nLab
cyclic set

A cyclic set is a presheaf on a particular category $\Lambda$ defined by 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 shape category for simplicial sets (the simplex category) can be identified with the full subcategory of $\mathrm{Cat}$ on the finite nonempty ordinals $[n]$. Likewise, the shape category for symmetric sets (FinSet) can be identified with the full subcategory of $\mathrm{Cat}$ on their localizations $[n{]}^{-1}[n]$. The shape category $\Lambda$ is the full subcategory of $\mathrm{Cat}$ whose objects are the categories $[n{]}_{\Lambda }$ which are freely generated by the graph $0\to 1\to 2\to \dots \to n\to 0$. If the overall composition $0\to 0$ is set equal to identity we obtain symmetric sets again.

Mike: I copied (and attempted to clarify) the above from symmetric set, but I don’t think I believe it. If you invert the composite $0\to 1\to 2$ in $[2]$, then the objects $0$ and $2$ become isomorphic and are both a retract of $1$. This localization has exactly one nondegenerate, nonidentity self-map, which exchanges $0$ and $2$. But shouldn’t the object ”$2$” in $\Lambda$ have a $\mathbb{Z}/3$ worth of self-maps?

Zoran Škoda: Thanks, Mike, I corrected the cyclic part, the symmetric was OK before. But even $[0]$ has an object with infinity worth of self-maps. If the new map $n\to 0$ is taken into account, then all $n+1$ objects of cyclic $[n+1{]}_{\Lambda }$ will be on the same footing: from point $k$ one has identity, going forward one step, 2 steps, 3 steps, and so on, and one is allowed to cross the boundary $k+n-k$, doing more than $n-k$ steps, even $n-1$ step coming all through to your predecessor $k-1$.

We can also explain cyclic sets and more general objects in terms of standard generators.

A $Z$-cyclic (synonym: paracyclic object) object in category $C$ is a simplicial object $F_{•}$ in $C$ together with a sequence of isomorphisms $t_n:F_n\to F_n$, $n\ge 1$, such that

$$\begin{array}{cc}{\partial }_i{t}_n=t_{n-1}{\partial }_{i-1},i>0,& {\sigma }_i{t}_n=t_{n+1}{\sigma }_{i-1},i>0,\\ {\partial }_0{t}_n={\partial }_n,& {\sigma }_0{t}_n=t_{n+1}^2{\sigma }_n,\end{array}$$\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 ${\partial }_i$ are boundaries, ${\sigma }_i$ are degeneracies. A $Z$-cocyclic (paracocyclic) object in $C$ is a $Z$-cyclic object in $C^{\mathrm{op}}$. $Z$-(co)cyclic object is (co)cyclic if, in addition, $t_n^{n+1}=1$