nLab cyclic object




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.


Let Λ\Lambda denote the cyclic category of Alain Connes. A cyclic object in a category CC is a CC-valued presheaf on Λ\Lambda. (Dually, a cocyclic object is a copresheaf on the cyclic category.)

Equivalently, this 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.


See the references at cyclic category and at cyclic set and cyclic space.

Last revised on June 27, 2021 at 05:27:41. See the history of this page for a list of all contributions to it.