An Equivalence of Categories between 3-Crossed Modules and Gray 4-Groups
Abstract
In this paper, we investigate the relation between the category of 3-crossed modules and the category of Gray-type 4-groups. The notion of a 3-crossed module was first introduced by Arvasi \textit{et al.}, motivated by the question of what kind of algebraic structure completely encodes a homotopy 4-type. On the other hand, from the point of view that higher groups are equivalent to algebraic realizations of higher categories -- as exemplified by the relationship between 2-crossed modules and Gray 3-groups established by Sarikaya--Ulualan -- it had not been clear how the 3-crossed modules of Arvasi \textit{et al.} relate to any higher category. In our previous paper, we proposed a new definition of a 3-crossed module and observed that it admits a natural interpretation in terms of higher categories. In this paper, we make this interpretation precise: we introduce a 4-category, which reduces to a semistrict braided monoidal 2-category when restricted to a single object and a single 1-morphism, and prove that the category of our 3-crossed modules is equivalent to the category of Gray 4-groups, defined as single-object versions of this 4-category in which all morphisms are invertible. Furthermore, while Gray 4-groups provide a conceptual framework for 4-dimensional higher structures, they are often computationally intractable. Our equivalence establishes 3-crossed modules as a concrete, group-theoretic calculus for Gray 4-groups, providing a powerful computational tool for studying surface knots, state-sum invariants, and higher gauge theories.
Related
BibTeX
@misc{fukuda2026equivalence,
title = {{An Equivalence of Categories between 3-Crossed Modules and Gray 4-Groups}},
author = {Masaki Fukuda and Tommy Shu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.24034},
eprint = {2609.24034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24034},
}
RIS
TY - UNPB
TI - An Equivalence of Categories between 3-Crossed Modules and Gray 4-Groups
AU - Masaki Fukuda
AU - Tommy Shu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.24034
UR - https://arxiv.org/abs/2609.24034
AB - In this paper, we investigate the relation between the category of 3-crossed modules and the category of Gray-type 4-groups. The notion of a 3-crossed module was first introduced by Arvasi \textit{et al.}, motivated by the question of what kind of algebraic structure completely encodes a homotopy 4-type. On the other hand, from the point of view that higher groups are equivalent to algebraic realizations of higher categories -- as exemplified by the relationship between 2-crossed modules and Gray 3-groups established by Sarikaya--Ulualan -- it had not been clear how the 3-crossed modules of Arvasi \textit{et al.} relate to any higher category. In our previous paper, we proposed a new definition of a 3-crossed module and observed that it admits a natural interpretation in terms of higher categories. In this paper, we make this interpretation precise: we introduce a 4-category, which reduces to a semistrict braided monoidal 2-category when restricted to a single object and a single 1-morphism, and prove that the category of our 3-crossed modules is equivalent to the category of Gray 4-groups, defined as single-object versions of this 4-category in which all morphisms are invertible. Furthermore, while Gray 4-groups provide a conceptual framework for 4-dimensional higher structures, they are often computationally intractable. Our equivalence establishes 3-crossed modules as a concrete, group-theoretic calculus for Gray 4-groups, providing a powerful computational tool for studying surface knots, state-sum invariants, and higher gauge theories.
ER -