### 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.

Original language | English |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 99-123 |

Number of pages | 25 |

DOIs | |

Publication status | Published - Jan 1 2018 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2226 |

ISSN (Print) | 0075-8434 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Lecture Notes in Mathematics*(pp. 99-123). (Lecture Notes in Mathematics; Vol. 2226). Springer Verlag. https://doi.org/10.1007/978-3-319-98137-6_7

**(Hopf) bimonoids in duoidal categories.** / Böhm, G.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Lecture Notes in Mathematics.*Lecture Notes in Mathematics, vol. 2226, Springer Verlag, pp. 99-123. https://doi.org/10.1007/978-3-319-98137-6_7

}

TY - CHAP

T1 - (Hopf) bimonoids in duoidal categories

AU - Böhm, G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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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PB - Springer Verlag

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