{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{E}$ be a [[topos|6837RS7N82]] with [[subobject classifier|NKP5TPQ79X]] $\\top \\colon 1 \\to \\Omega$.\nLet $\\wedge \\colon \\Omega \\times \\Omega \\to \\Omega$ be the characteristic morphism of the subobject $(\\top, \\top) \\colon 1 \\to \\Omega \\times \\Omega$.\nA *Lawvere–Tierney topology* on $\\mathcal{E}$ is a morphism $j \\colon \\Omega \\to \\Omega$ such that:\n\n1. $j \\circ \\top = \\top$;\n2. $j \\circ j = j$;\n3. $j \\circ \\wedge = \\wedge \\circ (j \\times j)$.\n\n## Remarks\n\nExample: the identity $\\id_\\Omega$ satisfies all three equations.\nExample: the double negation $\\neg\\neg \\colon \\Omega \\to \\Omega$ is a Lawvere–Tierney topology in every topos.\nNon-example: in $\\mathbf{Set}$, with $\\Omega = \\set{0, 1}$ and $\\top = 1$, the negation $\\neg$ sends $1$ to $0$, so $\\neg \\circ \\top \\ne \\top$ and clause 1 fails.\nAn object $F$ is a $j$-sheaf when, for every monomorphism $m \\colon A \\to B$ whose characteristic morphism satisfies $j \\circ \\chi_m = \\top \\circ {!}$, where ${!} \\colon B \\to 1$ is the unique morphism, every morphism $A \\to F$ extends uniquely along $m$.\nThe $j$-sheaves form a topos $\\mathrm{Sh}_j(\\mathcal{E})$, and the inclusion has a left adjoint that preserves finite limits.\nAnother name for a Lawvere–Tierney topology is a local operator.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "E6H1CTT439",
    "kind": "definition",
    "msc": [
      "18B25",
      "18F10",
      "03G30"
    ],
    "name": "Lawvere–Tierney topology",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lawvere%E2%80%93Tierney_topology"
      }
    ],
    "tag": "C9XM3X631W",
    "uses": [
      "6837RS7N82",
      "NKP5TPQ79X"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/C9XM3X631W/",
  "links": [
    {
      "field": "defines",
      "target": "E6H1CTT439"
    },
    {
      "field": "uses",
      "target": "6837RS7N82"
    },
    {
      "field": "uses",
      "target": "NKP5TPQ79X"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/C9XM3X631W.md",
  "msc": [
    "18B25",
    "18F10",
    "03G30"
  ],
  "name": "Lawvere–Tierney topology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C9X/C9XM3X631W.md",
  "status": null,
  "tag": "C9XM3X631W",
  "uri": "m2://C9XM3X631W",
  "url": "https://tamnd.github.io/math2/wiki/lawvere-tierney-topology-definition/",
  "writing": {
    "tag": "C9XM3X631W",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}