The monoidal center construction and bimodules

AbstractLet C be a cocomplete monoidal category such that the tensor product in C preserves colimits in each argument. Let A be an algebra in C. We show (under some assumptions including “faithful flatness” of A) that the center of the monoidal category (ACA,⊗A) of A–A-bimodules is equivalent to the center of C (hence in a sense trivial): Z(ACA)≅Z(C). Assuming A to be a commutative algebra in the center Z(C), we compute the center Z(CA) of the category of right A-modules (considered as a subcategory of ACA using the structure of A∈Z(C). We find Z(CA)≅dysZ(C)A, the category of dyslectic right A-modules in the braided category Z(C)