Hopf polyads, Hopf categories and Hopf group monoids viewed as Hopf monads

TL;DR

This paper develops a functorial framework linking Hopf polyads, Hopf categories, and Hopf group monoids as Hopf monads within a monoidal bicategory setting, unifying various structures in a categorical context.

Contribution

It introduces a functorial construction associating a monoidal bicategory to any monoidal bicategory, connecting different Hopf structures as Hopf monads in this framework.

Findings

Hopf polyads are Hopf monads in Span|Cat

Hopf group monoids are Hopf monads in Span|V

Hopf categories are Hopf monads in Span|V

Abstract

We associate, in a functorial way, a monoidal bicategory $\mathsf{Span}| \mathcal V$ to any monoidal bicategory $\mathcal V$. Two examples of this construction are of particular interest: Hopf polyads (due to Brugui\`eres) can be seen as Hopf monads in $\mathsf{Span}| \mathsf{Cat}$ while Hopf group monoids in a braided monoidal category $V$ (in the spirit of Turaev and Zunino), and Hopf categories over $V$ (by Batista, Caenepeel and Vercruysse) both turn out to be Hopf monads in $\mathsf{Span}| V$. Hopf group monoids and Hopf categories are Hopf monads on a distinguished type of monoidales fitting the framework studied recently by B\"ohm and Lack. These examples are related by a monoidal pseudofunctor $V\to \mathsf{Cat}$.