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) = n+f(x), where two functions f and g represent the same morphism when their difference is divisible by n.
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.
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.
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.
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- Connes, Alain (2002), "Noncommutative Geometry Year 2000", in Fokas, A., Highlights of mathematical physics, pp. 49–110, arXiv:math/0011193, ISBN 0-8218-3223-9, retrieved 15 May 2011
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