Timeline for How should I think about presentable $\infty$-categories?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Oct 8, 2019 at 1:37 | comment | added | Charles Rezk | @Patriot I am also surprised that I never saw locally presentable categories until I encountered presentable $\infty$-categories, which for me really happened when I saw Dan Dugger's paper "Universal homotopy theories" arxiv.org/abs/math/0007070 (which is you can see is the ultimate source of the answer I gave below). | |
| Oct 7, 2019 at 8:47 | vote | accept | Patriot | ||
| Oct 7, 2019 at 8:46 | comment | added | Patriot | I'll take a look, @AlexanderCampbell. I'm just surprised that I've never seen the adjective 'presentable' in my 1-categorical upbringing before, only to suddenly see it pop up everywhere in the $\infty$-categorical world. | |
| Oct 4, 2019 at 21:31 | answer | added | Charles Rezk | timeline score: 42 | |
| Oct 4, 2019 at 20:46 | answer | added | Mike Shulman | timeline score: 12 | |
| Oct 4, 2019 at 16:28 | comment | added | Pedro Sánchez Terraf | For regular cardinals: You know that a union of countably many countable sets is countable. Generalizing this, $\kappa$ is regular iff every union of a collection of size $<\kappa$ of sets of size $<\kappa$ has size $<\kappa$. (Thus the “countable” example is saying that the least uncountable cardinal is regular.) | |
| Oct 4, 2019 at 15:06 | answer | added | Tim Campion | timeline score: 8 | |
| Oct 4, 2019 at 14:36 | comment | added | Tim Campion | This is particularly useful in the $\infty$-categorical setting, where it can be daunting to explicitly construct a functor -- so anytime you get functors "for free" (as from the adjoint functor theorem), it's a big win. | |
| Oct 4, 2019 at 14:31 | comment | added | Tim Campion | In some sense, the thing that makes locally presentable categories / presentable $\infty$-categories tick is that they fulfill the promise of the adjoint functor theorem just about optimally: if $F: C \to D$ is a functor between locally presentable categories which preserves colimits, then $F$ is a left adjoint, while if $F$ preserves limits and is accessible (i.e. preserves sufficiently-filtered colimits), then $F$ is a right adjoint. This manifests itself in particular in the very nice theory of localization for locally presentable categories. | |
| Oct 4, 2019 at 10:57 | comment | added | Alexander Campbell | It might help to first think about locally presentable categories, the theory of which is nearly 50 years old. | |
| Oct 4, 2019 at 9:50 | answer | added | Najib Idrissi | timeline score: 12 | |
| Oct 4, 2019 at 9:29 | comment | added | Praphulla Koushik | I like the way this question is written.. I hope some one gives a direct answer than suggesting to read something.... | |
| Oct 4, 2019 at 9:01 | history | asked | Patriot | CC BY-SA 4.0 |