{
  "id": "2210.10096",
  "version": "v1",
  "published": "2022-10-18T18:51:07.000Z",
  "updated": "2022-10-18T18:51:07.000Z",
  "title": "An algebraic model for the free loop space",
  "authors": [
    "Manuel Rivera"
  ],
  "categories": [
    "math.AT",
    "math.QA"
  ],
  "abstract": "We describe an algebraic chain level construction that models the passage from an arbitrary topological space to its free loop space. The input of the construction is a categorical coalgebra, i.e. a curved coalgebra satisfying certain properties, and the output is a chain complex. The construction is a modified version of the coHochschild complex of a differential graded (dg) coalgebra. When applied to the chains on an arbitrary simplicial set $X$, appropriately interpreted, it yields a chain complex that is naturally quasi-isomorphic to the singular chains on the free loop space of the geometric realization of $X$. We relate this construction to a twisted tensor product model for the free loop space constructed using the adjoint action of a dg Hopf algebra model for the based loop space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-18T18:51:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free loop space",
      "algebraic model",
      "dg hopf algebra model",
      "chain complex",
      "algebraic chain level construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}