{
  "abstract": "In this work we investigate a relative notion of protomodularity. We say that a finitely complete category $\\mathsf{C}$ satisfies the \\defn{$\\mathscr{M}$-version of protomodularity} when the change-of-base functors (concerning the fibration of points) are conservative with respect to a class of morphisms $\\mathscr{M}$ of $\\mathsf{C}$. If $\\mathscr{M}$ is the class of all morphisms or the class of all monomorphisms of $\\mathsf{C}$, then this notion is precisely that of an ``ordinary'' protomodular category. We analyse what conditions the class $\\mathscr{M}$ should have to obtain the relative $\\mathscr{M}$-versions of some well-known properties of protomodular categories, such as the Split Short Five Lemma (when $\\mathsf{C}$ is also pointed) and their relation with Mal'tsev categories. We also give diverse examples of categories satisfying the $\\mathscr{M}$-version of protomodularity for different choices of $\\mathscr{M}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09739",
  "author_total": 2,
  "authors": [
    {
      "name": "Maria Manuel Clementino"
    },
    {
      "name": "Diana Rodelo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4f28esxgbfjtt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09739",
  "fulltext_url": null,
  "id": "4f28esxgbfjtt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09739",
  "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": "W7220976861",
  "pdf_url": "https://arxiv.org/pdf/2610.09739",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4f28esxgbfjtt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09739",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09739"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "How protomodular is your favourite category?",
  "title_text": "How protomodular is your favourite category?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/4f28esxgbfjtt",
  "venue": "arXiv",
  "versions": []
}