Timeline for How should I think about presentable $\infty$-categories?
Current License: CC BY-SA 4.0
9 events
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| Aug 31, 2022 at 19:56 | comment | added | Z. M | Maybe it is better to add an example of presentable category which is not compactly generated, i.e. an uncountable cardinal would involve. | |
| Oct 8, 2019 at 1:33 | comment | added | Charles Rezk | ... Frankly, 'locally presentable' is crappy terminology even on its own terms (the fact that 'presentable' objects are all so with respect to some single chosen set is not at all implied by the term). Jacob wisely chose to jettison the "locally" when defining the $\infty$ version, which allows you to reinterpret the "presentable" thing as the $\infty$-category itself, rather than its objects. I would prefer to call the 1-categorical version "presentable categories" without the 'locally', and would happily support a movement to do so. | |
| Oct 8, 2019 at 1:30 | comment | added | Charles Rezk | @Patriot Re: terminology, the "presentable" in 'locally presentable category' doesn't really mean the same thing as the "presentable" in 'presentable $\infty$-category'. As I understand it, "locally" is category-theorists-speak for 'condition put on each of the objects or hom-sets of a category'. (Think of 'locally small category' for instance.) The "presentable" in this case means as in Najib's answer: 'I am a colimit of a small diagram from some (unspecificed but fixed) set of objects".... | |
| Oct 7, 2019 at 8:51 | comment | added | Patriot | Thanks for the response. As for the terminology, from what I know the terms 'locally presentable' and 'presentable' are awkwardly used interchangeably in $\infty$-categorical contexts. "(...) if you don't impose them, then nothing would prevent you from choosing all objects as generators, which is somewhat useless." Do you have an example of a concrete issue that would arise if I'd ignore the set-theoretic constraints and just take some kind of proper class of generators? | |
| Oct 5, 2019 at 6:44 | history | edited | Najib Idrissi | CC BY-SA 4.0 |
added 159 characters in body
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| Oct 5, 2019 at 6:43 | comment | added | Najib Idrissi | @MikeShulman You're absolutely right, I'll edit my answer. | |
| Oct 4, 2019 at 20:37 | comment | added | Mike Shulman | Actually, A.3.7.6 says that locally presentable $\infty$-categories are exactly the ones that come from combinatorial simplicial model categories. And a combinatorial model category is one whose underlying 1-category is... locally presentable. So as useful as that theorem is, it doesn't really explain the importance of local presentability as such; it only reduces the $\infty$-categorical case to the 1-categorical one. | |
| Oct 4, 2019 at 9:58 | comment | added | Najib Idrissi | Now, is the definition "aesthetic"... I'll let people with stronger opinions on the beauty of math than me answer. For me the beauty is in the proofs and the connections between the different definitions. | |
| Oct 4, 2019 at 9:50 | history | answered | Najib Idrissi | CC BY-SA 4.0 |