nLab
cyclic object

Contents

Idea

The concept of cyclic object is the generalization of that of cyclic sets where Set may be replaced with any other category.

Cyclic objects are used in the description of the cyclic structure on Hochschild homology/Hochschild cohomology and hence for the discussion of cyclic homology/cyclic cohomology.

Definition

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.

Last revised on April 6, 2018 at 18:02:15. See the history of this page for a list of all contributions to it.