{
  "abstract": "The weak, stable, and ordinary Lusternik--Schnirelmann categories of a finite $T_0$-space $X$ satisfy $\\operatorname{cat}_w(X)\\leq \\operatorname{cat}_s(X)\\leq \\operatorname{cat}(X)$. We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If $P$ is weakly contractible but noncontractible and the deletion of one point makes $P$ contractible, then adjoining $m\\geq 2$ incomparable maximal points produces a connected finite space with category triple $(1,2,m)$. Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on $m+9$ points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple $(a,b,c)$ with $1\\leq a<b\\leq 2a$ and $c\\geq b$, as well as every triple $(a,a,c)$ with $2\\leq a\\leq c$. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39615",
  "author_total": 2,
  "authors": [
    {
      "name": "David Mosquera-Lois"
    },
    {
      "name": "Kohei Tanaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1548tw64m044k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39615",
  "fulltext_url": null,
  "id": "1548tw64m044k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39615",
  "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.CO"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
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  ],
  "openalex": "W7215097458",
  "pdf_url": "https://arxiv.org/pdf/2609.39615",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1548tw64m044k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39615",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39615"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Weak, stable, and ordinary Lusternik-Schnirelmann category of finite spaces",
  "title_text": "Weak, stable, and ordinary Lusternik-Schnirelmann category of finite spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/1548tw64m044k",
  "venue": "arXiv",
  "versions": []
}