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 $HH(\mathcal{B})$.