{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24034",
  "author_total": 2,
  "authors": [
    {
      "name": "Masaki Fukuda"
    },
    {
      "name": "Tommy Shu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/azqma29sm5ga4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24034",
  "fulltext_url": null,
  "id": "azqma29sm5ga4",
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  "landing_url": "https://arxiv.org/abs/2609.24034",
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  "licence_label": "arXiv licence",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
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      "arxiv",
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      "2610|Homotopy and Cohomology in Algebraic Topology"
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  "openalex": "W7214051456",
  "pdf_url": "https://arxiv.org/pdf/2609.24034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/azqma29sm5ga4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:45Z",
      "native_id": "2609.24034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24034"
    }
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "An Equivalence of Categories between 3-Crossed Modules and Gray 4-Groups",
  "title_text": "An Equivalence of Categories between 3-Crossed Modules and Gray 4-Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/azqma29sm5ga4",
  "venue": "arXiv",
  "versions": []
}