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 [math]\displaystyle{ f: C \to D }[/math] is a functor such that the classifying space [math]\displaystyle{ B(d \downarrow f) }[/math] of the comma category [math]\displaystyle{ d \downarrow f }[/math] is contractible for any object d in D, then f induces a homotopy equivalence [math]\displaystyle{ BC \to BD }[/math].

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

[math]\displaystyle{ \cdots \to \pi_{i+1} BD \to \pi_i B(d \downarrow f) \to \pi_i BC \to \pi_i BD \to \cdots. }[/math]

In general, the homotopy fiber of [math]\displaystyle{ Bf: BC \to BD }[/math] is not naturally the classifying space of a category: there is no natural category [math]\displaystyle{ Ff }[/math] such that [math]\displaystyle{ FBf = BFf }[/math]. Theorem B constructs [math]\displaystyle{ Ff }[/math] in a case when [math]\displaystyle{ f }[/math] 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.