{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "BAH89SW049"
    },
    {
      "field": "uses",
      "target": "YMK0D06F4B"
    },
    {
      "field": "defines",
      "target": "Q3TVJ4A3D4"
    }
  ],
  "body": "\n## Summary\n\nA Grothendieck topology on a [[category|A2DE66937D]] says which families of morphisms into an object count as coverings of that object.\nIt is stated with [[sieves|CNCXNWKC6R]] and abstracts the open covers of a topological space.\nA category with a Grothendieck topology is a [[site|NYE0Z94MME]], and one can define [[sheaves|RT8A8FJ566]] and sheaf cohomology on it.\nGrothendieck introduced it to define étale cohomology of schemes, where the \"open sets\" are étale maps rather than subsets.\n\n## Explanations\n\n### L1 human\n\nOn a topological space, a sheaf is defined by gluing data along open covers.\nA Grothendieck topology keeps only what is needed for this: a rule saying which collections of arrows into an object cover it.\nThe rule must contain the trivial cover, be stable when restricted to smaller pieces, and allow covers of covers.\nFor example, on the open sets of a space $X$, a family of open subsets of $U$ covers $U$ when its union is $U$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18F10",
      "14F20"
    ],
    "name": "Grothendieck topology",
    "same_as": {
      "wikidata": "Q1062242",
      "wikipedia": "Grothendieck topology"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Grothendieck_topology"
      }
    ],
    "tag": "GDRFSJ8E2B"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/GDRFSJ8E2B/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/GDRFSJ8E2B.md",
  "msc": [
    "18F10",
    "14F20"
  ],
  "name": "Grothendieck topology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GDR/GDRFSJ8E2B.md",
  "status": null,
  "tag": "GDRFSJ8E2B",
  "uri": "m2://GDRFSJ8E2B",
  "url": "https://tamnd.github.io/math2/wiki/grothendieck-topology/",
  "writing": {
    "tag": "GDRFSJ8E2B",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}