Constructing symmetric monoidal bicategories functorially

Abstract

We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise naturally in this way, and applying our general method is much easier than explicitly verifying the coherence laws of a monoidal bicategory for each example. Abstracting from earlier work in this direction, we express the construction as a functor between locally cubical bicategories that preserves monoid objects; this ensures that it also preserves monoidal functors, transformations, adjunctions, and so on. Examples include the monoidal bicategories of algebras and bimodules, categories and profunctors, sets and spans, open Markov processes, parametrized spectra, and various functors relating them.