Contents

category theory

topos theory

# Contents

## Definition

A cyclic set is a presheaf on the cyclic category (which is often called Connes' cyclic category though it is cocyclic, with the usual contravariant confusion), which is intermediate between a symmetric set and a simplicial set. With Set replaced by a general category one speaks of a cyclic object.

The concept of cyclic sets/objects is used in the description of the cyclic structure on Hochschild homology/Hochschild cohomology and hence for the discussion on cyclic homology/cyclic cohomology.

Just like the category of shapes for simplicial sets (the simplex category) may be identified with the full subcategory of Cat on the finite nonempty ordinals $[n]$; and like the shape category for symmetric sets (FinSet) may be identified with the full subcategory of Cat on their localizations $[n]^{-1}[n]$, so the cycle category $\Lambda$, is the full subcategory of Cat whose objects are the categories $[n]_\Lambda$ which are freely generated by the graph $0\to 1\to 2\to\ldots\to n\to 0$. If the overall composition $0\to 0$ is set equal to identity we obtain symmetric sets again.

We can also explain cyclic sets and more general cyclic objects in terms of standard generators.

A $\mathbf{Z}$-cyclic (synonym: paracyclic object) in a category $C$ is a simplicial object $F_\bullet$ in $C$ together with a sequence of isomorphisms $t_n : F_n \rightarrow F_n$, $n\geq 1$, such that

$\array{ \partial_i t_n = t_{n-1} \partial_{i-1},\,\, i \gt 0, & \sigma_i t_n = t_{n+1} \sigma_{i-1},\,\, i \gt0, \\ \partial_0 t_n = \partial_n, & \sigma_0 t_n = t_{n+1}^2 \sigma_n, }$

where $\partial_i$ are boundaries, $\sigma_i$ are degeneracies. A $\mathbf Z$-cocyclic (paracocyclic) object in $C$ is a $\mathbf{Z}$-cyclic object in $C^{\mathrm{op}}$. $\mathbf Z$-(co)cyclic object is (co)cyclic if, in addition, $t_n^{n+1} = 1$

## Properties

### As a classifying topos

The category of cyclic sets, being a presheaf category is a topos, and hence is the classifying topos for some geometric theory. This turns out to be the theory of abstract circles (Moerdijk 96). A further analysis can be found in (Caramello Wentzlaff 14). Accordingly there is an infinity-action of the circle group on the geometric realization of a cyclic set (see also Drinfeld 03).

### Model category structure and $S^1$-equivariant homotopy theory

There is a model category-structure on the category of cyclic sets, which makes it a presentation for $S^1$-equivariant homotopy theory (Spalinski 95, Blumberg 04).

## References

The definition is originally due to

• Alain Connes, Cohomologie cyclique et foncteurs $Ext^n$, C.R.A.S. 269 (1983), Série I, 953-958

Connections to simplicial sets are in:

The identification of the category of cyclic sets as the classifying topos for abstract circles is due to

The resulting circle-action on the (geometric realization of) cyclic sets is also discussed in

The homotopy theory of cyclic sets and its relation to $S^1$-equivariant homotopy theory is discussed in

• J. Spalinski, Strong homotopy theory of cyclic sets, J. of Pure and Appl. Alg. 99 (1995), 35–52.

• Andrew Blumberg, A discrete model of $S^1$-homotopy theory (arXiv:math/0411183)

An old query is archived in $n$Forum here.

There are fairly recent slides by Spalinski on the subject here, which also discuss relationships with dihedral sets? and quaternionic set?s, as studied by Loday.