### Abstract

Based on the novel notion of ‘weakly counital fusion morphism’, regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields (Böhm et al. Trans. Amer. Math. Soc. 367, 8681–8721, 2015) and multiplier bimonoids in braided monoidal categories (Böhm and Lack, J. Algebra 423, 853–889, 2015). Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

Original language | English |
---|---|

Pages (from-to) | 47-111 |

Number of pages | 65 |

Journal | Applied Categorical Structures |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1 2018 |

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### Keywords

- Braided monoidal category
- Multiplier bialgebra
- Weak bialgebra

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Applied Categorical Structures*,

*26*(1), 47-111. https://doi.org/10.1007/s10485-017-9481-3

**Weak Multiplier Bimonoids.** / Böhm, G.; Gómez-Torrecillas, José; Lack, Stephen.

Research output: Contribution to journal › Article

*Applied Categorical Structures*, vol. 26, no. 1, pp. 47-111. https://doi.org/10.1007/s10485-017-9481-3

}

TY - JOUR

T1 - Weak Multiplier Bimonoids

AU - Böhm, G.

AU - Gómez-Torrecillas, José

AU - Lack, Stephen

PY - 2018/2/1

Y1 - 2018/2/1

N2 - Based on the novel notion of ‘weakly counital fusion morphism’, regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields (Böhm et al. Trans. Amer. Math. Soc. 367, 8681–8721, 2015) and multiplier bimonoids in braided monoidal categories (Böhm and Lack, J. Algebra 423, 853–889, 2015). Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

AB - Based on the novel notion of ‘weakly counital fusion morphism’, regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields (Böhm et al. Trans. Amer. Math. Soc. 367, 8681–8721, 2015) and multiplier bimonoids in braided monoidal categories (Böhm and Lack, J. Algebra 423, 853–889, 2015). Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

KW - Braided monoidal category

KW - Multiplier bialgebra

KW - Weak bialgebra

UR - http://www.scopus.com/inward/record.url?scp=85018670689&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018670689&partnerID=8YFLogxK

U2 - 10.1007/s10485-017-9481-3

DO - 10.1007/s10485-017-9481-3

M3 - Article

AN - SCOPUS:85018670689

VL - 26

SP - 47

EP - 111

JO - Applied Categorical Structures

JF - Applied Categorical Structures

SN - 0927-2852

IS - 1

ER -