[Paper Review] String Diagrams For Double Categories and Equipments

David Jaz Myers|arXiv (Cornell University)|Dec 8, 2016

Homotopy and Cohomology in Algebraic Topology9 references12 citations

TL;DR

This paper introduces string diagram calculus for double categories and equipments, establishing foundational results on companions and conjoints via graphical reasoning. It proves that companions are unique up to unique isomorphism commuting with bends, and shows that companions, conjoints, and adjunctions are interdefinable via zig-zag identities and kink identities in the diagrammatic calculus.

ABSTRACT

A popular graphical calculus for monoidal categories makes computations tactile and intuitive. Complicated diagram chases can be expressed in a few pictures and discovered by playing with a shoelace. Joyal and Street's proof of the soundness of this calculus says that any deformation of a diagram, any bending of the strings, describes the same morphism. In this paper, we extend the graphical calculus to double categories and proarrow equipments in order to bring the string diagrammatic method to formal category theory. Our main theorem proves this calculus sound with the help of Dawson and Pare's results on composition in double categories.

Motivation & Objective

• To develop a coherent string diagram calculus for double categories and equipments.
• To formalize the relationship between companions, conjoints, and adjunctions in proarrow 2-categories.
• To prove uniqueness of companions up to isomorphism that commute with bends.
• To establish the equivalence between companion/conjoint structure and left adjunction in the proarrow category.
• To provide a diagrammatic proof of zig-zag identities using kink and bending operations.

Proposed method

• Uses string diagrams to represent 1-morphisms, 2-cells, and their compositions in double categories.
• Applies bending operations to define units and counits from adjunctions.
• Employs kink identities to verify that isomorphisms between companions commute with bends.
• Applies zig-zag (triangle) identities by pulling strings straight in the diagrammatic calculus.
• Uses the kink identities of the adjunction to verify that the constructed cells satisfy the required axioms.
• Demonstrates that companion structure arises naturally from left adjunctions via diagrammatic duality.

Experimental results

Research questions

• RQ1 How can string diagrams be used to reason about companions and conjoints in double categories?
• RQ2 What conditions ensure that two companions of a given morphism are isomorphic in a way that respects their bends?
• RQ3 To what extent are companions, conjoints, and left adjunctions interdefinable in the proarrow 2-category?
• RQ4 How do kink identities and zig-zag identities interact in the diagrammatic proof of adjunction properties?
• RQ5 What is the role of isomorphism in ensuring uniqueness of companions up to coherent structure?

Key findings

• The companion of a morphism is unique up to a unique isomorphism that commutes with its bends.
• An isomorphism between two companions commutes with their bends if and only if it satisfies the kink identities.
• If a morphism has a companion and a conjoint, then it is a left adjoint in the proarrow 2-category.
• Conversely, if a morphism is a left adjoint, then its right adjoint realizes the conjoint and companion structure.
• The zig-zag identities for adjunctions follow directly from pulling strings straight in the string diagram calculus.
• The kink identities of the adjunction ensure that the isomorphism between companions commutes with the bends.