nLab paracyclic object

Definition

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

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

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.

Related concepts

• cyclic object