Hopf-cyclic coefficients in the braided setting

Considering the monoidal category \mathcal{C} obtained as modules over a Hopf algebra H in a rigid braided category \mathcal{B} , we prove decomposition results for the Hochschild and cyclic homology categories HH(\mathcal{C}) and HC(\mathcal{C}) of \mathcal{C} . This is accomplished by defining a notion of a (stable) anti-Yetter–Drinfeld module with coefficients in a (stable) braided module over \mathcal{B} . When the stable braided module is HH(\mathcal{B}) , we recover HH(\mathcal{C}) and HC(\mathcal{C}) . The decomposition of HC(\mathcal{C}) now follows from that of HH(\mathcal{B}) .