Hopf modules for autonomous pseudomonoids and the monoidal centre

TL;DR

This paper extends Hopf algebra theory to autonomous pseudomonoids in monoidal bicategories, establishing a structure theorem for Hopf modules and analyzing the monoidal centre and Drinfel'd double in this context.

Contribution

It introduces a Hopf module construction for autonomous pseudomonoids and proves an equivalence with left autonomy, advancing the understanding of centres and doubles in higher category theory.

Findings

The Hopf module construction exists if and only if the pseudomonoid is left autonomous.

The lax centre of a left autonomous pseudomonoid can be characterized as an Eilenberg-Moore construction.

The Drinfel'd double of a coquasi-Hopf algebra is realized as the centre in this framework.

Abstract

In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel'd double. If $A$ is a map pseudomonoid in a monoidal bicategory \M, the analogue of the category of Hopf modules for $A$ is an Eilenberg-Moore construction for a certain monad in $\mathbf{Hom}(\M^{\mathrm{op}},\mathbf{Cat})$. We study the existence of the internalisation of this notion, called the Hopf module construction, by extending the completion under Eilenberg-Moore objects of a 2-category to a endo-homomorphism of tricategories on $\mathbf{Bicat}$. Our main result is the equivalence between the existence of a left dualization for $A$ ({\em i.e.}, $A$ is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case the Hopf module construction for $A$ always exists. We use these results to study the lax centre of a left autonomous map pseudomonoid. We show that the lax centre is the Eilenberg-Moore construction for a certain monad on $A$ (one existing if the other does). If $A$ is also right autonomous, then the lax centre equals the centre. We look at the examples of the bicategories of \V-modules and of comodules in \V, and obtain the Drinfel'd double of a coquasi-Hopf algebra $H$ as the centre of $H$.