{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "EXD1197NPH"
    },
    {
      "field": "uses",
      "target": "YV297TR3SC"
    }
  ],
  "body": "\n## Summary\n\nA geometric morphism $f \\colon \\mathcal{E} \\to \\mathcal{F}$ between [[toposes|6837RS7N82]] is a pair of [[adjoint functors|TZHF6ZM92S]] $f^* \\dashv f_*$, with $f_* \\colon \\mathcal{E} \\to \\mathcal{F}$, whose left adjoint $f^* \\colon \\mathcal{F} \\to \\mathcal{E}$ preserves finite limits.\nIt is the notion of map between toposes that matches continuous maps between spaces.\nA continuous map $g \\colon X \\to Y$ gives a geometric morphism $\\mathrm{Sh}(X) \\to \\mathrm{Sh}(Y)$, with $g_*$ the direct image and $g^*$ the inverse image of sheaves.\nPoints of a topos, and the classifying toposes of geometric theories, are described through geometric morphisms.\n\n## Explanations\n\n### L1 human\n\nA continuous map of spaces lets one push sheaves forward and pull sheaves back, and these two operations are adjoint.\nPulling back keeps finite limits, such as products and equalities of sections.\nA geometric morphism is any such adjoint pair between toposes, so it acts as a continuous map between generalized spaces.\nFor example, a point $x$ of a space $X$ gives a geometric morphism $\\mathbf{Set} \\to \\mathrm{Sh}(X)$ whose inverse image sends a sheaf to its stalk at $x$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18F10",
      "18B25"
    ],
    "name": "Geometric morphism",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Topos"
      }
    ],
    "tag": "N5R1WAEPFB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/N5R1WAEPFB/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/N5R1WAEPFB.md",
  "msc": [
    "18F10",
    "18B25"
  ],
  "name": "Geometric morphism",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/N5R/N5R1WAEPFB.md",
  "status": null,
  "tag": "N5R1WAEPFB",
  "uri": "m2://N5R1WAEPFB",
  "url": "https://tamnd.github.io/math2/wiki/geometric-morphism/",
  "writing": {
    "tag": "N5R1WAEPFB",
    "kind": "concept",
    "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
      }
    ]
  }
}