{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "77K0BZQ4MK"
    },
    {
      "field": "uses",
      "target": "7E7W0K3N8W"
    },
    {
      "field": "uses",
      "target": "9HZA9B51PZ"
    },
    {
      "field": "uses",
      "target": "AHKJ361FDB"
    },
    {
      "field": "uses",
      "target": "CT9GYCJV4V"
    },
    {
      "field": "uses",
      "target": "GTA8ZFGZ3N"
    },
    {
      "field": "uses",
      "target": "P84GBDM0QR"
    },
    {
      "field": "uses",
      "target": "PY4EWJ8BQA"
    },
    {
      "field": "uses",
      "target": "SHRXX7EW76"
    },
    {
      "field": "defines",
      "target": "9JA1N3CH80"
    },
    {
      "field": "about",
      "target": "AHKJ361FDB"
    }
  ],
  "body": "\n## Summary\n\nThe homology of a [[chain complex|WQ82C0M3QB]] $(C_\\bullet, d)$ is the sequence of objects $H_n(C) = \\ker d_n / \\im d_{n+1}$.\nIt measures the failure of the complex to be [[exact|JB7ZBFBA9C]]: $H_n(C) = 0$ exactly when the complex is exact at $C_n$.\nEach [[chain map|HTK9WQZJ1G]] induces maps on homology, so homology is a functor.\nThe homology of the singular chain complex of a space is its [[singular homology|M0Q5CFJ7ZH]].\nThe same construction gives [[simplicial homology|C6J40QF82R]], group homology and derived functors.\n\n## Explanations\n\n### L1 human\n\nIn a chain complex, every element in the image of $d_{n+1}$ is killed by $d_n$.\nThe elements killed by $d_n$ are called cycles, and those in the image of $d_{n+1}$ are called boundaries.\nHomology is \"cycles modulo boundaries\": it collects the cycles that are not explained by a boundary.\nIn topology, a loop around a hole is a cycle that is not the boundary of any surface, and it gives a nonzero homology class.\nFor the complex $0 \\to \\Z \\xrightarrow{2} \\Z \\to 0$ in degrees $1$ and $0$, the homology is $0$ in degree $1$ and $\\Z/2\\Z$ in degree $0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18G35"
    ],
    "name": "Homology of a chain complex",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Homology_(mathematics)"
      }
    ],
    "tag": "2J4JJJYP22"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/2J4JJJYP22/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/2J4JJJYP22.md",
  "msc": [
    "18G35"
  ],
  "name": "Homology of a chain complex",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2J4/2J4JJJYP22.md",
  "status": null,
  "tag": "2J4JJJYP22",
  "uri": "m2://2J4JJJYP22",
  "url": "https://tamnd.github.io/math2/wiki/homology-of-a-chain-complex/",
  "writing": {
    "tag": "2J4JJJYP22",
    "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
      }
    ]
  }
}