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Quillen's Theorem A is formulated as follows:

Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category $F/y$ is contractible. Then $F$ induces a weak equivalence between the nerves $N(X)\to N(Y)$.

I am not a topologist, but it seems can prove the following statement, which I believe is much more powerful:

Relative Theorem A. Let $F:X\to Y$ and $G:Y\to Z$ be functors between small categories. Suppose for each $z\in Z$ the induced functor $(G\circ F)/z\to G/z$ induces a weak equivalence on the nerves. Then $F$ induces a weak equivalence on the nerves $N(X)\to N(Y)$.

When $G$ is the identity functor we recover the usual Theorem A. This seems to be so basic that it must be in textbooks. Is it well-known?

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  • $\begingroup$ Isn't this Grothendieck's version? See Théorème 2.1.13 in [Cisinski, 2003, Le localisateur fondemental minimal]. $\endgroup$
    – Zhen Lin
    Mar 31, 2016 at 22:30
  • $\begingroup$ @ZhenLin Sorry for duplicating your comment, I'm a slow typer. $\endgroup$
    – Karol Szumiło
    Mar 31, 2016 at 22:38

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The condition that appears in the assumption of what you call "Relative Theorem A" was introduced by Grothendieck in Pursuing Stacks. It is a part of the definition of a basic localizer, i.e. a class of functors between small categories that behaves like the class of weak homotopy equivalences.

Grothendieck posed a conjecture that weak homotopy equivalences form the smallest basic localizer which was eventually proven by Cisinski here. In particular, Théorème 2.1.13 shows (already a well-known) fact that weak homotopy equivalences indeed form a basic localizer, i.e. that the "Relative Theorem A" holds.

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  • $\begingroup$ Thank you very much for the reference, in fact Remarque 1.1.4 in Cisinski's paper is precisely my point. His proof however relies on the homotopy colimits of Bousfield-Kan. I want to do the opposite: first to prove relative theorem A independently from homotopy colimits, then use it to construct homotopy colimits. Do you know of an independent proof? $\endgroup$
    – Anton Mellit
    Apr 1, 2016 at 11:46
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    $\begingroup$ At the moment I can't think of a significantly different argument. What would you count as an argument avoiding homotopy colimits? For example, do you consider Quillen's proof of Theorem A independent of homotopy colimits? It uses diagonals of bisimplicial sets which are homotopy colimits and Bousfield--Kan construction reduces general homotopy colimits to this special case. All these things are intimately related to each other so it may not be possible to separate them completely. $\endgroup$
    – Karol Szumiło
    Apr 4, 2016 at 11:29

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