cyclic object

A **cyclic object** in a category $C$ is a simplicial object ${F}_{\u2022}$ together with a sequence of isomorphisms ${t}_{n}:{F}_{n}\to {F}_{n}$, $n\ge 1$, such that

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

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)