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 $f: C \to D$ 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 the homotopy equivalence $BC \to BD$.

Quillen's Theorem B — If $f: C \to D$ is a functor that induces the homotopy equivalence $B (d' \downarrow f) \to B(d \downarrow f)$ for any morphism $d' \to d$, then there is the 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 is a substitute for this problem.

References

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

• 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, MR 0338129
• 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, Zbl 1125.19300
• C. Weibel "The K-book: An introduction to algebraic K-theory"