A Reduced Tensor Product of Braided Fusion Categories containing a Symmetric Fusion Category

In this talk I will construct a reduced tensor product of braided fusion categories containing a symmetric fusion category $\mathcal{A}$. This tensor product takes into account the relative braiding with respect to objects of $\mathcal{A}$ in these braided fusion categories. The resulting category is again a braided fusion category containing $\mathcal{A}$. This tensor product is inspired by the tensor product of $G$-equivariant once-extended three-dimensional quantum field theories, for a finite group $G$.
To define this reduced tensor product, we equip the Drinfeld centre $\mathcal{Z}(\mathcal{A})$ of the symmetric fusion category $\mathcal{A}$ with an unusual tensor product, making $\mathcal{Z}(\mathcal{A})$ into a 2-fold monoidal category. Using this 2-fold structure, we introduce a new type of category enriched over the Drinfeld centre to capture the braiding behaviour with respect to $\mathcal{A}$ in the braided fusion categories, and use this encoding to define the reduced tensor product.