Quillen's theorems A and B

In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen.

The precise statements of the theorems are as follows.[1]

Quillen's Theorem A — If $f:C\to D$ is a functor such that the classifying space $B(d\downarrow f)$ of the comma category $d\downarrow f$ is contractible for any object d in D, then f induces a homotopy equivalence $BC\to BD$ .

Quillen's Theorem B — If $f:C\to D$ is a functor that induces a homotopy equivalence $B(d'\downarrow f)\to B(d\downarrow f)$ for any morphism $d\to d'$ in D, then there is an induced long exact sequence:

\cdots \to \pi _{i+1}BD\to \pi _{i}B(d\downarrow f)\to \pi _{i}BC\to \pi _{i}BD\to \cdots .

In general, the homotopy fiber of $Bf:BC\to BD$ is not naturally the classifying space of a category: there is no natural category $Ff$ such that $FBf=BFf$ . Theorem B constructs $Ff$ in a case when $f$ is especially nice.