{
  "abstract": "We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories -- such as Atiyah--Segal $K$-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner--Meier -- analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space $X$ and a coefficient system $\\mathscr{A} \\colon \\mathrm{Orb}^{\\mathrm{op}} \\to \\mathrm{Pr}^{\\mathrm{L}}$, the categories $\\mathrm{LS}^{\\mathrm{glo}}(X,\\mathscr{A})$ and $\\mathrm{LS}^{\\mathrm{gen}}(X,\\mathscr{A})$ of globally equivariant and genuine local systems, the latter generalizing the genuine stable category $\\mathrm{Sp}^G$ of a compact Lie group $G$ to non-constant coefficients. When the coefficients come from an oriented abelian group stack $A$ over a locally complex periodic base, we also construct the category $\\mathrm{LS}^\\mathrm{temp}(X,A)$ of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah--Segal completion theorem. We show that global sections induce an equivalence $\\mathrm{LS}^{\\mathrm{temp}}(BU(1),A) \\simeq \\mathrm{QCoh}(A)$ and we prove that $\\mathrm{LS}^\\mathrm{temp}(X,A)$ is a smashing localization of $\\mathrm{LS}^\\mathrm{gen}(X,A)$ if $X$ is an orbispace.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23749",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikolai Konovalov"
    },
    {
      "name": "Artem Prikhodko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ey53xc73c6smt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23749",
  "fulltext_url": null,
  "id": "ey53xc73c6smt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23749",
  "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"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7214002229",
  "pdf_url": "https://arxiv.org/pdf/2609.23749",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ey53xc73c6smt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23749",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23749"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Local coefficients for genuine equivariant cohomology I",
  "title_text": "Local coefficients for genuine equivariant cohomology I",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/ey53xc73c6smt",
  "venue": "arXiv",
  "versions": []
}