Could not include topos theory - contents
A cyclic set is a presheaf on a particular category 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 on the finite nonempty ordinals . Likewise, the shape category for symmetric sets (FinSet) can be identified with the full subcategory of on their localizations . The category of shapes, , is the full subcategory of whose objects are the categories which are freely generated by the graph . If the overall composition 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 -cyclic (synonym: paracyclic object) object in category is a simplicial object in together with a sequence of isomorphisms , , such that
where are boundaries, are degeneracies. A -cocyclic (paracocyclic) object in is a -cyclic object in . -(co)cyclic object is (co)cyclic if, in addition,
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).
The definition is originally due to
Connections to simplicial sets are in:
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
J. Spalinski, Strong homotopy theory of cyclic sets, J. of Pure and Appl. Alg. 99 (1995), 35–52.
An old query is archived in Forum here.