Oplax Hopf Algebras

TL;DR

This paper introduces oplax Hopf monoids in braided monoidal bicategories, generalizing classical Hopf monoids, and explores their relation to Hopf V-categories and Frobenius V-categories within this framework.

Contribution

It defines oplax Hopf monoids in bicategories, connects them to Hopf V-categories, and introduces Frobenius V-categories as Frobenius objects in the same setting.

Findings

Hopf V-categories are a special case of oplax Hopf monoids

Frobenius V-categories are characterized as Frobenius objects in the bicategory

Generalization of Hopf monoids to bicategorical context

Abstract

We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [B\"oh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.