On the category of weak bialgebras

TL;DR

This paper explores the structure of weak bialgebras using duoidal categories, establishing new adjunctions and equivalences with categories of small categories and groupoids, and characterizing their algebraic properties.

Contribution

It introduces a categorical framework for weak bialgebras, defines a new category wba, and proves adjunctions and equivalences with categories of small categories and groupoids.

Findings

Existence of a right adjoint to the free vector space functor for weak bialgebras.

Equivalences between small categories/groupoids and pointed cosemisimple weak bialgebras/Hopf algebras.

Characterization of weak bialgebras as bimonoids in duoidal categories.

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.