nLab cyclic set



Category Theory

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topos theory



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Cohomology and homotopy

In higher category theory




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 Z\mathbf{Z}-cyclic object (synonym: paracyclic object)in a category CC is a simplicial object F F_\bullet in CC together with a sequence of isomorphisms t n:F nF nt_n : F_n \rightarrow F_n, n1n\geq 1, such that

it n=t n1 i1,i>0, σ it n=t n+1σ i1,i>0, 0t n= n, σ 0t n=t n+1 2σ n,\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 i\partial_i are boundaries, σ i\sigma_i are degeneracies. A Z\mathbf Z-cocyclic (paracocyclic) object in CC is a Z\mathbf{Z}-cyclic object in C opC^{\mathrm{op}}. Z\mathbf Z-(co)cyclic object is (co)cyclic if, in addition, t n n+1=1t_n^{n+1} = 1


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 1S^1-equivariant homotopy theory

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


The original definition:

Textbook account (in the generality of cyclic spaces):


  • Cary Malkiewich, A visual introduction to cyclic sets and cyclotomic spectra, 2015 (pdf)

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 1S^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 1S^1-homotopy theory (arXiv:math/0411183)

An old query is archived in nnForum 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.

Last revised on May 28, 2023 at 08:59:06. See the history of this page for a list of all contributions to it.