Timeline for Decomposing a (co)limit by decomposing the indexing diagram
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 4, 2020 at 9:00 | comment | added | D.-C. Cisinski | @PhilTosteson You are right. The localization of E at cocartesian morphism is literally the 2-colimit (i.e. the colimit in the $\infty$-category obtained by inverting equivalences of categories in the $1$-category of small categories). This does not coincide with the colimit (in the $1$-category of small categories) in general. Rune's answer together with Zhen Lin's show that the comparison map from the $2$-colimit to the $1$-colimit, although not an equivalence, is colimit-final. I also thought that the question was about $1$-colimits though. | |
| Sep 4, 2020 at 6:04 | comment | added | Phil Tosteson | @DenisNardin Indeed, that's why I chose this example-- since the homotopy quotient disagrees with the ordinary one. I thought that the question and this answer were meant to be specifically 1 categorical | |
| Sep 4, 2020 at 5:15 | comment | added | Denis Nardin | @PhilTosteson The colimit of the constant diagram in the $\infty$-category of $\infty$-categories is $BG$, not the point. This is just another way in which colimits are better behaved in $\infty$-categories :). | |
| Sep 4, 2020 at 1:01 | comment | added | Phil Tosteson | I may be misinterpreting your statement/making a mistake, but I don't think that $I$ is always the localization of $E$. For instance consider the case where $J$ is a one object groupoid $BG$ and $J \to Cat$ is the constant diagram with value the trivial category. Then the colimit is the trivial category, but $E$ is $BG$, and all localizations of $E$ are isomorphic to $E$. | |
| Sep 3, 2020 at 23:27 | vote | accept | Tim Campion | ||
| Sep 3, 2020 at 23:26 | comment | added | Tim Campion | I wonder if the proofs in HTT of the relevant statements actually depend on the more technical statement of 4.2.3.10. It would be nice if they didnt'! | |
| Sep 3, 2020 at 23:22 | comment | added | Tim Campion | Nice! Thanks, Rune. I'll probably accept this one -- though I also like Dylan's presentation of essentially the same argument in his answer below. And of course, it would be nice to work out an analog of Zhen Lin's argument in a higher setting -- those types of arguments in terms of homsets tend to generalize more easily to the enriched setting. | |
| Sep 3, 2020 at 23:11 | history | answered | Rune Haugseng | CC BY-SA 4.0 |