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).

The cyclic category Λ has one object Λ_n for each natural number n = 0, 1, 2, ...

The morphisms from Λ_m to Λ_n 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 Λ_m to Λ_n 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.

The number of morphisms from Λ_m to Λ_n 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.

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.

• Cyclic homology
• Simplex category

• 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

• Cycle category in nLab