{
  "abstract": "Let $\\mathsf{P}$ be a finite graded poset and $Δ_a^{\\mathsf{P}}$ the height-$a$ thickening of its diagonal, with projections $q_1,q_2$ to $\\mathsf{P}$. We study the \\emph{window convolution} $C_a=\\operatorname{Lan}_{q_1}\\circ q_2^\\ast$ on $\\mathrm{Shv}(\\mathsf{P};k)$, a discrete analogue of convolution against a thickening kernel. An interleaving distance needs the homotopy window convolution $\\mathbb{C}_a$ to compose as a flow, $\\mathbb{C}_a\\mathbb{C}_b\\simeq\\mathbb{C}_{a+b}$; where the meet assignment $Φ$ is total it is a functor and carries a comparison map. Finality is sufficient, and necessary at every minimal apex and wherever the finality defect of $Φ$ is essential; where $Φ$ is total at a minimal apex with unit windows, it is the failure of a length-two interval above the apex to have a single interior element. The flow fails at every branching length-two interval with minimal bottom element, and with it on the face poset of every finite regular cell complex of dimension $\\ge2$. It survives on tame posets, where $\\mathrm{id}\\Rightarrow\\mathbb{C}_a$ gives a canonical extended interleaving pseudometric on $\\operatorname{D^{b}}(\\mathrm{Shv}(\\mathsf{P};k))$; in the saturation cases computed here it takes no finite value above the length of $\\mathsf{P}$, and is finite if and only if the derived colimits agree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19904",
  "author_total": 1,
  "authors": [
    {
      "name": "Shinobu Yokoyama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/602dwdevc34ax.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19904",
  "fulltext_url": null,
  "id": "602dwdevc34ax",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19904",
  "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.AT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7203876220",
  "pdf_url": "https://arxiv.org/pdf/2608.19904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/602dwdevc34ax.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.19904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Meet obstructions and saturation for the constant window convolution on graded posets",
  "title_text": "Meet obstructions and saturation for the constant window convolution on graded posets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/602dwdevc34ax",
  "venue": "arXiv",
  "versions": []
}