{
  "abstract": "For $n\\ge1$, we study representations of a simplicial group $G$ in the full $\\infty$-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees $0,\\ldots,n-1$, retaining its full mapping spaces. The resulting functor category depends only on the $n$-truncated homotopy type of $\\mathbf{B}G$ and admits projectively bifibrant strict diagram and simplicial group-algebra module models. If restriction along a specified map $f:G\\to H$ is an equivalence, then $π_0(f)$ is a group isomorphism; the proof uses ordinary extension and restriction of scalars. Yet for every $n\\ge2$ and prime $q$ invertible in $k$, the nontrivial source $K(\\mathbb Z/q,n-1)$ has the same $n$-representation category with vertex evaluation as the trivial group. The group-algebra augmentation is a weak equivalence, so the constant functor is also an equivalence for untruncated representations. Thus $n$-representations depend only on the $n$-truncation of $\\mathbf{B}G$, but need not determine that truncation, even together with vertex evaluation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11576",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhangqi Pan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ab4r7dkn2nz8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11576",
  "fulltext_url": null,
  "id": "1ab4r7dkn2nz8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11576",
  "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.CT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7221199221",
  "pdf_url": "https://arxiv.org/pdf/2610.11576",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ab4r7dkn2nz8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11576",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11576"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Note on n-Representations of Simplicial Groups",
  "title_text": "A Note on n-Representations of Simplicial Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/1ab4r7dkn2nz8",
  "venue": "arXiv",
  "versions": []
}