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
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, $t_n^{n+1} = 1$
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).
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).
The definition is originally due to
Connections to simplicial sets are in:
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
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.
See also
Last revised on April 6, 2018 at 18:01:30. See the history of this page for a list of all contributions to it.