Gray categories with duals and their diagrams

Barrett, John W.; Meusburger, Catherine; Schaumann, Gregor

Abstract

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray category data. These can be viewed as a generalisation of ribbon diagrams. The Gray categories present two types of duals, which are extended to functors of 2-strict tricategories with natural isomorphisms, and correspond directly to symmetries of the diagrams. It is shown that these functors can be strictified so that the symmetries of a cube are realised exactly. A new condition on Gray categories with duals called the spatial condition is defined. A class of diagrams for which the evaluation for spatial Gray categories is invariant under homeomorphisms is exhibited. This relation between the geometry of the diagrams and structures in the Gray categories proves useful in computations and has applications in topological quantum field theory.

Citation

Barrett, J. W., Meusburger, C., & Schaumann, G. (2024). Gray categories with duals and their diagrams. Advances in Mathematics, 450, Article 109740. https://doi.org/10.1016/j.aim.2024.109740

• Journal Article Type: Article
• Acceptance Date: May 13, 2024
• Online Publication Date: Jun 5, 2024
• Publication Date: 2024-07
• Deposit Date: Jun 18, 2024
• Publicly Available Date: Jun 19, 2024
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Electronic ISSN: 1090-2082
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 450
• Article Number: 109740
• DOI: https://doi.org/10.1016/j.aim.2024.109740
• Keywords: Gray categories, Gray category duals, Diagrammatic calculus
• Public URL: https://nottingham-repository.worktribe.com/output/36291391
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S000187082400255X?via%3Dihub
• Additional Information: This article is maintained by: Elsevier; Article Title: Gray categories with duals and their diagrams; Journal Title: Advances in Mathematics; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.aim.2024.109740; Content Type: article; Copyright: © 2024 The Author(s). Published by Elsevier Inc.