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Gray categories with duals and their diagrams

John W. Barrett, Catherine Meusburger, Gregor Schaumann

TL;DR

The paper develops a comprehensive diagrammatic calculus for Gray categories with duals, encoding objects, morphisms, and duality data in three-dimensional cube diagrams. It introduces two dualities, $*$ and $$, and a spatial condition that generalizes ribbon-compatibility, along with a strictification theorem that makes these duals exact symmetries in a spatially strict Gray category. A major contribution is linking geometry to higher-categorical structure through three- and two-dimensional diagrammatics, including a rigorous treatment of invariance under isotopies and diagram mappings, and providing strong motivation from extended TQFTs and defects. The results yield tools for computations in higher categories, with clear connections to topological quantum field theory and ribbon-category analogies, while establishing a pathway toward strict, computation-friendly higher-categorical frameworks.

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 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 potential applications in topological quantum field theory.

Gray categories with duals and their diagrams

TL;DR

The paper develops a comprehensive diagrammatic calculus for Gray categories with duals, encoding objects, morphisms, and duality data in three-dimensional cube diagrams. It introduces two dualities, and , and a spatial condition that generalizes ribbon-compatibility, along with a strictification theorem that makes these duals exact symmetries in a spatially strict Gray category. A major contribution is linking geometry to higher-categorical structure through three- and two-dimensional diagrammatics, including a rigorous treatment of invariance under isotopies and diagram mappings, and providing strong motivation from extended TQFTs and defects. The results yield tools for computations in higher categories, with clear connections to topological quantum field theory and ribbon-category analogies, while establishing a pathway toward strict, computation-friendly higher-categorical frameworks.

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 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 potential applications in topological quantum field theory.

Paper Structure

This paper contains 25 sections, 28 theorems, 133 equations, 57 figures.

Table of Contents

  1. Introduction
  2. Category, 2-category and Gray category diagrams
  3. Piecewise-linear topology
  4. Comparison with the smooth case
  5. Categories and diagrams
  6. 2-category diagrams
  7. Invariance of the evaluation of 2-category diagrams
  8. Gray categories
  9. Example: 2Cat and 2Cat(1)
  10. Gray category diagrams
  11. Gray categories with duals
  12. 2-categories with duals
  13. Diagrammatic representation of the 2-category duals
  14. Gray categories with duals
  15. Diagrammatic representation of the Gray category duals
  16. ...and 10 more sections

Key Result

Theorem 1

(Theorem th_progressive_invariant) Let $D$, $D'$ be generic Gray category diagrams that are isotopic by a one-parameter family of isomorphisms of progressive diagrams. Then the evaluations of $D$ and $D'$ are equal.

Figures (57)

  • Figure 1: Category diagram (a) and its evaluation (b).
  • Figure 2: a), b), c), d), e) mappings of diagrams, b), c), e) homomorphisms of diagrams, c) isomorphism of diagrams, d) subdiagram, e) subdivision.
  • Figure 3: A 2-Category diagram together with its projection onto the $y$-axis.
  • Figure 4: The interchange law: associated 2-category diagrams with associated projections.
  • Figure 5: Diagrams for pseudo-natural transformations and modifications. a) Pseudonatural transformation $\mu\colon F\Rightarrow G$ with component morphisms $\mu_a\colon x\circ F(a)\to G(a)\circ x$. b) Modification $\Phi\colon \nu\Rrightarrow \mu$ between pseudo-natural transformations with component morphisms $\mu_a\colon x\circ F(a)\to G(a)\circ x$, $\nu_a\colon y\circ F(a)\to G(a)\circ y$. c) Compatibility of pseudo-natural transformation with the monoidal product. d) Naturality property of pseudo-natural transformations. e) Defining property of modifications.
  • ...and 52 more figures

Theorems & Definitions (110)

  • Theorem
  • Theorem : Theorem \ref{['graycatduals']}, Lemma \ref{['lemma:doublestar-is-one']}, Theorem \ref{['graydualnat']}
  • Theorem : Theorems \ref{['thm:strictification']} and \ref{['lem:graystrict']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: StreetH
  • Definition 2.7
  • ...and 100 more