Quillen's theorems A and B

The precise statements of the theorems are as follows.

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

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

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

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