Henriques, A

Penneys, D

A 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≃MoritaD)⟹(C′≃tensorD′).
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

Representations of fusion categories and their commutants

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