nLab
paracyclic object

A paracyclic (synonym: $Z$-cyclic) object in a category $C$ is a simplicial object $F_{•}$ 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. If $t_n^{n+1}=\mathrm{id}:F_n\to F_n$ then the paracyclic object is cyclic.

For example, a paracyclic object in Set is a paracyclic set.