# Quillen's theorems A and B

## Two theorems needed for Quillen's Q-construction in algebraic K-theory / From Wikipedia, the free encyclopedia

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 $d\to d'$, 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.