Quillen's theorems A and B

Short description: Two theorems needed for Quillen's Q-construction in algebraic K-theory

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.

Quillen's Theorem A — If $f:C\to D$ is a functor such that the classifying space $B(d\downarrow f)$ of the comma category $d\downarrow f$ is contractible for any object d in D, then f induces a homotopy equivalence $BC\to BD$.

Quillen's Theorem B — If $f:C\to D$ is a functor that induces a homotopy equivalence $B(d^{\prime }\downarrow f)\to B(d\downarrow f)$ for any morphism $d\to d^{\prime }$ in D, then there is an induced long exact sequence:

$\cdots \to {\pi }_{i+1}BD\to {\pi }_{i}B(d\downarrow f)\to {\pi }_{i}BC\to {\pi }_{i}BD\to \cdots .$

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. doi:10.1016/j.aim.2018.01.018. 
• Quillen, Daniel (1973), "Higher algebraic K-theory. I", Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math, 341, Berlin, New York: Springer-Verlag, pp. 85–147, doi:10.1007/BFb0067053, ISBN 978-3-540-06434-3 
• Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN 978-0-8176-4736-0 
• Weibel, Charles (2013). The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. 145. AMS. ISBN 978-0-8218-9132-2. http://sites.math.rutgers.edu/~weibel/Kbook.html.