Timeline for Pullback of localizations
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 24 at 23:39 | vote | accept | user39598 | ||
| Sep 24 at 23:39 | history | bounty ended | user39598 | ||
| Sep 24 at 9:25 | comment | added | Maxime Ramzi | Sorry I misread the original post and thought you already knew about the converse. I have added an explanation for this. | |
| Sep 24 at 9:25 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
added 1550 characters in body
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| Sep 24 at 1:34 | comment | added | user39598 | @MaximeRamzi My understanding is that in kerodon.net/tag/02LW, Lurie shows that if E-> B is a coCartesian fibration with contractible fibers, then it is a localization. Why does the converse hold? | |
| Sep 23 at 20:08 | comment | added | Z. M | @user39598 For the first: by kerodon.net/tag/02LW (and letting $U'=\operatorname{id}$), you can test whether a coCartesian fibration is a localization by testing at the fiber at every object of the base. | |
| Sep 23 at 20:05 | comment | added | Maxime Ramzi | For 1), tge logic is as follows : coCartesian fibrations with weakly contractible fibers are closed under pullbacks (because both "cocartesian fibration" and "weakly contractible fibers" are!), and Lurie characterizes "cocartesian fibrations that are localizations" as exactly those, so they are also closed under pullbacks, and in particular the pullback of one such is a localization (since it is a cocartesian fibration which is a localization) | |
| Sep 23 at 20:04 | comment | added | Maxime Ramzi | For 2), I wrote this up as Prop 1.4 in this note : sites.google.com/view/maxime-ramzi-en/notes/fun-with-pushouts | |
| Sep 23 at 19:02 | comment | added | user39598 | @MaximeRamzi: Great! Do you have references for the first two claims that 1) any coCartesian fibration which is a localization is a universal localization and 2) (co)reflexive localizations are closed under pullbacks? Or can you add some more details? I don't understand your logic in 1) and for 2), I know that faithful functors are closed under pullbacks (which are the left/right adjoints) but don't know the statement for (co)reflective localizations. | |
| Sep 23 at 17:59 | comment | added | Kevin Carlson | This is fun because it's almost the same example as the standard one people use to show that regular epis of categories are not always universal. | |
| Sep 23 at 17:31 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |