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Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, the comma category $(u \downarrow b)$ is aspheric.

Question. Consider a pullback diagram in $\mathbf{Cat}$, $$\require{AMScd} \begin{CD} \mathcal{E} @>{v}>> \mathcal{F} \\ @V{p}VV @VV{q}V \\ \mathcal{A} @>>{u}> \mathcal{B} \end{CD}$$ where $q : \mathcal{F} \to \mathcal{B}$ (hence also $p : \mathcal{E} \to \mathcal{A}$) is a Grothendieck fibration and $u : \mathcal{A} \to \mathcal{B}$ is aspheric. Is $v : \mathcal{E} \to \mathcal{F}$ also aspheric?


Of course, an aspheric functor is the same thing as a homotopy coinitial functor, so Thomason's homotopy colimit theorem implies that $N (v) : N (\mathcal{E}) \to N (\mathcal{F})$ is a weak homotopy equivalence, which is a strictly weaker conclusion.

I believe I have a proof that the answer to my question is yes, so what I am really looking for is an answer in the literature. I would not be surprised if it appears in Pursuing Stacks, but perhaps someone more familiar with the document knows for sure?

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up vote 6 down vote accepted

You may find this in Maltsiniotis'Astérisque 301: it is equivalent to prove the dual version (using Proposition 1.1.22), and then, Example 3.2.2 show that this is a particular case of Corollary 3.2.13 of loc. cit. This kind of ideas is developped systematically in his paper Structures d’asphéricité, foncteurs lisses, et fibrations.

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This seems to be a special case of Prop. 4.1.2.15 in Higher Topos Theory:

Let $p: X \to S$ be a coCartesian fibration of simplicial sets. Then $p$ is smooth.

Take $X$ and $S$ to be the nerves of the opposites of $\mathcal{F}$ and $\mathcal{B}$. The map $p$ being smooth means in particular that pulling back along $p$ preserves cofinal maps (see Remark 4.1.2.10).

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