{
  "id": "2105.01143",
  "version": "v1",
  "published": "2021-05-03T19:50:35.000Z",
  "updated": "2021-05-03T19:50:35.000Z",
  "title": "Traces for factorization homology in dimension 1",
  "authors": [
    "David Ayala",
    "John Francis"
  ],
  "comment": "29 pages",
  "categories": [
    "math.AT",
    "math.CT",
    "math.KT",
    "math.QA"
  ],
  "abstract": "We construct a circle-invariant trace from the factorization homology of the circle \\[ {\\sf trace} \\colon \\int^\\alpha_{{\\mathbb S}^1} \\\\underline{\\sf End}(V) \\longrightarrow \\uno \\] associated to a dualizable object $V\\in {\\boldsymbol{\\mathfrak X}}$ in a symmetric monoidal $\\infty$-category. This proves a conjecture of To\\\"en--Vezzosi~\\cite{toen.vezzosi} on existence of circle-invariant traces. Underlying our construction is a calculation of the factorization homology over the circle of the walking adjunction in terms of the paracyclic category of Getzler--Jones~\\cite{getzler.jones}: \\[ \\int_{{\\mathbb S}^1} {\\sf Adj} ~\\simeq~ {\\bDelta_{\\circlearrowleft}}^{\\triangleleft\\!\\triangleright} ~. \\] This calculation exhibits a form of Poincar\\'e duality for 1-dimensional factorization homology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-03T19:50:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "factorization homology",
      "circle-invariant trace",
      "poincare duality",
      "paracyclic category",
      "calculation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}