nLab
cyclic object

A 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

$$\begin{array}{cc}{\partial }_i{t}_n=t_{n-1}{\partial }_{i-1},i>0,& {\sigma }_i{t}_n=t_{n+1}{\sigma }_{i-1},i>0,\\ {\partial }_0{t}_n={\partial }_n,& {\sigma }_0{t}_n=t_{n+1}^2{\sigma }_n,\\ t_{n+1}^n=\mathrm{id}\end{array}$$\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,\\ t^n_{n+1} = \mathrm{id} }

where ${\partial }_i$ are boundaries, ${\sigma }_i$ are degeneracies. Equivalently, it is a $C$-presheaf on the Connes’ category of cycles $\Lambda$.