Finitely semisimple spherical categories and modular categories are self-dual

TL;DR

This paper proves that finitely semisimple spherical categories, including modular categories, are equivalent to their duals, by showing their universal coend is self-dual as a Weak Hopf Algebra.

Contribution

It establishes the self-duality of finitely semisimple spherical categories and their universal coends, extending to modular categories.

Findings

Finitely semisimple spherical categories are self-dual.

Universal coend of such categories is a self-dual Weak Hopf Algebra.

Includes modular categories as a special case.

Abstract

We show that every essentially small finitely semisimple k-linear additive spherical category in which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as a Weak Hopf Algebra.