{
  "abstract": "For a crossed module $χ: G \\to H$, we introduce the notion of $χ$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $χ= id: G \\to G$ with the conjugation action recovers $G$-crossed braided categories; taking $χ: G \\to \\{*\\}$ for abelian $G$ yields $G$-graded braided categories; taking $χ: \\{*\\} \\to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $χ$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $χ$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $χ$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $χ$-crossed ribbon or admit an orthogonal $G$-decomposition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07626",
  "author_total": 3,
  "authors": [
    {
      "name": "Azat M. Gainutdinov"
    },
    {
      "name": "Ingo Runkel"
    },
    {
      "name": "Bangxin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0k5jkkp0wbw1d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07626",
  "fulltext_url": null,
  "id": "0k5jkkp0wbw1d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07626",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7212010522",
  "pdf_url": "https://arxiv.org/pdf/2609.07626",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0k5jkkp0wbw1d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07626",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07626"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Crossed-module crossed braided categories",
  "title_text": "Crossed-module crossed braided categories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/0k5jkkp0wbw1d",
  "venue": "arXiv",
  "versions": []
}