{
  "abstract": "A dagger on a category assigns to every morphism $f \\colon X \\to Y$ a morphism $f^\\dagger \\colon Y \\to X$, contravariantly and involutively, in the way the transpose reverses a real matrix. We show that on the categories of linear algebra and differential geometry, daggers encode metric and Riemannian structures. As is known, on a single vector space, an anti-involution of the endomorphism ring is the adjoint of a form that is determined up to a scalar. But more generally, on the category of all finite-dimensional vector spaces, a dagger is the adjoint for an involution of the scalars together with a unique unimodular Hermitian form on every space. The same holds for finitely generated projective modules over a commutative ring, and hence, through the Serre-Swan correspondence, for vector bundles, where a dagger is a choice of bundle metrics. The dagger structure expresses notions of orthogonality categorically: orthogonal complements and normal bundles acquire universal properties, and Hermitian modules form a dagger compact category. Moreover, daggers on Lie groups, Lie groupoids and their infinitesimal counterparts capture unitary representations and metric connections: a connection is metric precisely when its parallel transport is a dagger functor, or, infinitesimally, when its splitting of the Atiyah sequence commutes with the daggers. Finally, we classify the daggers on the locally graded category of differential operators of a Lie-Rinehart algebra. They correspond to metric-divergence structures, consisting of a real structure, a divergence, and a unimodular Hermitian form on every finitely generated projective module, and they act on operators as formal adjoints; Green's formula expresses the difference between an operator and its adjoint as a divergence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02257",
  "author_total": 2,
  "authors": [
    {
      "name": "Jón Hákon Garðarsson"
    },
    {
      "name": "Paolo Perrone"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ak1wnd7xebvsw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02257",
  "fulltext_url": null,
  "id": "ak1wnd7xebvsw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02257",
  "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.CT"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7220457487",
  "pdf_url": "https://arxiv.org/pdf/2610.02257",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ak1wnd7xebvsw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02257",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02257"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Dagger Categories in Riemannian Geometry",
  "title_text": "Dagger Categories in Riemannian Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/ak1wnd7xebvsw",
  "venue": "arXiv",
  "versions": []
}