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 . 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. Quillen's Theorem A — If is a functor such that the classifying space of the comma category is contractible for any object d in D, then f induces a homotopy equivalence . Quillen's Theorem B — If is a functor that induces a homotopy equivalence for any morphism , then there is an induced long exact sequence: In general, the homotopy fiber of is not naturally the classifying space of a category: there is no natural category such that . Theorem B constructs in a case when is especially nice.