On the category of weak bialgebras

Gabriella Böhm, José Gómez-Torrecillas, Esperanza López-Centella

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5 Citations (Scopus)

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 languageEnglish
Pages (from-to)801-844
Number of pages44
JournalJournal of Algebra
Volume399
DOIs
Publication statusPublished - 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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    Böhm, G., Gómez-Torrecillas, J., & López-Centella, E. (2014). On the category of weak bialgebras. Journal of Algebra, 399, 801-844. https://doi.org/10.1016/j.jalgebra.2013.09.032