The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the ‘many-object’ setting of Hopf categories. To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories. Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.
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Acknowledgments. JV wants to thank Paolo Saracco for interesting and motivating discussions on the interaction between Hopf and Frobenius properties. He also thanks the FNRS (grant number F.4502.18 ) for support through the MIS grant “Antipode”. This work was initiated when both MB and CV were working as postdoctoral researchers at the Université Libre de Bruxelles within the framework of the ARC grant “Hopf algebras and the Symmetries of Non-commutative Spaces” funded by the “ Fédération Wallonie-Bruxelles ”. CV was supported by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI). All authors thank the referees for their careful reading and useful comments.
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