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

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?

-
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 $C$. – Akhil Mathew Jun 3 '12 at 18:24
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 $C$. – Akhil Mathew Jun 3 '12 at 18:33
@Akhil: Isn't this reasoning circular? How do you prove Cofinality without Quillen? – Martin Brandenburg Jun 3 '12 at 18:40
@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. – Akhil Mathew Jun 3 '12 at 20:49
@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.) – Akhil Mathew Jun 3 '12 at 20:51