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Quillen's theorems A and B

## Summary

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

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

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

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

## References

1. ^ Weibel 2013, Ch. IV. Theorem 3.7 and Theorem 3.8
• 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. arXiv:1703.04689. 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, vol. 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
• Weibel, Charles (2013). The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. Vol. 145. AMS. ISBN 978-0-8218-9132-2.