{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "XHHTS6TDYZ"
    }
  ],
  "body": "\n## Summary\n\nThe geometric realization $\\abs{K}$ of a simplicial set $K$ is a topological space.\nIt has one topological $n$-simplex for each $n$-simplex of $K$, glued as the face and degeneracy maps prescribe.\nIt is a CW complex with one $n$-cell for each nondegenerate $n$-simplex.\nRealization is left adjoint to the [[singular simplicial set|RWBMXQW6S5]] functor, and the two functors identify the homotopy theory of simplicial sets with that of spaces.\n\n## Explanations\n\n### L1 human\n\nA simplicial set is a list of points, edges, triangles and higher simplices with instructions for how they meet.\nThe realization builds the actual shape: it takes a real triangle for each listed triangle, a real segment for each edge, and glues them together.\nDegenerate simplices, which are collapsed copies of lower ones, are squashed and add nothing new.\nFor example, one vertex and one nondegenerate edge from that vertex to itself realize to a circle.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "55U10",
      "18N50"
    ],
    "name": "Geometric realization",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Simplicial_set"
      }
    ],
    "tag": "TG6WHHR7CF"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/TG6WHHR7CF/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/TG6WHHR7CF.md",
  "msc": [
    "55U10",
    "18N50"
  ],
  "name": "Geometric realization",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TG6/TG6WHHR7CF.md",
  "status": null,
  "tag": "TG6WHHR7CF",
  "uri": "m2://TG6WHHR7CF",
  "url": "https://math2.tamnd.com/wiki/geometric-realization/",
  "writing": {
    "tag": "TG6WHHR7CF",
    "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
      }
    ]
  }
}
