{
  "abstract": "We introduce several flavours of semi-strict 4-category as enriched categories. This is done using the framework of enrichment over skew monoidal categories of Gray-categories and their associated multicategories, as developed in our previous work. We give the axioms for semi-strict 4-categories in elementary terms and describe a wide range of examples. In particular, we answer the question of what exactly Gray-categories and the higher transformations between them form: namely, a left-sharp 4-category G-Cat$^4$. In addition, we describe examples arising from braided monoidal 2-categories, monoidal Gray-categories (herein defined) and Gray-categories with structure. We use the concepts introduced here and the relationships between them to unify various concepts in the literature, such as Crans' 4-teisi. In the second part of the paper, we investigate homotopical aspects. In particular, we classify the equivalence n-cells in the left-sharp 4-category G-Cat$^4$ for each n. Using this, we show that the left-sharp skew monoidal structure on the category of Gray-categories induces a symmetric monoidal closed structure on the homotopy category of Gray-categories.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31448",
  "author_total": 2,
  "authors": [
    {
      "name": "John Bourke"
    },
    {
      "name": "Gabriele Lobbia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/642tsabetvcam.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31448",
  "fulltext_url": null,
  "id": "642tsabetvcam",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31448",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7223259699",
  "pdf_url": "https://arxiv.org/pdf/2610.31448",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/642tsabetvcam.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.31448",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31448"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Semi-strict 4-categories as enriched categories",
  "title_text": "Semi-strict 4-categories as enriched categories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/642tsabetvcam",
  "venue": "arXiv",
  "versions": []
}