{
  "abstract": "In this paper, we construct the $(\\infty,\\infty)$-category $\\mathsf{Span}_\\infty(\\mathcal{C})$ of spans, also known as correspondences, in any given $(\\infty,1)$-category $\\mathcal{C}$ with finite limits. This yields new models for the span $(\\infty,n)$-categories for $n \\in \\mathbb{N} \\cup \\{\\infty\\}$. We characterize the mapping $(\\infty, n-1)$-categories in these $(\\infty,n)$-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span $(\\infty,n)$-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where \"higher\" refers to both higher categorical and \"higher monoidal\" structures, i.e., $\\mathsf{E}_k$-algebras in $(\\infty, n)$-categories for $n, k>1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29495",
  "author_total": 3,
  "authors": [
    {
      "name": "Jonte Gödicke"
    },
    {
      "name": "Quoc P. Ho"
    },
    {
      "name": "Walker H. Stern"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33dt7fbsmmc0b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29495",
  "fulltext_url": null,
  "id": "33dt7fbsmmc0b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29495",
  "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.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
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  "openalex": "W7204892024",
  "pdf_url": "https://arxiv.org/pdf/2608.29495",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33dt7fbsmmc0b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29495",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29495"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The $(\\infty,\\infty)$-category of spans",
  "title_text": "The (∞,∞)-category of spans",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/33dt7fbsmmc0b",
  "venue": "arXiv",
  "versions": []
}