Cartesian Gray-Monoidal Double Categories

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 do...

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Bibliographic Details
Published in:arXiv.org 2023-07
Main Author: Morehouse, Edward
Format: Article
Language:English
Subjects:
Braiding
Cartesian coordinates
Categories
Graphical representations
Symmetry
Online Access:Get full text
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Summary: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.
ISSN:2331-8422