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.
is a functor such that the classifying space
is contractible for any object d in D, then f induces a homotopy equivalence
is a functor that induces a homotopy equivalence
in D, 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
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