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added to action groupoid a section on action oo-groupoids
since it is an important example, I added a section action oo-groupoids also to fiber sequence
Thanks alot. But what I was having in mind is to have the data of action quasicategory directly in terms of Kan complex which is acting and the quasicategory which is acted upon. Doing abstract infinity limit involving an action into indirectly defined quasicategory of automorphisms is too far from practical use.
Yeah, I know, it can be hard to compute. There are of course explicit tools to compute the Grothendieck construction (some mentioned at (infinity,1)-Grothendieck construction) but it can still be hard, sure.
But I think for low categorical degree, one can handle it: that universal fibration $Z \to \infty nGrpd$ should be really just the fiber of the codomain fibration for $n Grpd$ over one of its legs, i.e. the pullbacl
$\array{ Z &\to& * \\ \downarrow && \downarrow \\ n Grpd^{\Delta[1]} &\to& n Grpd \\ \downarrow \\ n Grpd } \,.$And then the ordinary pullback of that should be the action $n$-groupoid.
So in other words I am saying that the action n-groupoid should be computed just as the homotopy pullback of the point
$\array{ V//G &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G &\to & \infty Grpd }$using the tools described at homotopy pullback, only that instead of using the path object $\infty Grpd^I$ we use the arrow category $\infty Grpd^{\Delta[1]}$.
So in other words: the thing should really be the lax (“comma”) pullback of the point, simply. I expect that this is really what Lurie’s universal fibration $Z \to \infty Grpd$ achieves, but I have not found the time to really check this.
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