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 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.

Properties

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.

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.