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Keywords:
Dagger 2-categories; pseudonatural transformations; unitarity; Morita theory
Dagger 2-categories; pseudonatural transformations; unitarity; Morita theory
Summary:
We suggest two approaches to a definition of unitarity for pseudonatural transformations between unitary pseudofunctors on pivotal dagger 2-categories. The first is to require that the 2-morphism components of the transformation be unitary. The second is to require that the dagger of the transformation be equal to its inverse. We show that the 'inverse' making these definitions equivalent is the right dual of the transformation in the 2-category Fun$(C,D)$ of pseudofunctors $C \rightarrow D$, pseudonatural transformations, and modifications. We show that the subcategory $\rm{Fun}_u(C,D) \subset \rm{Fun}(C,D)$ whose objects are unitary pseudofunctors and whose 1-morphisms are unitary pseudonatural transformations is a pivotal dagger 2-category. We apply these results to obtain a Morita-theoretical classification of unitary pseudonatural transformations between fibre functors on the category of representations of a compact quantum group.
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