{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *class of weak equivalences* in $\\mathcal{C}$ is a class $W$ of morphisms of $\\mathcal{C}$ such that:\n\n1. every isomorphism of $\\mathcal{C}$ is in $W$;\n2. (two out of three) if $f \\colon X \\to Y$ and $g \\colon Y \\to Z$ are morphisms and two of $f$, $g$, $g \\circ f$ are in $W$, then so is the third.\n\nA pair $(\\mathcal{C}, W)$ is a *category with weak equivalences*, and the members of $W$ are its *weak equivalences*.\nThe weak equivalences of a model category are the members of the class $W$ of its model structure, which satisfies these two clauses.\n\n## Remarks\n\nExample: in the category of topological spaces, the [[weak homotopy equivalences|4Q3WY1RX91]] form a class of weak equivalences.\nHomeomorphisms induce isomorphisms on all homotopy groups, and two out of three follows from functoriality of $\\pi_0$ and of $\\pi_n(-, x)$ at all base points.\nExample: in the category of chain complexes of modules over a ring, the [[quasi-isomorphisms|C9JEC2WQKV]] form a class of weak equivalences, by functoriality of homology.\n\nNon-example: in the category of sets, the class of injective maps is closed under composition and contains the isomorphisms, but it fails two out of three.\nWith $f \\colon \\set{0} \\to \\set{0, 1}$ the inclusion and $g \\colon \\set{0, 1} \\to \\set{0}$, both $f$ and $g \\circ f = \\id$ are injective, but $g$ is not.\n\nSome authors, for homotopical categories, ask the stronger two out of six property.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "E2870TEDG7",
    "kind": "definition",
    "msc": [
      "18N40",
      "55U35"
    ],
    "name": "Weak equivalence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)"
      }
    ],
    "tag": "NVXMVBS1ES",
    "uses": [
      "A2DE66937D",
      "4Q3WY1RX91",
      "C9JEC2WQKV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/NVXMVBS1ES/",
  "links": [
    {
      "field": "defines",
      "target": "E2870TEDG7"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "4Q3WY1RX91"
    },
    {
      "field": "uses",
      "target": "C9JEC2WQKV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NVXMVBS1ES.md",
  "msc": [
    "18N40",
    "55U35"
  ],
  "name": "Weak equivalence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NVX/NVXMVBS1ES.md",
  "status": null,
  "tag": "NVXMVBS1ES",
  "uri": "m2://NVXMVBS1ES",
  "url": "https://tamnd.github.io/math2/wiki/weak-equivalence-definition/",
  "writing": {
    "tag": "NVXMVBS1ES",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}