#
nLab

cyclic object

Contents
# 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 $\Lambda$ 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.

## References

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

Last revised on June 27, 2021 at 01:27:41.
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