{
  "abstract": "We introduce the notion of a unitary disk-like $n$-category, which is a disk-like $n$-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like $n$-category is defined to be the local field data of a fully extended $(n+1)$D unitary TQFT, we propose a complete finite unitary disk-like $n$-category as the definition of a finite $(n+1)$-Hilbert space for all $n$. For $n=1$ and $n=2$ we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For $n=1$ we recover a unitary refinement of Schommer-Pries' classification of oriented $(1+1)$D TQFTs in terms of $\\mathrm{H}^*$-Morita equivalence classes of $\\mathrm{H}^*$-algebras, and for $n=2$ we categorify this to classify oriented $(2+1)$D unitary TQFTs by $\\mathrm{H}^*$-Morita equivalence classes of $\\mathrm{H}^*$-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like $n$-categories.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19713",
  "author_total": 1,
  "authors": [
    {
      "name": "Greyson Wesley"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmd7xzd698vsh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19713",
  "fulltext_url": null,
  "id": "dmd7xzd698vsh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19713",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7213595020",
  "pdf_url": "https://arxiv.org/pdf/2609.19713",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmd7xzd698vsh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.19713",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19713"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Unitary TQFTs, unitary disk-like $n$-categories, and higher Hilbert spaces",
  "title_text": "Unitary TQFTs, unitary disk-like n-categories, and higher Hilbert spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/dmd7xzd698vsh",
  "venue": "arXiv",
  "versions": []
}