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 .
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 "S"1 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 "B"Λ of the cyclic category is a classifying space "BS"1of the circle group "S"1.
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".