Representations of fusion categories and their commutants

bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant 𝒞' of a fully faithful representation 𝒞→Bim(R) of a unitary fusion category 𝒞 . Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if 𝒞 and 𝒟 are Morita equivalent unitary fusion categories, then their commutant categories 𝒞' and 𝒟' are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: ( 𝒞 ≃ _Morita 𝒟 ) ⟹ ( 𝒞' ≃ _tensor 𝒟' ). This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional 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 C^* -category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and Bim(R) admit distinguished positive structures, and that fully faithful representations 𝒞→Bim(R) automatically respect these positive structures. This is the published version of arXiv:2004.08271 .