{
  "abstract": "A Grothendieck topology on a category determines both a subform of covering sieves and a quotient form obtained by identifying locally equivalent sieves. We place these two constructions in a single short exact sequence. For this purpose we introduce distributivity forms: indexed meet-semilattices equipped with distinguished indexed joins over which finite indexed meets distribute, and a strong version in which these joins also satisfy Beck--Chevalley. Over a fixed base, the resulting categories have all small limits and have kernels and cokernels relative to the closed ideal of fibrewise constant-top morphisms. Their monomorphisms are the fibrewise injective morphisms, and their relative cokernels, together with the top-reflecting morphisms, form an orthogonal factorization system. Cokernels compose but need not be stable under pullback, whereas kernels need not compose. We call a short exact sequence with prescribed middle term an extensional presentation, and prove that Grothendieck topologies on a category are precisely the extensional presentations of its maximally distributive form of sieves. Further applications recover universal productive closure operators and Lawvere--Tierney topologies, functorial non-Archimedean group topologies, and functorial linear topologies on commutative rings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14671",
  "author_total": 2,
  "authors": [
    {
      "name": "Roy Ferguson"
    },
    {
      "name": "Zurab Janelidze"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f43fmxhe88v5q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14671",
  "fulltext_url": null,
  "id": "f43fmxhe88v5q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14671",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.14671",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f43fmxhe88v5q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14671",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14671"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Grothendieck Topologies Are Extensional Presentations of the Form of Sieves",
  "title_text": "Grothendieck Topologies Are Extensional Presentations of the Form of Sieves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/f43fmxhe88v5q",
  "venue": "arXiv",
  "versions": []
}