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.[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 the homotopy equivalence .

Quillen's Theorem B  If is a functor that induces the homotopy equivalence for any morphism , then there is the 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 is a substitute for this problem.

References

  1. Weibel 2013, Ch. IV. Theorem 3.7 and Theorem 3.8


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