Quillen's theorems A and B explained

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

Bf: BC \to BD is not naturally the classifying space of a category: there is no natural category Ff such that FBf = BFf . Theorem B constructs Ff in a case when f is especially nice.

References

• Ara. Dimitri. Maltsiniotis. Georges. April 2018. Un théorème A de Quillen pour les ∞-catégories strictes I : La preuve simpliciale. Advances in Mathematics. 328. 446–500. 10.1016/j.aim.2018.01.018. 1703.04689.
  • • Book: Weibel, Charles. The K-book: an introduction to algebraic K-theory. AMS. 2013. 978-0-8218-9132-2. Graduate Studies in Math. 145.