{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $W$ be a class of morphisms of $\\mathcal{C}$.\nA *localization of $\\mathcal{C}$ at $W$* is a category $\\mathcal{C}[W^{-1}]$ together with a [[functor|28NQ0KRHD3]] $\\gamma \\colon \\mathcal{C} \\to \\mathcal{C}[W^{-1}]$ such that:\n\n1. $\\gamma(w)$ is an isomorphism for every $w \\in W$;\n2. for every functor $F \\colon \\mathcal{C} \\to \\mathcal{D}$ such that $F(w)$ is an isomorphism for every $w \\in W$, there is a unique functor $G \\colon \\mathcal{C}[W^{-1}] \\to \\mathcal{D}$ with $F = G \\circ \\gamma$.\n\n## Remarks\n\nBy the universal property, a localization is unique up to a unique isomorphism of categories compatible with $\\gamma$.\nIt always exists if size issues are ignored.\nIts objects are those of $\\mathcal{C}$, and its morphisms are equivalence classes of zigzags of morphisms of $\\mathcal{C}$ whose backward arrows are in $W$.\nThe hom-sets of this construction may fail to be small; conditions such as a calculus of fractions or a model structure give control over them.\nMany authors use a weaker universal property, asking that precomposition with $\\gamma$ be an equivalence between functor categories onto the functors that invert $W$.\n\nExample: let $\\mathcal{C}$ have two objects $0$ and $1$ and one non-identity morphism $a \\colon 0 \\to 1$, and let $W = \\set{a}$.\nThe category $\\mathcal{L}$ with objects $0, 1$ and exactly one morphism between any two objects is a localization, with $\\gamma$ the identity on objects.\nIndeed $\\gamma(a)$ is invertible, and a functor $F$ inverting $a$ factors uniquely by sending the morphism $1 \\to 0$ of $\\mathcal{L}$ to $F(a)^{-1}$.\n\nNon-example: the identity functor $\\mathcal{C} \\to \\mathcal{C}$ satisfies clause 2 for the functors that invert $a$, but it fails clause 1, since $a$ is not an isomorphism in $\\mathcal{C}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "DNFR3YP0XF",
    "kind": "definition",
    "msc": [
      "18E35"
    ],
    "name": "Localization of a category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Localization_of_a_category"
      }
    ],
    "tag": "KSN3AJ6EJD",
    "uses": [
      "A2DE66937D",
      "28NQ0KRHD3"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/KSN3AJ6EJD/",
  "links": [
    {
      "field": "defines",
      "target": "DNFR3YP0XF"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "28NQ0KRHD3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KSN3AJ6EJD.md",
  "msc": [
    "18E35"
  ],
  "name": "Localization of a category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KSN/KSN3AJ6EJD.md",
  "status": null,
  "tag": "KSN3AJ6EJD",
  "uri": "m2://KSN3AJ6EJD",
  "url": "https://tamnd.github.io/math2/wiki/localization-of-a-category-definition/",
  "writing": {
    "tag": "KSN3AJ6EJD",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}