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''.

See also

* Cyclic homology
*Simplex category

References

*
*
*
*

External links

Cycle category
in nLab

Categories in category theory
Homology theory

{{categorytheory-stub