{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2FR1BWY3H6"
    }
  ],
  "body": "\n## Summary\n\nA dg category, or differential graded category, is a category whose morphism sets are [[chain complexes|WQ82C0M3QB]] and whose composition is compatible with the differentials.\nIt is a category [[enriched|BPH9SJ30Y0]] over chain complexes of modules over a commutative ring.\nDg categories serve as enhancements of triangulated categories: they keep homotopical information that the [[triangulated category|T2V7GJJ2EN]] loses, such as functorial cones.\nThey are a main tool in noncommutative algebraic geometry and in the study of derived categories of sheaves.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary linear category, the maps between two objects form a vector space.\nIn a dg category they form a chain complex, with maps of each degree and a differential that sends a map of degree $n$ to a map of degree $n + 1$.\nThe main example has chain complexes as objects and all graded maps between them as morphisms.\nTaking cohomology in degree $0$ of each morphism complex gives an ordinary category, here the [[homotopy category of chain complexes|9Y5Y92R4D5]].\nA dg category with one object is the same thing as a differential graded algebra.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18G35",
      "18D20",
      "16E45"
    ],
    "name": "dg category",
    "same_as": {
      "wikidata": "Q5275347",
      "wikipedia": "Differential graded category"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Differential_graded_category"
      }
    ],
    "specializes": [
      "BPH9SJ30Y0"
    ],
    "tag": "KTN8HTSY3R"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/KTN8HTSY3R/",
  "links": [
    {
      "field": "specializes",
      "target": "BPH9SJ30Y0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KTN8HTSY3R.md",
  "msc": [
    "18G35",
    "18D20",
    "16E45"
  ],
  "name": "dg category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KTN/KTN8HTSY3R.md",
  "status": null,
  "tag": "KTN8HTSY3R",
  "uri": "m2://KTN8HTSY3R",
  "url": "https://tamnd.github.io/math2/wiki/dg-category/",
  "writing": {
    "tag": "KTN8HTSY3R",
    "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
      }
    ]
  }
}