{
  "id": "math/0110101",
  "version": "v2",
  "published": "2001-10-09T18:12:44.000Z",
  "updated": "2004-11-22T17:16:15.000Z",
  "title": "Algebraic theories in homotopy theory",
  "authors": [
    "Bernard Badzioch"
  ],
  "comment": "19 pages, published version",
  "categories": [
    "math.AT"
  ],
  "abstract": "An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict $T$-algebra is a product preserving functor $A: T\\to Spaces$. Lawvere showed that for a suitable choice of T giving such an algebra amounts to providing the space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie algebra... Given a functor $X: T\\to Spaces$ which preserves products up to a weak equivalence we show that $X$ is more or less canonically weakly equivalent to a strict $T$-algebra $LX$. Thus any `homotopy' algebraic structure on the space $X(t_1)$ can be rigidified to a strict algebraic structure on a space weakly equivalent to $X(t_1)$. This fact can be interpreted as a generalization of the results establishing equivalence of homotopy theories of loop spaces and simplicial groups, products of Eilenberg-Mac Lane spaces and abelian monoids etc.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-11-22T17:16:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P48",
      "18G55"
    ],
    "keywords": [
      "algebraic theory",
      "homotopy theory",
      "strict algebraic structure",
      "eilenberg-mac lane spaces",
      "simplicial groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....10101B"
    }
  }
}