[Paper Review] Constructing symmetric monoidal bicategories functorially

Linde Wester Hansen, Michael Shulman|arXiv (Cornell University)|Oct 21, 2019

Homotopy and Cohomology in Algebraic Topology34 references32 citations

TL;DR

The paper provides a functorial method to lift monoidal, braided, and symmetric structures from monoidal double categories to their underlying bicategories, enabling systematic constructions of monoidal bicategories, functors, and transformations.

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.

Motivation & Objective

• Motivate the importance of symmetric monoidal bicategories across contexts and ease of coherence verification.
• Introduce a general lifting method that transfers monoidal structure from double categories to their underlying bicategories.
• Provide a functorial framework that preserves monoidal objects and morphisms, including functors and transformations.
• Show applicability through diverse examples including algebras with bimodules, categories with profunctors, and spans.
• Explain how companions/conjoints enable the lifting and ensure coherence in the resulting bicategories.

Proposed method

• Define and work with symmetric monoidal double categories and their lax/colax/strong monoidal functors.
• Introduce companions and conjoints to lift tight 1-cells to loose ones in the underlying bicategory.
• Prove that the underlying bicategory L(D) becomes a monoidal/braided/symmetric monoidal bicategory under suitable conditions.
• Develop a functor L from monoidal double categories to monoidal bicategories that preserves monoidal structures and morphisms.
• Establish that L extends to a functor between locally cubical bicategories, preserving monoidal objects and cells.
• Illustrate with multiple examples to demonstrate the construction in practice.

Experimental results

Research questions

• RQ1 How can monoidal, braided, and symmetric structures on double categories be transferred to the underlying bicategories functorially?
• RQ2 What lifting conditions and coherence data ensure that L(D) inherits a monoidal (or braided/symmetric) bicategory structure?
• RQ3 Can the lifting be extended to monoidal functors, transformations, and adjunctions in a way that preserves composition?
• RQ4 In which concrete settings (e.g., algebras and bimodules, profunctors, spans, open Markov processes, parametrized spectra) does the method yield new monoidal bicategories?
• RQ5 How do companions and conjoints facilitate the functorial lifting and maintain coherence?

Key findings

• A general theorem: if D is a monoidal double category with loosely strong companions, then L(D) is a monoidal bicategory (and braided/symmetric if D is so).
• The construction L is extended to a functor between the locally cubical bicategories of monoidal double categories and bicategories, preserving monoidal objects and morphisms.
• The framework yields not only monoidal bicategories but also monoidal functors, transformations, and composites in a coherent, functorial way.
• The method applies to a range of examples, including algebras and bimodules, categories and profunctors, and spans, as well as open Markov processes and parametrized spectra.
• Coherence and lifting are handled via companions/conjoints, which provide unique (up to isomorphism) liftings of tight arrows to loose arrows, enabling the monoidal structure to pass to L(D).
• The approach can accommodate lax, colax, or strong monoidal functors and transformations, with the output ensuring the appropriate level of strictness in the lifted structure.