TITLE: Is there a topos theoretic interpretation/proof of Quillen's Theorem A?
QUESTION [17 upvotes]: 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?

REPLY [4 votes]: 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.