of maps, then it's stabilization should be a stable localization of Fun(J,Spectra) at a set of maps, no? (I'm asking, I don't really know.) If you compute this on the usual presentation of infty-Cat, you should just get spectra back. | |
| Aug 25, 2016 at 12:42 | comment | added | Dylan Wilson | It feels like loops on some pointed category will be the space of automorphisms of the distinguished object, no? So the stabilization should just be spectra back again... To get a more interesting answer one probably needs to at least consider lax pullbacks in your definition of `loops', but I dunno | |
| Aug 25, 2016 at 12:20 | comment | added | Denis Nardin | Note that in general you don't have a functor $\Sigma^\infty$ from a category to its stabilization, but only a functor $\Omega^\infty$ going in the other direction ($\Sigma^\infty$ exists when the category is presentable by the adjoint functor theorem). Moreover I don't think that the category of spectrally enriched categories is stable. Interesting question though. | |
| Aug 25, 2016 at 12:15 | history | asked | Yuri Sulyma | CC BY-SA 3.0 |