Representations of fusion categories and their commutants

A bicommutant category is a higher categorical analog of von Neumann algebra. We study the categories which arise as commutant $${\mathcal {C}}'$$ fully faithful representation {C}}\rightarrow \textrm{Bim}(R)$$ unitary fusion {C}}$$ . Using results Izumi, Popa, and Tomatsu about existence uniqueness representations (multi)fusion categories, we prove that if {D}}$$ are Morita equivalent then their {D}}'$$ categories. In particular, they tensor categories: $$\begin{aligned} \Big (\,\,{\mathcal {C}}\,\,\simeq _{\textrm{Morita}}\,\,{\mathcal {D}}\,\,\Big ) \qquad \Longrightarrow {C}}' \,\,\simeq _{\textrm{tensor}}\,\,{\mathcal {D}}'\,\,\Big ). \end{aligned}$$ This categorifies well-known result according to commutants (in some representations) finite dimensional $$\textrm{C}^*$$ -algebras isomorphic algebras, provided ‘big enough’. also introduce notion positivity for bi-involutive For dagger property (the being -category). But extra structure. show $$\textrm{Bim}(R)$$ admit distinguished positive structures, automatically respect these structures. published version arXiv:2004.08271