nLab
cyclic object

A cyclic object in a category C is a simplicial object F together with a sequence of isomorphisms t n:F nF n, n1, such that

it n=t n1 i1,i>0, σ it n=t n+1σ i1,i>0, 0t n= n, σ 0t n=t n+1 2σ n, t n+1 n=id\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 i are boundaries, σ i are degeneracies. Equivalently, it is a C-presheaf on the Connes’ category of cycles Λ.

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