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). 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:
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.
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