{
  "abstract": "We show that under mild assumptions a functor is a prefibration if and only if it has terminal objects in its fibers and admits precartesian liftings of monomorphisms. We use this to show that the following conditions on a pointed category $\\mathbb{C}$ are equivalent: (a) The category of morphisms of $\\mathbb{C}$ admits generic split extensions; (b) Each functor category of $\\mathbb{C}$ with finite domain category admits generic split extensions; (c) $\\mathbb{C}$ admits normalizers and generic split extensions; (d) The kernel functor from the category split extensions in $\\mathbb{C}$ to $\\mathbb{C}$ is a prefibration. In addition we show that the category of morphisms of a pointed protomodular $\\mathbb{C}$ admits generic split extensions if and only if for each morphism $f:X\\to Z$ the functor sending each object $B$ to the isomorphism class of split extensions in $\\mathbb{C}^{\\mathbf{2}}$ of $(B,B,1_B)$ with kernel $(X,Z,f)$, is representable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35122",
  "author_total": 1,
  "authors": [
    {
      "name": "James Richard Andrew Gray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5s434e7yqaapa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35122",
  "fulltext_url": null,
  "id": "5s434e7yqaapa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35122",
  "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"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7214700458",
  "pdf_url": "https://arxiv.org/pdf/2609.35122",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5s434e7yqaapa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35122",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35122"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "More on action representability and normalizers",
  "title_text": "More on action representability and normalizers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5s434e7yqaapa",
  "venue": "arXiv",
  "versions": []
}