{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1F833GKJ9V"
    },
    {
      "field": "uses",
      "target": "1MJW4YPQW6"
    },
    {
      "field": "uses",
      "target": "6ZRRH96VZT"
    },
    {
      "field": "uses",
      "target": "7E7W0K3N8W"
    },
    {
      "field": "uses",
      "target": "7ZGQKWYKQS"
    },
    {
      "field": "uses",
      "target": "8FEGZWG3JH"
    },
    {
      "field": "uses",
      "target": "8MCYV0P04T"
    },
    {
      "field": "uses",
      "target": "8V296XR2M3"
    },
    {
      "field": "uses",
      "target": "9HZA9B51PZ"
    },
    {
      "field": "uses",
      "target": "9JA1N3CH80"
    },
    {
      "field": "uses",
      "target": "9R9MVA4PWR"
    },
    {
      "field": "uses",
      "target": "AHKJ361FDB"
    },
    {
      "field": "uses",
      "target": "AK2XQC04PC"
    },
    {
      "field": "uses",
      "target": "C84CHVTZ4J"
    },
    {
      "field": "uses",
      "target": "CT9GYCJV4V"
    },
    {
      "field": "uses",
      "target": "CTVVXCWTX7"
    },
    {
      "field": "uses",
      "target": "F2W20ZNWC7"
    },
    {
      "field": "uses",
      "target": "HEVF97NZ14"
    },
    {
      "field": "uses",
      "target": "JMCKACGCQ0"
    },
    {
      "field": "uses",
      "target": "N67R59ZVC4"
    },
    {
      "field": "uses",
      "target": "P84GBDM0QR"
    },
    {
      "field": "uses",
      "target": "PBE65H0506"
    },
    {
      "field": "uses",
      "target": "PCEVB6XV6J"
    },
    {
      "field": "uses",
      "target": "PY4EWJ8BQA"
    },
    {
      "field": "uses",
      "target": "Q8AYGD93CW"
    },
    {
      "field": "uses",
      "target": "VNMBSYX9PY"
    },
    {
      "field": "uses",
      "target": "W57F9ENPMV"
    },
    {
      "field": "uses",
      "target": "W6AVEAW2WR"
    },
    {
      "field": "uses",
      "target": "WXRH0CYKYF"
    },
    {
      "field": "uses",
      "target": "YGHYD555Y7"
    },
    {
      "field": "uses",
      "target": "ZV7VGE2SQ2"
    },
    {
      "field": "about",
      "target": "8V296XR2M3"
    },
    {
      "field": "about",
      "target": "N67R59ZVC4"
    },
    {
      "field": "instance_of",
      "target": "99ZVSM254K"
    },
    {
      "field": "instance_of",
      "target": "JBSSXP4FFY"
    },
    {
      "field": "specializes",
      "target": "A7W468BQSM"
    },
    {
      "field": "defines",
      "target": "AEF4CBRAMN"
    }
  ],
  "body": "\n## Summary\n\nAn abelian category is an [[additive category|B2VP96NJSK]] in which every morphism has a [[kernel|S417YNEJ6M]] and a [[cokernel|H8W38933NC]].\nIn it, monomorphisms and epimorphisms behave as in abelian groups.\nIn it, exact sequences, homology and the standard diagram lemmas make sense.\nModules over a ring and sheaves of abelian groups on a space are abelian categories.\nAbelian categories were introduced by Buchsbaum and by Grothendieck to give a common framework for homological algebra.\n\n## Explanations\n\n### L1 human\n\nAbelian groups have a rich toolkit: subgroups, quotients, images, and exact sequences that track how they fit together.\nAn abelian category is any category where the same toolkit works, described only through maps.\nIn such a category every injective-like map is the kernel of something, and every surjective-like map is the cokernel of something.\nSo a theorem of homological algebra proved once for abelian categories applies at once to modules, sheaves and representations.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18E10"
    ],
    "name": "Abelian category",
    "same_as": {
      "wikidata": "Q318737",
      "wikipedia": "Abelian category"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Abelian_category"
      }
    ],
    "specializes": [
      "B2VP96NJSK"
    ],
    "tag": "Y9CZ3MTNF1"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/Y9CZ3MTNF1/",
  "links": [
    {
      "field": "specializes",
      "target": "B2VP96NJSK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Y9CZ3MTNF1.md",
  "msc": [
    "18E10"
  ],
  "name": "Abelian category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y9C/Y9CZ3MTNF1.md",
  "status": null,
  "tag": "Y9CZ3MTNF1",
  "uri": "m2://Y9CZ3MTNF1",
  "url": "https://tamnd.github.io/math2/wiki/abelian-category/",
  "writing": {
    "tag": "Y9CZ3MTNF1",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}