Relative centers and tensor products of tensor and braided fusion categories

TL;DR

This paper explores the structure and properties of relative tensor products and centers of module categories over braided fusion categories, providing new insights into their tensor and braiding structures with applications to group representations.

Contribution

It introduces the notion of the relative center of a module category and studies the tensor and braiding structures of relative tensor products in braided fusion categories.

Findings

Expressed the category of representations of fiber products of finite groups via relative tensor products.

Showed how to multiply braided fusion categories from pre-metric groups.

Analyzed relative centers and tensor products over Muger centers.

Abstract

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 module 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 Muger centers. We finish with an in-depth example hinting at a categorification of conjugacy classes for finite groups.