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Quillen's Theorem A says that a functor $F:C \to D$ (between 1-categories) induces a homotopy equivalence of classifying spaces $BC \simeq BD$ if for every object $d$ in $D$ the fiber category $F/d$ has a contractible classifying space.

Quick reminder: the objects of $F/d$ are pairs $(c,\gamma)$ consisting of an object $c$ in $C$ and a morphism $\gamma:Fc \to d$ in $D$; and morphisms $(c,\gamma) \to (c',\gamma')$ in $F/d$ are given by all $f:c \to c'$ in $C$ satisfying $\gamma = \gamma' \circ Ff$ in $D$.

Here are two natural extensions of this result:

  1. If a group $\Gamma$ acts on $C$ and $D$ with $F$ an $\Gamma$-equivariant functor, then $F$ induces a $\Gamma$-equivariant homotopy equivalence between $C$ and $D$ provided its fibers are $\Gamma$-equivariantly contractible. I must confess that the only published version that I am aware of restricts to the case where $C$ and $D$ are $\Gamma$-posets, due to Thevanaz and Webb (pdf here)
  2. If we're working with a (lax/oplax) 2-functor between $2$ categories, then Quillen's Theorem A holds provided we define the lax fiber 2-categories $F/d$ correctly; I learned this from Bullejos and Cegarra's paper (pdf, see Thm 2)

My question is, is the union of 1 and 2 published somewhere? Which is to say, if I have a (lax/oplax) functor $F:C \to D$ between $2$-categories which is equivariant with respect to the action of some group $\Gamma$, and if its fiber 2-categories are $\Gamma$-equivariantly contractible, can I automatically claim that $F$ induces a $\Gamma$-equivariant homotopy equivalence between classifying spaces?

What I'm hoping to avoid is the scenario where the result is decipherable "only from the corresponding result for sheaves on $(\infty,1)$-topoi" or something similar. For reasons both psychological and mathematical, I'd like to restrict the categorical depth to $\leq 2$ if at all possible.

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  • $\begingroup$ You say you want to avoid $\infty$-categories, but as nobody has provided such an answer, perhaps it's worth mentioning that Quillen's Theorem A has a natural strengthening $\infty$-categorically: an $\infty$-functor satisfies the the hypotheses of Quillen's Theorem A if and only if it is cofinal. See HTT 4.1.3. One place the equivariant (over a finite group) version of this is carried out is (in in the more general setting of parametrized $\infty$-category theory) in Thm 6.7 of Jay Shah's thesis Parametrized Higher Category Theory. $\endgroup$
    – Tim Campion ♦
    Jan 9, 2021 at 5:34
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    $\begingroup$ For a more point-set-style approach to equivariant cofinality, see Theorem 2.25 of Dotto and Moi's Homotopy theory of $G$-diagrams and equivariant excision. $\endgroup$
    – Tim Campion ♦
    Jan 9, 2021 at 5:36
  • $\begingroup$ Did you ever find a reference for the 1-categorical version? I need it for a paper and I'm not sure my target audience would easily be able to see it follows from Tim's references which prove more general.things. $\endgroup$
    – Benjamin Steinberg
    Aug 16, 2022 at 13:14
  • $\begingroup$ @BenjaminSteinberg sorry, no. I managed to bypass the need for it after several failed attempts at googling $\endgroup$
    – Vidit Nanda
    Sep 6, 2022 at 20:27
  • $\begingroup$ @ViditNanda thanks $\endgroup$
    – Benjamin Steinberg
    Sep 6, 2022 at 20:57

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