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 denote the cycle category of Alain Connes. A cyclic object in a category is a -valued presheaf on . Equivalently it is a simplicial object together with a sequence of isomorphisms , , such that
where are boundaries, are degeneracies.
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