nLab cyclic set

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.

Cyclic sets and more generally cyclic objects can be described in terms of standard generators:

A $\mathbf{Z}$-cyclic object (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).

Related concepts

• cyclic category

• cyclic cohomology

• cyclic space, cyclotomic spectrum

• skew-simplicial set, symmetric set, simplicial set

References

The original definition:

• Alain Connes, Cohomologie cyclique et foncteurs $Ext^n$, C.R.A.S. 296 (1983), Série I, 953-958 MR777584 Zbl0534.18009 (pdf, pdf).

• Pierre Cartier, Section 1.6 of: Homologie cyclique : rapport sur des travaux récents de Connes, Karoubi, Loday, Quillen…, Séminaire Bourbaki: volume 1983/84, exposés 615-632, Astérisque, no. 121-122 (1985), Exposé no. 621 (numdam:SB_1983-1984__26__123_0)

Textbook account (in the generality of cyclic spaces):

• Jean-Louis Loday, Cyclic Spaces and $S^1$-Equivariant Homology (doi:10.1007/978-3-662-21739-9_7)

  Chapter 7 in: Cyclic Homology, Grundlehren 301, Springer 1992 (doi:10.1007/978-3-662-21739-9)

• Jean-Louis Loday, Section 3 of: Free loop space and homology, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics 24, EMS 2015 (arXiv:1110.0405, ISBN:978-3-03719-153-8)

Exposition:

• Cary Malkiewich: A visual introduction to cyclic sets and cyclotomic spectra, talk at Young Topologists Meeting Lausanne, Switzerland (2015) [pdf, pdf]

Discussion of the relation to simplicial sets:

• Alain Connes, Caterina Consani, Cyclic structures and the topos of simplicial sets, 1309.0394

• Julia E. Bergner, Walker H. Stern, Cyclic Segal Spaces, arXiv:2409.11945 (2024). (abstract)

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

• Ieke Moerdijk, Cyclic sets as a classifying topos, 1996 (pdf)

• Olivia Caramello, Nicholas Wentzlaff, Cyclic theories, 2014 (arXiv:1406.5479)

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

• Vladimir Drinfeld, On the notion of geometric realization (arXiv:0304064)

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 sets, as studied by Loday.