Could not include topos theory - contents
A cyclic set is a presheaf on a particular category $\Lambda$ 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 $Cat$ on the finite nonempty ordinals $[n]$. Likewise, the shape category for symmetric sets (FinSet) can be identified with the full subcategory of $Cat$ on their localizations $[n]^{-1}[n]$. The category of shapes, $\Lambda$, is the full subcategory of $\mathrm{Cat}$ 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 objects in terms of standard generators.
A $\mathbf{Z}$-cyclic (synonym: paracyclic object) object in 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.