• Source: Cyclic category

In mathematics, the cyclic category or cycle category or category of cycles is a category of finite cyclically ordered sets and degree-1 maps between them. It was introduced by Connes (1983).




Definition


The cyclic category Λ has one object Λn for each natural number n = 0, 1, 2, ...
The morphisms from Λm to Λn are represented by increasing functions f from the integers to the integers, such that f(x+m+1) = f(x)+n+1, where two functions f and g represent the same morphism when their difference is divisible by n+1.
Informally, the morphisms from Λm to Λn can be thought of as maps of (oriented)
necklaces with m+1 and n+1 beads. More precisely, the morphisms can be identified with homotopy classes of degree 1 increasing maps from S1 to itself that map the subgroup Z/(m+1)Z to Z/(n+1)Z.




Properties


The number of morphisms from Λm to Λn is (m+n+1)!/m!n!.
The cyclic category is self dual.
The classifying space BΛ of the cyclic category is a classifying space BS1of the circle group S1.




Cyclic sets


A cyclic set is a contravariant functor from the cyclic category to sets. More generally a cyclic object in a category C is a contravariant functor from the cyclic category to C.




See also


Cyclic homology
Simplex category




References


Connes, Alain (1983), "Cohomologie cyclique et foncteurs Extn" (PDF), Comptes Rendus de l'Académie des Sciences, Série I (in French), 296 (23): 953–958, MR 0777584, archived from the original (PDF) on 4 March 2016, retrieved 15 May 2011
Connes, Alain (2002), "Noncommutative Geometry Year 2000" (PDF), in Fokas, A. (ed.), Highlights of mathematical physics, pp. 49–110, arXiv:math/0011193, Bibcode:2000math.....11193C, ISBN 0-8218-3223-9, retrieved 15 May 2011
Kostrikin, A. I.; Shafarevich, I. R. (1994), Algebra V: Homological algebra, Encyclopaedia of Mathematical Sciences, vol. 38, Springer, pp. 60–61, ISBN 3-540-53373-7
Loday, Jean-Louis (1992), Cyclic homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Berlin, New York: Springer-Verlag, ISBN 978-3-540-53339-9, MR 1217970




External links


Cycle category in nLab

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