nLab
cyclic object

Let Λ\Lambda denote the cycle category of Alain Connes. A cyclic object in a category CC is a CC-valued presheaf on Λ\Lambda. Equivalently it 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, 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\partial_i are boundaries, σ i\sigma_i are degeneracies.

Revised on July 8, 2014 04:35:55 by Adeel Khan (132.252.63.38)