{
  "abstract": "We build on the recent development that toposes with enough points can be represented as categories of points equipped with ultraconvergence data (a.k.a. virtual ultracategories), characterizing usual classes of geometric morphisms --- surjections, embeddings, hyperconnected maps, and localic maps --- in terms of their points. In particular, the characterisation of geometric surjections relies on a notion of \"virtual ultraretractions\", which generalises usual retractions and filtered colimits, categorifies Johnstone's notion of subclosure for topological spaces, and provides a new characterisation of separating classes of points of a topos.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31445",
  "author_total": 2,
  "authors": [
    {
      "name": "Saadia Gabriel"
    },
    {
      "name": "Yuksel Errol"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2jj02xb4bqjr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31445",
  "fulltext_url": null,
  "id": "2jj02xb4bqjr9",
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  "landing_url": "https://arxiv.org/abs/2609.31445",
  "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"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7214670964",
  "pdf_url": "https://arxiv.org/pdf/2609.31445",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2jj02xb4bqjr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31445",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31445"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Geometric morphisms of virtual ultracategories",
  "title_text": "Geometric morphisms of virtual ultracategories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/2jj02xb4bqjr9",
  "venue": "arXiv",
  "versions": []
}