Skew monoidales, skew warpings and quantum categories

TL;DR

This paper introduces skew monoidales in monoidal bicategories, linking them to quantum categories and groupoids, and explores their properties and connections with opmonoidal monads using lax warping techniques.

Contribution

It generalizes skew-monoidal categories to monoidal bicategories and characterizes quantum categories and groupoids as skew monoidales, expanding the theoretical framework.

Findings

Quantum categories correspond to skew monoidales in comodules.

Quantum groupoids are skew monoidales with invertible associativity.

Connections established between opmonoidal monads and skew monoidales.

Abstract

Kornel Szlach\'anyi recently used the term skew-monoidal category for a particular laxified version of monoidal category. He showed that bialgebroids $H$ with base ring $R$ could be characterized in terms of skew-monoidal structures on the category of one-sided $R$-modules for which the lax unit was $R$ itself. We define skew monoidales (or skew pseudo-monoids) in any monoidal bicategory $\mathscr M$. These are skew-monoidal categories when $\mathscr M$ is $\mathrm{Cat}$. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories in the sense of Day-Street with base comonoid $C$ in a suitably complete braided monoidal category $\mathscr V$ are precisely skew monoidales in $\mathrm{Comod} (\mathscr V)$ with unit coming from the counit of $C$. Quantum groupoids are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping defined recently by Booker-Street to modify monoidal structures.