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(Hopf) Bimonoids in Duoidal Categories

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Hopf Algebras and Their Generalizations from a Category Theoretical Point of View

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2226))

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Abstract

This chapter deals with unspecified duoidal —so in particular braided monoidal –categories; at which level of generality there is no bijection between the morphisms in the category and all natural transformations between the induced functors. Those monads are identified which correspond to monoids; and those opmonoidal functors are identified which correspond to comonoids. This leads to an equivalence between bimonoids in a duoidal category and bimonads on it of a certain kind. A characterization of those bimonoids is given whose induced bimonad is a Hopf monad. Examples include Hopf monoids in braided monoidal categories—such as classical Hopf algebras and Hopf group algebras—small categories, Hopf algebroids over commutative (but not arbitrary) base algebras, weak Hopf algebras of Chap. 6, Hopf monads of Chap. 3 and certain coalgebra-enriched categories called Hopf categories.

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Authors and Affiliations

  1. Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary

    Gabriella Böhm

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  1. Gabriella Böhm
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Böhm, G. (2018). (Hopf) Bimonoids in Duoidal Categories. In: Hopf Algebras and Their Generalizations from a Category Theoretical Point of View. Lecture Notes in Mathematics, vol 2226. Springer, Cham. https://doi.org/10.1007/978-3-319-98137-6_7

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