{
  "id": "2002.00038",
  "version": "v1",
  "published": "2020-01-31T19:50:53.000Z",
  "updated": "2020-01-31T19:50:53.000Z",
  "title": "Fibration theorems for TQ-completion of structured ring spectra",
  "authors": [
    "Nikolas Schonsheck"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan \"fibration lemma\" under appropriate conditions. We work in the context of algebraic structures that can be described as algebras over an operad $\\mathcal{O}$ in symmetric spectra. Our main result is that completion with respect to topological Quillen homology (or TQ-completion, for short) preserves homotopy fibration sequences provided that the base and total $\\mathcal{O}$-algebras are connected. Our argument essentially boils down to proving that the natural map from the homotopy fiber to its TQ-completion tower is a pro-$\\pi_*$ isomorphism. More generally, we also show that similar results remain true if we replace \"homotopy fibration sequence\" with \"homotopy pullback square.\"",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-31T19:50:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P43",
      "55P48",
      "55P60",
      "55U35",
      "18G55"
    ],
    "keywords": [
      "structured ring spectra",
      "fibration theorems",
      "tq-completion",
      "similar results remain true",
      "preserves homotopy fibration sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}