{
  "id": "1304.5676",
  "version": "v2",
  "published": "2013-04-20T23:08:04.000Z",
  "updated": "2015-01-07T11:31:18.000Z",
  "title": "Bundles of spectra and algebraic K-theory",
  "authors": [
    "John Lind"
  ],
  "comment": "v2: minor changes in section 7. 36 pages, comments welcome",
  "categories": [
    "math.AT",
    "math.KT"
  ],
  "abstract": "A parametrized spectrum E is a family of spectra E_x continuously parametrized by the points x of a topological space X. We take the point of view that a parametrized spectrum is a bundle-theoretic geometric object. When R is a ring spectrum, we consider parametrized R-module spectra and show that they give cocycles for the cohomology theory determined by the algebraic K-theory K(R) of R in a manner analogous to the description of topological K-theory K^0(X) as the Grothendieck group of vector bundles over X. We prove a classification theorem for parametrized spectra, showing that parametrized spectra over X whose fibers are equivalent to a fixed R-module M are classified by homotopy classes of maps from X to the classifying space BAut_R(M) of the A_\\infty space of R-module equivalences from M to M. In proving the classification theorem for parametrized spectra, we define of the notion of a principal G fibration where G is an A_\\infty space and prove a similar classification theorem for principal G fibrations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-20T23:08:04.000Z",
      "comment": "36 pages, comments welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-07T11:31:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic k-theory",
      "parametrized spectrum",
      "similar classification theorem",
      "bundle-theoretic geometric object",
      "vector bundles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.5676L"
    }
  }
}