A 3-categorical perspective on G-crossed braided categories

Abstract

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 by elements of a group $G$ correspond to $G$-crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of $3$-categories $\mathcal{C}$ equipped with a 3-functor $\mathrm{B}G\to \mathcal{C}$ which is essentially surjective on objects and $1$-morphisms is equivalent to the $2$-category of $G$-crossed braided categories. This provides a uniform approach to various constructions of $G$-crossed braided categories.

arXiv comment66 pages, many figures

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Cited by in corpus (10)

• Higher-group symmetry in finite gauge theory and stabilizer codes
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Symmetry Transmutation and Anomaly Matching
• Fusion 3-Categories for Duality Defects
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Soft symmetries of topological orders
• Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
• Fermionic defects of topological phases and logical gates
• Disclinations, dislocations, and emanant flux at Dirac criticality
• A Graded Schur Lemma and a graded-monoidal structure for induced modules over graded-commutative algebras