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

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 Bruguières) 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 Böhm and Lack. These examples are related by a monoidal pseudofunctor $V\to \mathsf{Cat}$.

16 pages v2: the same results presented more pedagogically in 26 pages, final version to appear in TAC