nLab cyclic object

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

Equivalently, this is a simplicial object $F_\bullet$ together with a sequence of isomorphisms $t_n : F_n \rightarrow F_n$, $n\geq 1$, such that

$$\array{ \partial_i t_n = t_{n-1} \partial_{i-1},\,\, i \gt 0, & \sigma_i t_n = t_{n+1} \sigma_{i-1},\,\, i \gt0, \\ \partial_0 t_n = \partial_n, & \sigma_0 t_n = t_{n+1}^2 \sigma_n,\\ t^n_{n+1} = \mathrm{id} }$$

where $\partial_i$ are boundaries, $\sigma_i$ are degeneracies.

Related concepts

• cyclic space

• cyclic set

• cyclotomic space, cyclotomic spectrum

References

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