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.

In general, the homotopy fiber of B f : B C → B D is not naturally the classifying space of a category: there is no natural category F f such that F B f = B F f . Theorem B is a substitute for this problem.