A skew-duoidal Eckmann-Hilton argument and quantum categories

TL;DR

This paper establishes a general connection between skew monoidal structures and monads, applying it to quantum categories and bialgebroids, revealing that quantum categories can be viewed as monads within a specific bicategory framework.

Contribution

It introduces a skew-duoidal Eckmann-Hilton argument that clarifies the relationship between skew monoidal structures and monads, especially in the context of quantum categories.

Findings

Quantum categories are monads in a particular bicategory M.

Quantum categories can be characterized as skew monoidal structures on objects in M.

The paper generalizes the relationship between monoidal structures and monads in bicategories.

Abstract

A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories were originally defined as monoidal comonads on endomorphism objects in a particular monoidal bicategory M. Then they were shown also to be skew monoidal structures (with an appropriate unit) on objects in M. Now we see in what kind of M quantum categories are merely monads.