{
  "abstract": "An inclusion of nonvoid simplicial complexes induces an arrow of polyhedral products. Applying Ayzenberg's polyhedral join to both complexes gives a symmetric relative join operad. Its endpoint suboperads recover Abramyan--Panov substitution and Ayzenberg composition. Neither endpoint operad nor either relative join operad is finitely generated. The suboperad with nonvoid lower complex arises as the nonempty power-set quotient of the subset-inclusion operad. When tensoring preserves colimits of nonempty finite diagrams, the suboperad acts on arrows by polyhedral colimits up to coherent natural isomorphism. The action induces a set-operad algebra on isomorphism classes of arrows. The dual construction produces Stanley--Reisner quotient arrows, and the colimit action refines Eldridge's loop-space decomposition to arrows. PL ball--boundary pairs form a suboperad of the relative join operad, and minimal interior faces give an operad morphism. Principal pairs have odd-dimensional spherical moment-angle homotopy fibers and yield a closure result for Eldridge's loop-space class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.02440",
  "author_total": 1,
  "authors": [
    {
      "name": "Mathieu Vallée"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ps9x2v8db8d2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.02440",
  "fulltext_url": null,
  "id": "8ps9x2v8db8d2",
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  "landing_url": "https://arxiv.org/abs/2605.02440",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "metadata_licence": "CC0-1.0",
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      "2610|Homotopy and Cohomology in Algebraic Topology"
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  "openalex": "W7160247704",
  "pdf_url": "https://arxiv.org/pdf/2605.02440",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ps9x2v8db8d2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2605.02440",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.02440"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The relative join operad and polyhedral products",
  "title_text": "The relative join operad and polyhedral products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/8ps9x2v8db8d2",
  "venue": "arXiv",
  "versions": []
}