Timeline for Cofinality for coends?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 4, 2020 at 5:07 | comment | added | Tim Campion | @RoaldKoudenburg Thanks! As mentioned in the CW answer below, Alex Campbell had the same idea, and it turned out to be very enlightening! | |
| Mar 3, 2020 at 22:58 | answer | added | Tim Campion | timeline score: 10 | |
| Mar 3, 2020 at 17:36 | comment | added | Roald Koudenburg | Another approach to cofinality for coends might be to consider them as left Kan extensions along the profunctor $H\colon J^\circ \times J \to 1$ given by the hom-sets of $J$. Precomposition with a double cell into $H$, in the double category of profunctors, preserves this left Kan extension whenever it satisfies some kind of Beck-Chevalley condition. The horizontal dual setting, preservation of right Kan extensions, is explained in Section 4 of link. | |
| Mar 2, 2020 at 10:44 | answer | added | D.-C. Cisinski | timeline score: 15 | |
| Mar 2, 2020 at 0:28 | comment | added | Tim Campion | @Denis-CharlesCisinski Thanks! This would make a great answer, especially because I don't immediately see a proof -- e.g. $Tw(I) \to Tw(J)$ is not typically a base change of $I \to J$... | |
| Mar 1, 2020 at 23:45 | comment | added | D.-C. Cisinski | For question 1, examples are provided by functors which are universally cofinal (i.e. cofinal after any base change); this work for $\infty$-categories as well. For instance, any functor which is smooth or proper with weakly contractile fibers. For instance, if $X$ is a quasi-category, the canonical functor from the category of simplices of $X$ to $X$ has the property you want. | |
| Mar 1, 2020 at 15:18 | comment | added | Dylan Wilson | It seems to me like the general finality criterion might be equivalent to the special case of the coend that computes Hom-sets in J, in case that's any easier... it seems hard at the moment to come up with examples though. | |
| Mar 1, 2020 at 11:54 | answer | added | fosco | timeline score: 3 | |
| Feb 29, 2020 at 21:39 | comment | added | Martin Brandenburg | An answer would probably also help with mathoverflow.net/questions/352607 | |
| Feb 29, 2020 at 20:37 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 29, 2020 at 20:29 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 29, 2020 at 20:23 | history | asked | Tim Campion | CC BY-SA 4.0 |