{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "CWKNRTVERF"
    },
    {
      "field": "uses",
      "target": "QKJPM71RMG"
    },
    {
      "field": "uses",
      "target": "Z18AE0R3FR"
    },
    {
      "field": "defines",
      "target": "G206NVQYVD"
    }
  ],
  "body": "\n## Summary\n\nWhen two [[morphisms|859F0TFVA3]] $f \\colon A \\to B$ and $r \\colon B \\to A$ satisfy $r \\circ f = 1_A$, the morphism $f$ is a section of $r$ and $r$ is a retraction of $f$.\nA morphism with a retraction is a split [[monomorphism|Y2NP1JFR86]], and a morphism with a section is a split [[epimorphism|RVXDSY04KG]].\nThese are one-sided forms of invertibility, and every functor preserves them.\nIn linear algebra and homological algebra, split monomorphisms and split epimorphisms describe direct summands.\n\n## Explanations\n\n### L1 human\n\nA section is a morphism that can be undone from one side, though perhaps not from the other.\nThink of $f$ as placing $A$ inside $B$ and of $r$ as folding $B$ back onto $A$ so that each placed point returns to where it came from.\nFor example, put $f \\colon \\R \\to \\R^2$, $f(x) = (x, 0)$, and $r \\colon \\R^2 \\to \\R$, $r(x, y) = x$.\nThen $r(f(x)) = x$ for every $x$, so $f$ is a section of $r$ and $r$ is a retraction of $f$.\nThe other composite $f(r(x, y)) = (x, 0)$ is not the identity, so neither map is an isomorphism.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A20"
    ],
    "name": "Section and retraction",
    "same_as": {
      "wikidata": "Q60727552",
      "wikipedia": "Section (category theory)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Section_(category_theory)"
      }
    ],
    "tag": "DXJH3DGXBC"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/DXJH3DGXBC/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/DXJH3DGXBC.md",
  "msc": [
    "18A20"
  ],
  "name": "Section and retraction",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DXJ/DXJH3DGXBC.md",
  "status": null,
  "tag": "DXJH3DGXBC",
  "uri": "m2://DXJH3DGXBC",
  "url": "https://tamnd.github.io/math2/wiki/section-and-retraction/",
  "writing": {
    "tag": "DXJH3DGXBC",
    "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
      }
    ]
  }
}