Relative centers and tensor products of tensor and braided fusion categories

In this paper we study the relative tensor product of module categories over braided fusion categories using, in part, the notion of the relative center of a bimodule category. In particular we investigate the canonical tensor category structure and braiding inherited by the relative tensor product when module categories are themselves tensor/braided. As a basic example we show that the category of representations of the fibre product of finite groups may be expressed in terms of the relative product and show how to multiply braided fusion categories arising from pre-metric groups. Also, we consider relative centers of braided fusion categories and look at the relative tensor product over Müger centers. We finish with an in-depth example hinting at a categorification of conjugacy classes for finite groups.