{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a commutative ring.\nA *dg category* over $k$ is a [[category|A2DE66937D]] $\\mathcal{C}$ [[enriched|BPH9SJ30Y0]] over the monoidal category of cochain complexes of $k$-modules.\nIn detail, for objects $x, y$ of $\\mathcal{C}$ there is a [[chain complex|WQ82C0M3QB]] $\\Hom(x, y) = \\bigoplus_{n \\in \\Z} \\Hom^n(x, y)$ of $k$-modules with differential $d$ of degree $+1$, such that:\n\n1. For objects $x, y, z$ there are $k$-bilinear composition maps $\\Hom^p(y, z) \\times \\Hom^q(x, y) \\to \\Hom^{p+q}(x, z)$, written $(g, f) \\mapsto g \\circ f$.\n2. Composition is associative.\n3. Each object $x$ has an identity $1_x \\in \\Hom^0(x, x)$ with $d(1_x) = 0$, which is a unit for composition.\n4. The Leibniz rule holds: $d(g \\circ f) = d(g) \\circ f + (-1)^{p} g \\circ d(f)$ for $g \\in \\Hom^p(y, z)$ and $f \\in \\Hom^q(x, y)$.\n\nClauses 1 and 4 say that composition is a map of complexes $\\Hom(y, z) \\otimes_k \\Hom(x, y) \\to \\Hom(x, z)$.\n\n## Remarks\n\nExample: the dg category $C_{dg}(k)$ has as objects the cochain complexes $X$ of $k$-modules, with $\\Hom^n(X, Y) = \\prod_{i} \\Hom_k(X^i, Y^{i+n})$.\nFor $f$ of degree $n$ the differential is $d(f) = d_Y \\circ f - (-1)^n f \\circ d_X$.\nOne checks $d^2(f) = d_Y^2 \\circ f - f \\circ d_X^2 = 0$, because the two middle terms $\\pm d_Y \\circ f \\circ d_X$ cancel, and the Leibniz rule follows from the same sign computation.\nThe closed morphisms of degree $0$ are the chain maps, and the exact ones are the null-homotopic chain maps.\n\nNon-example: over $k = \\Q$, use the same graded maps but drop the sign, so $d(f) = d_Y \\circ f + f \\circ d_X$.\nThen $d^2(f) = 2\\, d_Y \\circ f \\circ d_X$.\nLet $C$ be the complex with $C^0 = C^1 = \\Q$ and differential the identity, and let $f \\in \\Hom^{-1}(C, C)$ be the identity map $C^1 \\to C^0$.\nThen $d^2(f)$ is $2 \\ne 0$ on $C^0$, so $\\Hom(C, C)$ is not a complex.\n\nA dg category with one object is the same as a differential graded algebra.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "KTN8HTSY3R",
    "kind": "definition",
    "msc": [
      "18G35",
      "18D20",
      "16E45"
    ],
    "name": "dg category",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Differential_graded_category"
      }
    ],
    "tag": "2FR1BWY3H6",
    "uses": [
      "A2DE66937D",
      "BPH9SJ30Y0",
      "WQ82C0M3QB"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/2FR1BWY3H6/",
  "links": [
    {
      "field": "defines",
      "target": "KTN8HTSY3R"
    },
    {
      "field": "uses",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "BPH9SJ30Y0"
    },
    {
      "field": "uses",
      "target": "WQ82C0M3QB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2FR1BWY3H6.md",
  "msc": [
    "18G35",
    "18D20",
    "16E45"
  ],
  "name": "dg category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2FR/2FR1BWY3H6.md",
  "status": null,
  "tag": "2FR1BWY3H6",
  "uri": "m2://2FR1BWY3H6",
  "url": "https://tamnd.github.io/math2/wiki/dg-category-definition/",
  "writing": {
    "tag": "2FR1BWY3H6",
    "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
      }
    ]
  }
}