A Skew-Duoidal Eckmann-Hilton Argument and Quantum Categories

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.