Representations of fusion categories and their commutants

A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $mathcal{C}$ of a fully faithful representation $mathcal{C}tooperatorname{Bim}(R)$ of a unitary fusion category $mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $mathcal{C}$ and $mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $mathcal{C}$ and $mathcal{D}$ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: [ Big(,,mathcal{C},,simeq_{text{Morita}},,mathcal{D},,Big) qquadLongrightarrowqquad Big(,,mathcal{C},,simeq_{text{tensor}},,mathcal{D},,Big). ] This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional $rm C^*$-algebras are isomorphic von Neumann algebras, provided the representations are `big enough. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a $rm C^*$-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and $operatorname{Bim}(R)$ admit distinguished positive structures, and that fully faithful representations $mathcal{C}tooperatorname{Bim}(R)$ automatically respect these positive structures.