# Gray categories with duals and their diagrams

@article{Barrett2012GrayCW, title={Gray categories with duals and their diagrams}, author={John W. Barrett and Catherine Meusburger and Gregor Schaumann}, journal={arXiv: Quantum Algebra}, year={2012} }

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 Gray category functors with natural isomorphisms, and correspond directly to symmetries of the diagrams. It…

## Figures from this paper

figure 1 figure 2 figure 3 figure 4 figure 5 figure 6 figure 7 figure 8 figure 9 figure 10 figure 11 figure 12 figure 13 figure 14 figure 15 figure 16 figure 17 figure 18 figure 19 figure 20 figure 21 figure 22 figure 23 figure 24 figure 25 figure 26 figure 27 figure 28 figure 29 figure 30 figure 31 figure 32 figure 33 figure 34 figure 35 figure 36 figure 37 figure 38 figure 39 figure 40 figure 41 figure 42 figure 43 figure 44 figure 45 figure 46 figure 47 figure 48 figure 49 figure 50 figure 51 figure 52 figure 53 figure 54 figure 55 figure 56 figure 57 figure 58 figure 59 figure 60 figure 61 figure 62

## 41 Citations

Associative n-categories

- Mathematics
- 2018

We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces are locally modelled on manifold…

Topological state sum models in four dimensions, half-twists and their applications

- Mathematics
- 2017

Various mathematical tools are developed with the aim of application in mathematical physics.
In the first part, a new state sum model for four-manifolds is introduced which generalises the…

Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors

- Mathematics
- 2016

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal…

Evaluating TQFT invariants from G-crossed braided spherical fusion categories via Kirby diagrams with 3-handles

- Mathematics
- 2018

A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the…

On the Brauer Groups of Symmetries of Abelian Dijkgraaf–Witten Theories

- Mathematics
- 2015

Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer–Picard groups. We present a gauge…

Higher Categories and Topological Quantum Field Theories

- Mathematics
- 2019

Author(s): Cui, Xingshan | Advisor(s): Wang, Zhenghan | Abstract: We give a construction of Turaev-Viro type (3+1)-TQFT out of a G-crossed braided spherical fusion category for G a finite group. The…

Communications in Mathematical Physics Dichromatic State Sum Models for Four-Manifolds from Pivotal

- Mathematics
- 2017

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal…

A 3-categorical perspective on G-crossed braided categories

- Mathematics
- 2020

A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given…

Automated rewriting for higher categories and applications to quantum theory

- Computer Science
- 2016

A novel framework for rewriting in higher categories is proposed, and a result that in a quasistrict 4-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the butterfly equations is proved.

Pivotal Tricategories and a Categorification of Inner-Product Modules

- Mathematics
- 2014

This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using…

## References

SHOWING 1-10 OF 41 REFERENCES

Fusion categories and homotopy theory

- Mathematics
- 2009

We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to…

On fusion categories

- Mathematics
- 2002

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show…

Spherical Categories

- Mathematics
- 1993

This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the…

From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories

- Mathematics
- 2003

On unitary 2-representations of finite groups and topological quantum field theory

- Mathematics
- 2009

This thesis contains various results on unitary 2-representations of finite groups and their 2-characters, as well as on pivotal structures for fusion categories. The motivation is extended…

Extending combinatorial piecewise linear structures on stratified spaces. II

- Mathematics
- 1980

Let A1 be a stratified space and suppose that both the complement of the n-skeleton and the «-stratum have been endowed with combinatorial piecewise linear (PL) structures. In this paper we…

A tensor product for Gray-categories.

- Mathematics
- 1999

In this paper I extend Gray’s tensor product of 2-categories to a new tensor product of Gray-categories. I give a description in terms of generators and relations, one of the relations being an…

Higher-Dimensional Algebra V: 2-Groups

- Mathematics
- 2003

A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this…

Higher Dimensional Algebra: I. Braided Monoidal 2-Categories

- Mathematics
- 1995

Abstract We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of…