Some remarks on free products of rigid $C^*$-2-categories

Abstract

In this informal expository note, we present a universal, formulaic construction of the free product of rigid $C^*$-2-categories. This construction allows for a straightforward, purely categorical formulation of the free composition of subfactors and planar algebras considered by Bisch and Jones. As an application, we explain the results of Tarrago and Wahl on free wreath products of compact quantum groups in this categorical language.