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.
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, ISBN978-3-540-06434-3, MR 0338129
Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN978-0-8176-4736-0, Zbl 1125.19300
Weibel, Charles (2013). The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. Vol. 145. AMS. ISBN978-0-8218-9132-2.
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quillen, theorems, topology, branch, mathematics, quillen, theorem, gives, sufficient, condition, classifying, spaces, categories, homotopy, equivalent, quillen, theorem, gives, sufficient, condition, square, consisting, classifying, spaces, categories, homoto. 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 D displaystyle f C to D is a functor such that the classifying space B d f displaystyle B d downarrow f of the comma category d f displaystyle d downarrow f is contractible for any object d in D then f induces a homotopy equivalence B C B D displaystyle BC to BD Quillen s Theorem B If f C D displaystyle f C to D is a functor that induces a homotopy equivalence B d f B d f displaystyle B d downarrow f to B d downarrow f for any morphism d d displaystyle d to d then there is an induced long exact sequence p i 1 B D p i B d f p i B C p i B D 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 B f B C B D displaystyle Bf BC to BD is not naturally the classifying space of a category there is no natural category F f displaystyle Ff such that F B f B F f displaystyle FBf BFf Theorem B constructs F f displaystyle Ff in a case when f displaystyle f is especially nice References Edit Weibel 2013 Ch IV Theorem 3 7 and Theorem 3 8 Ara Dimitri Maltsiniotis Georges April 2018 Un theoreme A de Quillen pour les categories 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 AlgebraicK theory Modern Birkhauser Classics Paperback reprint of the 1996 2nd ed Boston MA Birkhauser 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 This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Quillen 27s theorems A and B amp oldid 1069838598, wikipedia, wiki, book, books, library,
