Timeline for The localization of the span category
Current License: CC BY-SA 4.0
13 events
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| May 2, 2022 at 23:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 2, 2022 at 22:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 3, 2021 at 23:38 | comment | added | D.-C. Cisinski | A version for which $C$ is allowed to be an $\infty$-category is discussed in §7.2 on calculus of fractions, in my book on higher categories mathematik.uni-regensburg.de/cisinski/CatLR.pdf | |
| Dec 3, 2021 at 23:38 | comment | added | D.-C. Cisinski | If you invert the $2$-cells in $C_{span}$, you will get the Dwyer-Kan localization of $C$ by $W$ (at least if you restrict to fibrant objects). This is not presented exactly in these terms, but this is somehow discussed in this paper of Michael Weiss: core.ac.uk/download/pdf/82257797.pdf which is quite relevant, but this is more recent paper of Joost Nuiten is more likely to be useful: arxiv.org/abs/1612.03800 | |
| Dec 3, 2021 at 21:39 | answer | added | Tim Campion | timeline score: 2 | |
| Dec 3, 2021 at 16:47 | comment | added | Connor Malin | @MikeShulman That sounds right; I think I would restrict the 2-morphisms to be only the commutative diagrams of spans where the middle map is a weak equivalence, and then I would hope there is a positive answer to "Is the localization of $(C,W)$ equivalent to a bicategorical localization of $(C_{\operatorname{span}},W_{\operatorname{span}})?$ | |
| Dec 3, 2021 at 1:50 | comment | added | Mike Shulman | Of course, the "category" of spans is actually a bicategory. Are you thinking of a sort of Hammock localization that applies to a "relative bicategory"? | |
| Dec 3, 2021 at 0:15 | history | edited | Connor Malin | CC BY-SA 4.0 |
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| Dec 2, 2021 at 23:10 | comment | added | Connor Malin | @ZhenLin Thanks for the relevant paper; the reason I am after such a result is because I want to be able to use zig-zags of weak equivalences for the vertical maps in the Hammock localizations | |
| Dec 2, 2021 at 22:35 | comment | added | Zhen Lin | I considered a more restrictive notion here where the induced morphism $y \to z \times x$ is also required to be a fibration. I forget the precise reasons, but they had better technical properties. Amusingly, I used them to get a simple description of the hammock localisation... | |
| Dec 2, 2021 at 19:37 | history | edited | Connor Malin | CC BY-SA 4.0 |
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| Dec 2, 2021 at 18:34 | history | edited | Connor Malin |
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| Dec 2, 2021 at 18:08 | history | asked | Connor Malin | CC BY-SA 4.0 |