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Adjointable monoidal functors and quantum groupoids
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Every monoidal functor G: C --> M has a canonical factorization through the category of bimodules over some monoid R in M such that the factor U: C -->_R M_R is strongly unital. Using this result and the characterization of the forgetful functors M_A -->_R M_R of bialgebroids A over R given by Schauenburg together with their bimonad description given by the author recently here we characterize the "long" forgetful functors M_A -->_R M_R --> M of both bialgebroids and weak bialgebras.
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Classification of symmetric fusion categories over $\mathbb{R}$
Every symmetric fusion category over R is equivalent to the semi-linear super representation category of a Z2-graded finite super group.
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