{
  "id": "math/0110109",
  "version": "v1",
  "published": "2001-10-10T14:46:19.000Z",
  "updated": "2001-10-10T14:46:19.000Z",
  "title": "Algebraic Geometry over model categories (a general approach to derived algebraic geometry)",
  "authors": [
    "Bertrand Toen",
    "Gabriele Vezzosi"
  ],
  "comment": "LaTeX,xy-pic, 51 pages",
  "categories": [
    "math.AG",
    "math.AT"
  ],
  "abstract": "For a (semi-)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to \"do algebraic geometry over model categories\". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory of E_{\\infty}-ring spectra. This first version is very preliminary and might be considered as a detailed research announcement. Some proofs, more details and more examples will be added in a forthcoming version.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-10-10T14:46:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14A20",
      "18G55",
      "55P43",
      "55U40",
      "18F10"
    ],
    "keywords": [
      "derived algebraic geometry",
      "general approach",
      "geometric stack",
      "symmetric monoidal base model category"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....10109T"
    }
  }
}