### Abstract

This chapter continues the survey of the theoretical background, turning to more specific topics. Monoidal categories are introduced and monads on them are studied. The general theory of lifting (explained in Chap. 2) is applied to the functors and natural transformations constituting the monoidal structure of the base category. A bijection is proven between the liftings of the monoidal structure of the base category to the Eilenberg-Moore category of a monad; and opmonoidal structures on the monad. An opmonoidal monad is termed a bimonad. If the base category is closed monoidal, then a sufficient and necessary condition is obtained for the lifting of the closed structure as well, in the form of the invertibility of a canonical natural transformation. A Hopf monad is defined as a bimonad for which this natural transformation is invertible.

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

Publisher | Springer Verlag |

Pages | 29-46 |

Number of pages | 18 |

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. 29-46). (Lecture Notes in Mathematics; Vol. 2226). Springer Verlag. https://doi.org/10.1007/978-3-319-98137-6_3

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

*Lecture Notes in Mathematics.*Lecture Notes in Mathematics, vol. 2226, Springer Verlag, pp. 29-46. https://doi.org/10.1007/978-3-319-98137-6_3

}

TY - CHAP

T1 - (Hopf) bimonads

AU - Böhm, G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - This chapter continues the survey of the theoretical background, turning to more specific topics. Monoidal categories are introduced and monads on them are studied. The general theory of lifting (explained in Chap. 2) is applied to the functors and natural transformations constituting the monoidal structure of the base category. A bijection is proven between the liftings of the monoidal structure of the base category to the Eilenberg-Moore category of a monad; and opmonoidal structures on the monad. An opmonoidal monad is termed a bimonad. If the base category is closed monoidal, then a sufficient and necessary condition is obtained for the lifting of the closed structure as well, in the form of the invertibility of a canonical natural transformation. A Hopf monad is defined as a bimonad for which this natural transformation is invertible.

AB - This chapter continues the survey of the theoretical background, turning to more specific topics. Monoidal categories are introduced and monads on them are studied. The general theory of lifting (explained in Chap. 2) is applied to the functors and natural transformations constituting the monoidal structure of the base category. A bijection is proven between the liftings of the monoidal structure of the base category to the Eilenberg-Moore category of a monad; and opmonoidal structures on the monad. An opmonoidal monad is termed a bimonad. If the base category is closed monoidal, then a sufficient and necessary condition is obtained for the lifting of the closed structure as well, in the form of the invertibility of a canonical natural transformation. A Hopf monad is defined as a bimonad for which this natural transformation is invertible.

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U2 - 10.1007/978-3-319-98137-6_3

DO - 10.1007/978-3-319-98137-6_3

M3 - Chapter

AN - SCOPUS:85056285422

T3 - Lecture Notes in Mathematics

SP - 29

EP - 46

BT - Lecture Notes in Mathematics

PB - Springer Verlag

ER -