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 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
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
such that
. Theorem B constructs
in a case when
is especially nice.