# 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

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

Created on March 19, 2009 00:53:43 by Zoran Škoda (195.37.209.180)