Cartesian Gray-Monoidal Double Categories
Edward Morehouse

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…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
