nLab
paracyclic object

A paracyclic (synonym: Z\mathbf{Z}-cyclic) object in a category CC is a simplicial object F F_\bullet together with a sequence of isomorphisms t n:F nF nt_n : F_n \rightarrow F_n, n1n\geq 1, 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,\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 i\partial_i are boundaries, σ i\sigma_i are degeneracies. If t n n+1=id:F nF nt_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)