[Paper Review] Supplying bells and whistles in symmetric monoidal categories

Brendan Fong, David I. Spivak|arXiv (Cornell University)|Aug 7, 2019

Homotopy and Cohomology in Algebraic Topology7 references11 citations

TL;DR

This paper formalizes the notion of a symmetric monoidal category 'supplying' an algebraic structure via a prop, such as comonoids or bimonoids, by defining compatibility with the monoidal product and unit. It proves that associators, unitors, and braiding are automatically homomorphisms for any supply, and that supplies extend naturally to strictifications, providing a foundational framework for structured string diagrams in categories like Rel.

ABSTRACT

It is common to encounter symmetric monoidal categories $\mathcal{C}$ for which every object is equipped with an algebraic structure, in a way that is compatible with the monoidal product and unit in $\mathcal{C}$. We define this formally and say that $\mathcal{C}$ supplies the algebraic structure. For example, the category $\mathsf{Rel}$ of relations between sets has monoidal structures given by both cartesian product and disjoint union, and with respect to either one it supplies comonoids. We prove several facts about the notion of supply, e.g. that the associators, unitors, and braiding of $\mathcal{C}$ are automatically homomorphisms for any supply, as are the coherence isomorphisms for any strong symmetric monoidal functor that preserve supplies. We also show that any supply of structure in a symmetric monoidal category can be extended to a supply of that structure on its strictification.

Motivation & Objective

• To formally define when a symmetric monoidal category supplies an algebraic structure encoded by a prop.
• To establish conditions under which strong monoidal functors preserve such supplies.
• To show that supplies in a category can be extended to its strictification.
• To unify and generalize categorical frameworks used in network theory, probability, and differentiation.
• To provide a systematic foundation for adding 'bells and whistles'—extra algebraic structures—to string diagram calculi.

Proposed method

• Introduces the concept of a 'supply' of a prop structure in a symmetric monoidal category, requiring compatible morphisms for each object.
• Defines 'homomorphic supply' and shows that it characterizes cartesian monoidal categories and biproducts.
• Uses string diagrams and commutative diagrams to express coherence conditions between monoidal structure and algebraic operations.
• Proves that coherence isomorphisms (associators, unitors, braiding) are automatically homomorphisms for any supply.
• Establishes that strong monoidal functors preserving supplies induce functors between subcategories of homomorphisms.
• Shows that any supply in a category induces a supply in its strictification via the canonical equivalence.

Experimental results

Research questions

• RQ1 When does a symmetric monoidal category naturally support a comonoid or bimonoid structure on every object in a monoidal-compatible way?
• RQ2 How can one formally define the compatibility of an algebraic structure (via a prop) with the monoidal product and unit in a category?
• RQ3 Under what conditions do strong monoidal functors preserve such supplied structures?
• RQ4 How do coherence isomorphisms (associators, unitors, braiding) interact with supplied structures?
• RQ5 Can a supply in a category be lifted to its strictification, and if so, how is this preserved?

Key findings

• The associators, unitors, and braiding in any symmetric monoidal category that supplies a prop structure are automatically homomorphisms for that structure.
• Any strong monoidal functor preserving a supply of a prop induces a well-defined functor between the subcategories of homomorphisms for that structure.
• A supply of a prop in a symmetric monoidal category induces a supply of the same structure in its strictification, and the strictification equivalence preserves the supply.
• The coherence maps (e.g., unitors and associators) are sent to homomorphisms under a supply-preserving functor, ensuring structural consistency.
• The condition that a category homomorphically supplies a comonoid is equivalent to it being cartesian monoidal, and supplying bimonoids implies the product is a biproduct.
• The natural isomorphism of strongators in a supply-preserving functor is equivalent to the commutativity of a key diagram involving the supply morphisms.