Jump to content

Quillen's theorems A and B

From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by Cmalk (talk | contribs) at 20:11, 25 June 2020 (Corrected the direction of the map d \to d' in the statement of Theorem B). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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 is a functor such that the classifying space of the comma category is contractible for any object d in D, then f induces a homotopy equivalence .

Quillen's Theorem B — If is a functor that induces a homotopy equivalence for any morphism , then there is an induced long exact sequence:

In general, the homotopy fiber of is not naturally the classifying space of a category: there is no natural category such that . Theorem B constructs in a case when is especially nice.

References

  1. ^ Weibel 2013, Ch. IV. Theorem 3.7 and Theorem 3.8
  • Ara, Dimitri; Maltsiniotis, Georges (2017-03-14). "A Quillen's Theorem A for strict ∞-categories I: the simplicial proof". arXiv:1703.04689 [math.AT].
  • 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.