A locally rigid semi-Hopf V-category is Hopf and Frobenius if and only if it has non-singular left and right integral families, equivalently if its integral spaces are the monoidal unit.
Oplax Hopf Algebras
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abstract
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [B\"oh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.
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2019 1verdicts
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A Larson-Sweedler Theorem for Hopf V-Categories
A locally rigid semi-Hopf V-category is Hopf and Frobenius if and only if it has non-singular left and right integral families, equivalently if its integral spaces are the monoidal unit.