{
  "id": "2008.01706",
  "version": "v1",
  "published": "2020-08-04T17:17:04.000Z",
  "updated": "2020-08-04T17:17:04.000Z",
  "title": "Complete filtered $L_\\infty$-algebras and their homotopy theory",
  "authors": [
    "Christopher L. Rogers"
  ],
  "comment": "43 pages. Comments are welcome",
  "categories": [
    "math.AT",
    "math.CT",
    "math.QA"
  ],
  "abstract": "We analyze a model for the homotopy theory of complete filtered $\\mathbb{Z}$-graded $L_\\infty$-algebras, which lends itself well to computations in deformation theory and homotopical algebra. We first give an explicit proof of an unpublished result of E. Getzler which states that the category ${\\widehat{\\mathsf{Lie}}\\lbrack 1\\rbrack_\\infty}$ of such $L_\\infty$-algebras and filtration-preserving $\\infty$-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. As a novel application, we use this result to show that, under some mild conditions, every $L_\\infty$-quasi-isomorphism between $L_\\infty$-algebras in ${\\widehat{\\mathsf{Lie}}\\lbrack 1\\rbrack_\\infty}$ has a filtration preserving homotopy inverse. Finally, building on previous joint work with V. Dolgushev, we prove that the simplicial Maurer--Cartan functor, which assigns a Kan simplicial set to each complete filtered $L_\\infty$-algebra, is an exact functor between the respective categories of fibrant objects. We interpret this as an optimal homotopy-theoretic generalization of the classical Goldman--Millson theorem from deformation theory. One immediate application is the ``$\\infty$-categorical'' uniqueness theorem for homotopy transferred structures previously sketched by the author in arXiv:1612.07868.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T17:17:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homotopy theory",
      "deformation theory",
      "fibrant objects",
      "optimal homotopy-theoretic generalization",
      "kan simplicial set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}