The Drinfeld Centre of a Symmetric Fusion Category is 2-Fold Monoidal

TL;DR

This paper proves that the Drinfeld centre of a symmetric fusion category possesses a bilax 2-fold monoidal structure with compatible braiding and symmetry, constructed purely from the fusion category itself.

Contribution

It demonstrates that the Drinfeld centre of a symmetric fusion category is a bilax 2-fold monoidal category without relying on Tannaka duality, enabling broader applicability.

Findings

The Drinfeld centre has two compatible monoidal structures.

The convolution and symmetric tensor products are bilax monoidal functors.

Braiding and symmetry are compatible with the bilax structure.

Abstract

We show that the Drinfeld centre of a symmetric fusion category is a bilax 2-fold monoidal category. That is, it carries two monoidal structures, the convolution and symmetric tensor products, that are bilax monoidal functors with respect to each other. We additionally show that the braiding and symmetry for the convolution and symmetric tensor products are compatible with this bilax structure. We establish these properties without referring to Tannaka duality for the symmetric fusion category. This has the advantage that all constructions are done purely in terms of the fusion category structure, making the result easy to use in other contexts.