{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a small category, and let $y \\colon \\mathcal{C} \\to [\\mathcal{C}^{\\mathrm{op}}, \\mathbf{Set}]$ be the [[Yoneda embedding|4DKB6AQ8J7]] into [[presheaves|AND2WHZC8P]].\nA presheaf $P$ on $\\mathcal{C}$ is an *ind-object* if there are a small [[filtered category|975PMY4FG8]] $\\mathcal{I}$ and a functor $X \\colon \\mathcal{I} \\to \\mathcal{C}$ with $P \\cong \\operatorname{colim}_{i} y(X_i)$.\nThe *Ind-completion* $\\mathrm{Ind}(\\mathcal{C})$ is the full subcategory of $[\\mathcal{C}^{\\mathrm{op}}, \\mathbf{Set}]$ whose objects are the ind-objects.\n\n## Remarks\n\nSo an ind-object is a filtered [[colimit|BJS1CGTKQ5]] of [[representable presheaves|KCCHH87H6E]].\nIn terms of the diagrams $X \\colon \\mathcal{I} \\to \\mathcal{C}$ and $Y \\colon \\mathcal{J} \\to \\mathcal{C}$, the morphisms are\n$$\\mathrm{Ind}(\\mathcal{C})(X, Y) = \\lim_{i} \\operatorname{colim}_{j} \\mathcal{C}(X_i, Y_j).$$\nThe functor $y$ gives a fully faithful functor $\\mathcal{C} \\to \\mathrm{Ind}(\\mathcal{C})$, and $\\mathrm{Ind}(\\mathcal{C})$ has all small filtered colimits.\nIt is universal: every functor from $\\mathcal{C}$ to a category with filtered colimits extends, uniquely up to isomorphism, to a filtered-colimit-preserving functor on $\\mathrm{Ind}(\\mathcal{C})$.\n\nExample: let $\\mathcal{C}$ be a skeleton of the category of finite sets.\nEvery set is the filtered colimit of its finite subsets, and this gives an equivalence between $\\mathrm{Ind}(\\mathcal{C})$ and the category of sets.\n\nNon-example: let $\\mathcal{C} = \\mathbf{1}$, the category with one object and one morphism.\nA presheaf on $\\mathbf{1}$ is a set, and the only representable is a one-point set.\nA filtered category is nonempty and connected, so every filtered colimit of one-point sets along identities is a one-point set.\nSo a presheaf with two elements, or with none, is not an ind-object.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "1DKFWGWHH5",
    "kind": "definition",
    "msc": [
      "18A30",
      "18C35"
    ],
    "name": "Ind-completion",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ind-completion"
      }
    ],
    "tag": "290G1NGAMR",
    "uses": [
      "975PMY4FG8",
      "AND2WHZC8P",
      "KCCHH87H6E",
      "BJS1CGTKQ5",
      "4DKB6AQ8J7"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/290G1NGAMR/",
  "links": [
    {
      "field": "defines",
      "target": "1DKFWGWHH5"
    },
    {
      "field": "uses",
      "target": "975PMY4FG8"
    },
    {
      "field": "uses",
      "target": "AND2WHZC8P"
    },
    {
      "field": "uses",
      "target": "KCCHH87H6E"
    },
    {
      "field": "uses",
      "target": "BJS1CGTKQ5"
    },
    {
      "field": "uses",
      "target": "4DKB6AQ8J7"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/290G1NGAMR.md",
  "msc": [
    "18A30",
    "18C35"
  ],
  "name": "Ind-completion",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/290/290G1NGAMR.md",
  "status": null,
  "tag": "290G1NGAMR",
  "uri": "m2://290G1NGAMR",
  "url": "https://tamnd.github.io/math2/wiki/ind-completion-definition/",
  "writing": {
    "tag": "290G1NGAMR",
    "kind": "definition",
    "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
      }
    ]
  }
}