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In the statement for general fibered categories, the way it was stated seemed to be assuming that $C$ was small, so I made the assumption explicit. I also clarified the arrows of $Cart(C)$ are just the cartesian functors.
I also added the statement that the construction gives a natural equivalence.
Random aside: the language I’ve seen clarifying which functor is meant by just saying its action on morphisms – e.g. that $Cart(-)$ acts on morphisms by pullback – is dissatisfying, but I don’t really know a better way to state that without being overly cumbersome. Is there a better way of phrasing it?
I have adjusted the first sentences of the Idea-section, which were essentially unreadable with their nested case distinctions.
Also I moved the proposition about the case of $\infty$-groupoids over $\infty$-groupoids up to (here) right after the statement of the $(\infty,0)$-Grothendieck construction (for it uses nothing else that would be discussed later and usefully completes the picture at this point)
It’s this Prop. that is used over at shape of an (infinity,1)-topos and I will finally link to it from there.
Here is a question:
Given an $\infty$-presheaf $\mathcal{X} \,\colon\, \mathcal{C}^{op} \xrightarrow{\;} Grpd_\infty$, its $\infty$-Grothendieck construction is naturally fibered over that of its $n$-truncation (for any $n \in \mathbb{N}$):
$\array{ \overset{c \in \mathcal{C}}{\int} \mathcal{X}(c) \\ \big\downarrow {}^{\mathrlap{P_n}} \\ \overset{c \in \mathcal{C}}{\int} \tau_n\mathcal{X}(c) }$This map itself is the $\infty$-Grothendieck construction of some $\infty$-functor
$\overset{c \in \mathcal{C}}{\int} \tau_n\mathcal{X}(c) \; \xrightarrow{\;\;\;\;\;\;\;} \; Grpd_\infty \,.$What is this $\infty$-functor? I suppose it should the the functor that assigns the system of $n$-connected covers of $\mathcal{X}$:
$(c,x) \;\mapsto\; cn_n\big( \mathcal{X}(c), x \big) \,.$Is this right? Is this discussed/proven anywhere?
I have put a note with some elaboration on this question into the Sandbox (using some tikz
-typesetting, which doesn’t display here on the $n$Forum).
added pointer to
for disucssion of the $\infty$-Grothendieck construction for $\infty$-functors to $\infty$-toposes from inverse images of $\infty$-groupoids.
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