Timeline for Decomposing a (co)limit by decomposing the indexing diagram
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 6, 2021 at 16:56 | comment | added | KotelKanim | @TimCampion might be the only thing... At the time we didn't know much oo-category theory beyond Lurie's books. As it was not (explicitly) there, we were excited to have this as a, somewhat unexpected, application of our theorem. The other answers here show of course much more direct and reasonable proofs. Nonetheless, it's always nice to get a citation! | |
| Mar 5, 2021 at 22:29 | comment | added | Tim Campion | @KotelKanim Thanks -- I somehow just noticed your comment here. One nice thing is that your paper is a citable source! | |
| Nov 15, 2020 at 11:59 | comment | added | KotelKanim | Though the answers below are of course very slick and comprehensive... | |
| Nov 15, 2020 at 11:53 | comment | added | KotelKanim | @TimCampion you might want to check out example 2.5 here: arxiv.org/pdf/1705.04933.pdf. Asaf Horev and I prove here what you ask for. | |
| Sep 4, 2020 at 5:38 | history | became hot network question | |||
| Sep 3, 2020 at 23:42 | comment | added | Tim Campion | @Denis-CharlesCisinski Thanks -- this may actually be what I need for my purposes -- I happen to be working with a directed union of indexing diagrams, it it will actually be nice not to have to check that the union is a homotopy colimit of $\infty$-categories. Lurie makes a note of this case in Rmk 4.2.3.9, and I think gives the other cases you mention later in the book, but without the naturality statement as far as I can tell. | |
| Sep 3, 2020 at 23:27 | vote | accept | Tim Campion | ||
| Sep 3, 2020 at 23:15 | answer | added | Dylan Wilson | timeline score: 5 | |
| Sep 3, 2020 at 23:11 | answer | added | Rune Haugseng | timeline score: 9 | |
| Sep 3, 2020 at 22:36 | comment | added | D.-C. Cisinski | There is theorem 7.3.16 in my book on higher categories (in the spirit of the result you quote from HTT but a little bit more usable). This is what explains decompositions of diagrams with Reedy-like considerations, as explained in corollary 7.4.4 proposition 7.4.5 of loc. cit. for instance. | |
| Sep 3, 2020 at 22:23 | answer | added | Zhen Lin | timeline score: 4 | |
| Sep 3, 2020 at 22:21 | comment | added | Maxime Ramzi | (Actually, it might not be that subtle in the $1$-categorical case; I think it mostly relies on the fact that Cat is cartesian closed, and on the analysis of hom-sets in a limit of categories) | |
| Sep 3, 2020 at 22:13 | comment | added | Maxime Ramzi | But he doesn't give a reference, and the claim doesn't seem that obvious to me (it seems a bit subtle, because of the possibly weird behaviour of colimits in Cat, as you point out) , so I guess it'd be good to have some clarification (should it only be to see if the proof goes through in more generality, which you seem to seek). | |
| Sep 3, 2020 at 22:02 | comment | added | Tim Campion | Ah, so he does, thanks! Sanity check: it works for the canonical "bad coequalizer" in Cat given by $(\bullet) \rightrightarrows (\uparrow) \to B \mathbb N$. So I suppose I'd buy it... | |
| Sep 3, 2020 at 21:55 | comment | added | Maxime Ramzi | In his answer to the question you reference, Peter seems to claim this is always true for $1$-categories. | |
| Sep 3, 2020 at 21:46 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 18 characters in body
|
| Sep 3, 2020 at 21:44 | comment | added | Tim Campion | This question seems to ask about an instance of this phenomenon. | |
| Sep 3, 2020 at 21:38 | history | asked | Tim Campion | CC BY-SA 4.0 |