### Abstract

Weak (Hopf) bialgebras are described as (Hopf) bimonoids in appropriate duoidal (also known as 2-monoidal) categories. This interpretation is used to define a category wba of weak bialgebras over a given field. As an application, the "free vector space" functor from the category of small categories with finitely many objects to wba is shown to possess a right adjoint, given by taking (certain) group-like elements. This adjunction is proven to restrict to the full subcategories of groupoids and of weak Hopf algebras, respectively. As a corollary, we obtain equivalences between the category of small categories with finitely many objects and the category of pointed cosemisimple weak bialgebras; and between the category of small groupoids with finitely many objects and the category of pointed cosemisimple weak Hopf algebras.

Original language | English |
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Pages (from-to) | 801-844 |

Number of pages | 44 |

Journal | Journal of Algebra |

Volume | 399 |

DOIs | |

Publication status | Published - Feb 1 2014 |

### Keywords

- Duoidal category
- Groupoid
- Hopf monoid
- Weak Hopf algebra
- Weak bialgebra

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Algebra*,

*399*, 801-844. https://doi.org/10.1016/j.jalgebra.2013.09.032