{
  "abstract": "We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space $X$, we prove that the Gabriel spectrum of ${\\mathbf{Sh}}(X)$ is homeomorphic to ${\\mathrm{Sk}}({\\mathrm{Sob}}(X))$, where ${\\mathrm{Sob}}(X)$ denotes the sobrification of $X$ and ${\\mathrm{Sk}}$ indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if $X$ is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ${\\mathbb R}^n$ is homeomorphic to the space of ideals of ${\\mathbb R}^n$ with a natural topology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25761",
  "author_total": 2,
  "authors": [
    {
      "name": "Manu Harsu"
    },
    {
      "name": "Eero Hyry"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4ee5p68qwy69.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25761",
  "fulltext_url": null,
  "id": "b4ee5p68qwy69",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25761",
  "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.AT"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7214079054",
  "pdf_url": "https://arxiv.org/pdf/2609.25761",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4ee5p68qwy69.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.25761",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25761"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Gabriel Spectrum of Persistence Categories",
  "title_text": "Gabriel Spectrum of Persistence Categories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/b4ee5p68qwy69",
  "venue": "arXiv",
  "versions": []
}