{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "4AG8ZWK2BB"
    },
    {
      "field": "uses",
      "target": "5NAGGDQQTM"
    },
    {
      "field": "uses",
      "target": "7Y69FN6M7Y"
    },
    {
      "field": "uses",
      "target": "FS8H2KHZ9X"
    },
    {
      "field": "uses",
      "target": "JRTB9SHAFD"
    },
    {
      "field": "uses",
      "target": "MDPRVV9DN9"
    },
    {
      "field": "uses",
      "target": "R15Z3DPF6A"
    },
    {
      "field": "uses",
      "target": "R56VTW55SG"
    },
    {
      "field": "uses",
      "target": "W4TT4W95GR"
    },
    {
      "field": "defines",
      "target": "F1CGWAW74E"
    }
  ],
  "body": "\n## Summary\n\nThe stalk of a sheaf at a point is the set of germs of sections at that point.\nA germ is a section defined near the point, where two sections are identified when they agree on some smaller neighbourhood.\nStalks capture the local behaviour of a sheaf at a single point.\nMany properties of sheaves and of maps between them, such as being zero, injective or surjective, can be checked stalk by stalk.\n\n## Explanations\n\n### L1 human\n\nZoom in on one point of a space and look only at what a sheaf does very close to it.\nTwo sections that agree on some small enough neighbourhood are counted as the same.\nFor the sheaf of continuous functions, the stalk at a point is the set of germs of continuous functions there.\nFor holomorphic functions near $0$ in $\\C$, a germ is the same as a power series with a positive radius of convergence.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "54B40",
      "14F06"
    ],
    "name": "Stalk",
    "same_as": {
      "wikidata": "Q17103642",
      "wikipedia": "Stalk (sheaf)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stalk_(sheaf)"
      }
    ],
    "tag": "JTAKAPTT8H"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/JTAKAPTT8H/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/JTAKAPTT8H.md",
  "msc": [
    "54B40",
    "14F06"
  ],
  "name": "Stalk",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JTA/JTAKAPTT8H.md",
  "status": null,
  "tag": "JTAKAPTT8H",
  "uri": "m2://JTAKAPTT8H",
  "url": "https://tamnd.github.io/math2/wiki/stalk/",
  "writing": {
    "tag": "JTAKAPTT8H",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}