category theory

topos theory

# Contents

## Definition

A cyclic set is a presheaf on a particular category $\Lambda$ defined by Alain Connes (which is usually called Connes' cyclic category though it is cocyclic, with the usual contravariant confusion), which is intermediate between a symmetric set and a simplicial set.

The category of shapes for simplicial sets (the simplex category) can be identified with the full subcategory of $Cat$ on the finite nonempty ordinals $[n]$. Likewise, the shape category for symmetric sets (FinSet) can be identified with the full subcategory of $Cat$ on their localizations $[n]^{-1}[n]$. The category of shapes, $\Lambda$, is the full subcategory of $\mathrm{Cat}$ 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 objects in terms of standard generators.

A $\mathbf{Z}$-cyclic (synonym: paracyclic object) object in 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). 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.

Revised on April 8, 2014 19:54:19 by Tim Porter (2.26.27.237)