{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(P, \\le)$ and $(Q, \\le)$ be [[partially ordered sets|011C29Q1S1]].\nA *Galois insertion* of $Q$ into $P$ is a [[Galois connection|MBVJ4JQXC5]] with lower adjoint $f \\colon P \\to Q$ and upper adjoint $g \\colon Q \\to P$ such that $f(g(y)) = y$ for all $y \\in Q$.\n\n## Remarks\n\nLet $G$ be a group, let $P$ be the set of subsets of $G$, and let $Q$ be the set of subgroups of $G$, both ordered by inclusion.\nLet $f(S)$ be the [[subgroup generated by|0RG9SPQSWM]] $S$, and let $g(H) = H$ as a subset.\nThen $f(S) \\subseteq H$ if and only if $S \\subseteq H$, and $f(g(H)) = H$, so this is a Galois insertion.\nThe pair of the inclusion $\\Z \\to \\R$ and the floor function $\\R \\to \\Z$ is a Galois connection but not a Galois insertion.\nThere the lower adjoint is the inclusion, and the inclusion of $\\lfloor 1/2 \\rfloor = 0$ is $0 \\ne 1/2$.\nFor a Galois connection, these conditions are equivalent: $f \\circ g$ is the identity of $Q$; $f$ is surjective; $g$ is injective.\nThe dual notion, with $g(f(x)) = x$ for all $x \\in P$, is a Galois coinsertion.\n\n## Explanations\n\n### L1 human\n\nA Galois insertion is a Galois connection in which the smaller side sits faithfully inside the larger side.\nGoing in and coming back out changes nothing.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "BPV58EPYCQ",
    "kind": "definition",
    "msc": [
      "06A15"
    ],
    "name": "Galois insertion",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Galois_connection"
      }
    ],
    "tag": "XK933YCW13",
    "uses": [
      "MBVJ4JQXC5"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/XK933YCW13/",
  "links": [
    {
      "field": "defines",
      "target": "BPV58EPYCQ"
    },
    {
      "field": "uses",
      "target": "MBVJ4JQXC5"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/XK933YCW13.md",
  "msc": [
    "06A15"
  ],
  "name": "Galois insertion",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XK9/XK933YCW13.md",
  "status": null,
  "tag": "XK933YCW13",
  "uri": "m2://XK933YCW13",
  "url": "https://tamnd.github.io/math2/wiki/galois-insertion-definition/",
  "writing": {
    "tag": "XK933YCW13",
    "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
      }
    ]
  }
}