# 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

$\begin{array}{cc}{\partial }_{i}{t}_{n}={t}_{n-1}{\partial }_{i-1},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i>0,& {\sigma }_{i}{t}_{n}={t}_{n+1}{\sigma }_{i-1},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i>0,\\ {\partial }_{0}{t}_{n}={\partial }_{n},& {\sigma }_{0}{t}_{n}={t}_{n+1}^{2}{\sigma }_{n},\end{array}$\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.

Revised on March 19, 2009 00:37:20 by Toby Bartels (71.104.234.95)