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

↑ 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"

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