# Relative version of Quillen's theorem A

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?

• Isn't this Grothendieck's version? See Théorème 2.1.13 in [Cisinski, 2003, Le localisateur fondemental minimal]. – Zhen Lin Mar 31 '16 at 22:30
• @ZhenLin Sorry for duplicating your comment, I'm a slow typer. – Karol Szumiło Mar 31 '16 at 22:38