Cartesian Gray-Monoidal Double Categories

TL;DR

This paper develops a cartesian structure for symmetric Gray-monoidal double categories using locally cubical Gray categories, introducing new concepts and graphical calculus to manage complex algebraic structures in higher-dimensional category theory.

Contribution

It introduces locally cubical Gray categories and extends Gray-monoidal double categories with cartesian structure, providing a graphical calculus for complex higher-dimensional categorical algebra.

Findings

Defined locally cubical Gray categories as 3D categorical structures.

Extended Gray-monoidal double categories with cartesian structure using doubly-lax functors.

Presented a graphical calculus for representing complex algebraic structures.

Abstract

In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally globular, Gray categories. The motivating example comprises double categories themselves, together with their functors, transformations, and modifications. A one-object locally cubical Gray category is a Gray-monoidal double category. Braiding, syllepsis, and symmetry for these is introduced in a manner analogous to that for 2-categories. Adding cartesian structure requires the introduction of doubly-lax functors of double categories to manage the order of copies. The resulting theory is algebraically rather complex, largely due to the bureaucracy of linearizing higher-dimensional boundary constraints. Fortunately, it has a relatively simple and compelling representation in the graphical calculus of surface diagrams, which we present.