{
  "abstract": "A pair of topological spaces $X$ and $Y$ is a \\emph{Fourier--Mukai (FM) pair} if there is an equivalence of $\\infty$-categories of derived sheaves of abelian groups \\[\\operatorname{Shv}(X;\\mathrm D(\\mathbb Z))\\simeq \\operatorname{Shv}(Y;\\mathrm D(\\mathbb Z)).\\] An FM pair $(X,Y)$ is \\emph{non-trivial} if $X$ and $Y$ are not homeomorphic. We consider the question of whether non-trivial FM pairs $(X,Y)$ exist for which $X$ and $Y$ are both \"nice\" spaces, which in this article means admitting a triangulation. We show that the answer is no if $X$ is second-countable and either (i) one-dimensional, (ii) a surface, or (iii) the 3-sphere. In order to prove this, we define two invariants of dualizable categories. The first invariant is an extension of \\emph{Tate--Hochschild cohomology} (aka singular Hochschild cohomology) to a class of dualizable categories satisfying a countability assumption. For a homology manifold $X$, we relate the Tate--Hochschild cohomology of $\\operatorname{Shv}(X;\\mathrm D(\\mathbb Z))$ to the homology at infinity of $X$ (in the sense of proper homotopy theory). The second invariant takes the form of a subcategory $\\mathcal D_{\\mathrm{Gor}}\\subseteq\\mathcal D$ called the \\emph{Gorenstein locus}, defined for every dualizable stable category $\\mathcal D$. For a triangularizable space $X$ and a field $k$, we show that the Gorenstein locus of $\\operatorname{Shv}(X;\\mathrm D(k))$ is equal to the full subcategory of sheaves supported on the $k$-homology manifold locus of $X$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13488",
  "author_total": 1,
  "authors": [
    {
      "name": "Oscar Harr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/63ts2egwkcaf2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13488",
  "fulltext_url": null,
  "id": "63ts2egwkcaf2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13488",
  "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.KT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7221521711",
  "pdf_url": "https://arxiv.org/pdf/2610.13488",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/63ts2egwkcaf2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13488",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13488"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Reconstruction of graphs and surfaces from derived categories",
  "title_text": "Reconstruction of graphs and surfaces from derived categories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/63ts2egwkcaf2",
  "venue": "arXiv",
  "versions": []
}