Timeline for Does $Pr^L_\kappa \in Pr^L$ behave like an "object classifier" or "universe"?
Current License: CC BY-SA 4.0
9 events
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| Jul 15, 2022 at 18:13 | comment | added | Tim Campion | @SimonHenry But now I think I see your point, and I am confused... I still want to say that (3) does imply that we have a bifibration in $CAT$, but I think you're right that the descent condition needs to be stated separately. | |
| Jul 15, 2022 at 17:58 | comment | added | Tim Campion | @SimonHenry What I mean is, I believe the following are equivalent for a functor $\xi : X \to C$ with $C \in Pr^L$: (1) $\xi$ is a presentable fibration; (2) $\xi$ is the unstraightening of a functor $\Xi : C \to Pr^L$ which preserves colimits and and is "accessible" in the sense that $C \xrightarrow \Xi Pr^L \to CAT$ preserves $\kappa$-filtered colimits for some $\kappa$; (3) $\xi : X \to C$ is an opfibration in the 2-category $Pr^L$ in the sense that (i) $X \in \Pr^L$, (ii) $\xi$ preserves colimits and (iii) the functor $X \to (\xi \downarrow C)$ has a left adjoint in $Pr^L/C$. | |
| Jul 15, 2022 at 17:51 | comment | added | Simon Henry | Well, it's a bit more complicated than cocartesian fibration. I'm sure what the exact condition is, but you at least want bifibration with both adjoint, and you need a descent condition to express préservation of colimits. | |
| Jul 15, 2022 at 17:05 | comment | added | Tim Campion | Oh of course... So it appears that $Pr^L$, if it lived in $Pr^L$, would classify precisely cocartesian fibrations in $Pr^L$ in the usual 2-categorical sense. So $Pr^L_\kappa$ should definitely be some kind of "size-restricted fibration classifier". | |
| Jul 15, 2022 at 16:58 | comment | added | Simon Henry | Under the condition you have it is both left and right adjoint: taking each object to the initial or terminal object of the fibers gives both adjoint. | |
| Jul 15, 2022 at 16:55 | comment | added | Tim Campion | @SimonHenry Ah, I think you're right... A colimit-preserving functor $F : C \to Pr^L$ such that the composite $C \to Pr^L \to CAT$ preserves $\kappa$-filtered colimits for some $\kappa$ is exactly a presentable fibration, and Gepner - Haugseng - Nikolaus show that in this case the Grothendieck construction $\int F$ of the functor is presentable, and the fibration $\int F \to C$ is an accessible functor. I don't think that $\int F \to C$ need be a left adjoint though, which is interesting -- we seem to get something bigger than the slice... | |
| Jul 15, 2022 at 16:40 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jul 15, 2022 at 16:29 | comment | added | Simon Henry | I think the case of colimit preserving functors $C \to Pr^L$ has a nice fibrational description. I would say something like a "left and right adjoint bifibration $D \to C$ that satisfies a descent condition with respect to colimit in $C$" (using that colimit in $Pr^L$ are limits in $Pr^R$). So, it might be a good idea to start by figuring out this case in details and then see what happen when one adds the further condition to take values in $Pr^L_\kappa$ and to preserves $\kappa$-compact objects... | |
| Jul 15, 2022 at 16:05 | history | asked | Tim Campion | CC BY-SA 4.0 |