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I think the title says it all. Quillen's Theorem A says that a functor $F\colon C\to D$ induces a homotopy equivalence of classifying spaces if each fiber category $F/d$ with $d$ an object of $D$ is contractible. Now Moerdijk showed that in some sense the classifying topos of a category is weakly equivalent to the classifying space of the category, so one would guess there is a topos theoretic interpretation/proof of the theorem.

Question: Is there a topos theoretic interpretation/proof of Quillen's Theorem A?

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    $\begingroup$ This is homotopy-theoretic rather than topos-theoretic, but there is an interpretation of Quillen's theorem A in terms of cofinality: a map of categories (or quasi-categories) has contractible fiber categories if and only if it is homotopy cofinal: i.e., computing a homotopy colimit along $D$ is the same as evaluating on along $C$. $\endgroup$
    – Akhil Mathew
    Commented Jun 3, 2012 at 18:24
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    $\begingroup$ If you believe this, then you can prove the theorem as follows: the homotopy of (the nerve of) $D$ is the homotopy colimit of $\ast$ indexed by $d \in D$. By cofinality, that is the same as the homotopy colimit of $\ast$ indexed by $c \in C$, which is the nerve of $C$. $\endgroup$
    – Akhil Mathew
    Commented Jun 3, 2012 at 18:33
  • $\begingroup$ @Akhil: Isn't this reasoning circular? How do you prove Cofinality without Quillen? $\endgroup$
    – Martin Brandenburg
    Commented Jun 3, 2012 at 18:40
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    $\begingroup$ @Martin: This can be proved directly using various explicitly models for homotopy colimits, see math.harvard.edu/~eriehl/266x/lectures.pdf. Alternatively one can do this $(\infty, 1)$-categorically: this is done in HTT for instance. The whole point is that (in Lurie's/Joyal's language) cofinality is equivalent to $F$ being an equivalence in the covariant model structure. This is something that can be checked on the homotopy fibers, which turn out to be precisely the nerves of the overcategories in question. $\endgroup$
    – Akhil Mathew
    Commented Jun 3, 2012 at 20:49
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    $\begingroup$ @Martin (contd): (This response is a bit dense, and I think the proof in HTT is a little more complicated than it needs to be. Feel free to email me if you'd like more details; they might also be in some of Joyal's writings.) $\endgroup$
    – Akhil Mathew
    Commented Jun 3, 2012 at 20:51

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I have no compelling answer to this question myself, but you may find relevant results and ideas in the work of Grothendieck, Maltsiniotis and Cisinski in homotopical algebra. Have you looked at Pursuing Stacks? In Maltsiniotis's Astérisque, there is a hint as to what a cohomological proof of Quillen's result would be. See http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf, page numbered 11 in the document. The text of Cisinski's Astérisque is available at http://www.math.univ-toulouse.fr/~dcisinsk/publications.html. It is called Les préfaisceaux comme modèles des types d'homotopie. He also has an old and never-published preprint —which may contain some typos— called Faisceaux localement asphériques, available at the same webpage. I hope this helps and is not off the point.

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  • $\begingroup$ Rick Jardine has an expository paper regarding Cisinski's work. It's freely available here: intlpress.com/HHA/v8/n1 $\endgroup$
    – Dan Ramras
    Commented Jun 4, 2012 at 16:17
  • $\begingroup$ I have seen Jardine's paper. I was still hoping someone knew of a connection via the work of Moerdijk in his lecture notes Classifying spaces and classifying toposes. $\endgroup$
    – Benjamin Steinberg
    Commented Jun 4, 2012 at 16:34
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Maybe Matias Del Hoyo's article http://arxiv.org/abs/0707.1718 helps you.

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  • $\begingroup$ I've seen this before. It's not what I'm looking for. $\endgroup$
    – Benjamin Steinberg
    Commented Jun 3, 2012 at 16:47

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