nLab
cyclic object

Let $\Lambda$ denote the cycle category of Alain Connes. A cyclic object in a category $C$ is a $C$-valued presheaf on $\Lambda$. Equivalently it is a simplicial object $F_\bullet$ together with a sequence of isomorphisms $t_n : F_n \rightarrow F_n$, $n\geq 1$, such that

$$\array{ \partial_i t_n = t_{n-1} \partial_{i-1},\,\, i \gt 0, & \sigma_i t_n = t_{n+1} \sigma_{i-1},\,\, i \gt0, \\ \partial_0 t_n = \partial_n, & \sigma_0 t_n = t_{n+1}^2 \sigma_n,\\ t^n_{n+1} = \mathrm{id} }$$

where $\partial_i$ are boundaries, $\sigma_i$ are degeneracies.