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8:30 AM
@RuneHaugseng It seems you're right. Let me ask another question: is there an easy proof that for any ∞-category I the map $\mathrm{colim}_{i∊I^{op}} I_{i/}\to I$ is an equivalence?
 
 
1 hour later…
9:55 AM
That's the lax (or maybe colax) colimit for the constant functor from I with value a point, so it definitely follows from my paper with David and Thomas, but there's probably an easier way to do it.
 
 
2 hours later…
12:12 PM
@DenisNardin If you find an easy proof, I'd be interested in hearing it.
 
1:11 PM
A way of reformulating it is that Tw(I)→I is a localization (that is I is obtained from Tw(I) by inverting all arrows that lie above equivalences), which seems eminently reasonable since Tw(I)→I is a cocartesian fibration with weakly contractible fibers, but I don't see an immediate proof (it is not a trivial marked fibration :( )
 
 
6 hours later…
7:01 PM
@AaronMazel-Gee Ok. Thanks for the clarifications. It seems that the property of being perfect can be substituted by the weaker assumption of the global sections commuting with cotensoring by S^1. About the classical definition, it's a fibrant replacement of the cyclic bar construction I think (which when applied to cofibrant Gamma spaces gives the same thing). The theorem relating both definitions is III.6.1 in Scholze-Nikolaus.
@DenisNardin Thanks. Indeed theorem III.6.1 gives the precise statement of the equivalence that I was looking for.
 

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