{
  "id": "2005.07490",
  "version": "v1",
  "published": "2020-05-15T12:19:59.000Z",
  "updated": "2020-05-15T12:19:59.000Z",
  "title": "The Karoubi envelope of the mirage of a subshift",
  "authors": [
    "Alfredo Costa",
    "Benjamin Steinberg"
  ],
  "categories": [
    "math.GR",
    "math.DS"
  ],
  "abstract": "We study a correspondence associating to each subshift $\\mathcal X$ of $A^{\\mathbb Z}$ a subcategory of the Karoubi envelope of the free profinite semigroup generated by $A$. The objects of this category are the idempotents in the mirage of $\\mathcal X$, that is, in the set of pseudowords whose finite factors are blocks of $\\mathcal X$. The natural equivalence class of the category is shown to be invariant under flow equivalence. As a corollary of our proof, we deduce the flow invariance of the profinite group that Almeida associated to each irreducible subshift. We also show, in a functorial manner, that the isomorphism class of the category is invariant under conjugacy. Finally, we see that the zeta function of $\\mathcal X$ is naturally encoded in the category. These results hold, with obvious translations, for relatively free profinite semigroups over many pseudovarieties, including all of the form $\\overline{\\mathsf H}$, with $\\mathsf H$ a pseudovariety of groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-15T12:19:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20M07",
      "37B10"
    ],
    "keywords": [
      "karoubi envelope",
      "relatively free profinite semigroups",
      "natural equivalence class",
      "zeta function",
      "pseudovariety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}