{
  "abstract": "We show a universal property of the span $\\infty$-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double $\\infty$-categories to $\\infty$-categories, and show that this functor admits a right adjoint defined by the double $\\infty$-categories of squares. As an application, we obtain a model for the localization of an $\\infty$-category in terms of a span $\\infty$-category and also give another proof of the model given by the relative nerve. Moreover, using the span-squares adjunction, we obtain new proofs of the equivalences between different models of algebraic $K$-theory, given by the $\\Q$-, the $\\oS$-, the cobordism model, and the squares construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.10052",
  "author_total": 2,
  "authors": [
    {
      "name": "George Raptis"
    },
    {
      "name": "Wolfgang Steimle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/21y7aq9pb05b7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.10052",
  "fulltext_url": null,
  "id": "21y7aq9pb05b7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.10052",
  "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.KT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7164150641",
  "pdf_url": "https://arxiv.org/pdf/2606.10052",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/21y7aq9pb05b7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2606.10052",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.10052"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The span-squares adjunction",
  "title_text": "The span-squares adjunction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/21y7aq9pb05b7",
  "venue": "arXiv",
  "versions": []
}