{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7E7W0K3N8W"
    },
    {
      "field": "uses",
      "target": "9JA1N3CH80"
    },
    {
      "field": "uses",
      "target": "AHKJ361FDB"
    },
    {
      "field": "uses",
      "target": "K8JG3C40HZ"
    },
    {
      "field": "uses",
      "target": "P84GBDM0QR"
    },
    {
      "field": "uses",
      "target": "PCEVB6XV6J"
    },
    {
      "field": "uses",
      "target": "SHRXX7EW76"
    },
    {
      "field": "uses",
      "target": "V3MP0RJ2NX"
    },
    {
      "field": "specializes",
      "target": "C9JEC2WQKV"
    },
    {
      "field": "defines",
      "target": "MHF34RK5ZT"
    }
  ],
  "body": "\n## Summary\n\nA chain map is a morphism between two [[chain complexes|WQ82C0M3QB]]: a family of maps $f_n \\colon C_n \\to D_n$, one in each degree, that commute with the differentials.\nChain complexes and chain maps form a category, which is abelian when the underlying category is abelian.\nEvery chain map induces maps on [[homology|2J4JJJYP22]], and continuous maps of spaces induce chain maps of singular chains.\n\n## Explanations\n\n### L1 human\n\nA chain complex is a row of objects with downward maps.\nA chain map from one such row to another is a ladder of vertical maps such that every square in the ladder commutes.\nGoing down then across gives the same result as going across then down.\nFor example, multiplication by $2$ in every degree is a chain map from any complex of abelian groups to itself.\nBecause the squares commute, a chain map sends cycles to cycles and boundaries to boundaries.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18G35"
    ],
    "name": "Chain map",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chain_complex"
      }
    ],
    "specializes": [
      "859F0TFVA3"
    ],
    "tag": "HTK9WQZJ1G"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HTK9WQZJ1G/",
  "links": [
    {
      "field": "specializes",
      "target": "859F0TFVA3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HTK9WQZJ1G.md",
  "msc": [
    "18G35"
  ],
  "name": "Chain map",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HTK/HTK9WQZJ1G.md",
  "status": null,
  "tag": "HTK9WQZJ1G",
  "uri": "m2://HTK9WQZJ1G",
  "url": "https://tamnd.github.io/math2/wiki/chain-map/",
  "writing": {
    "tag": "HTK9WQZJ1G",
    "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
      }
    ]
  }
}