{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1S9JGHT5HJ"
    },
    {
      "field": "uses",
      "target": "9KP392QCGV"
    },
    {
      "field": "uses",
      "target": "D1MYE6WVZ9"
    },
    {
      "field": "uses",
      "target": "GC9FN7TTBV"
    },
    {
      "field": "uses",
      "target": "HX2DMKTT3Y"
    },
    {
      "field": "uses",
      "target": "W3B4XH5JHZ"
    },
    {
      "field": "defines",
      "target": "B75H4VEARE"
    }
  ],
  "body": "\n## Summary\n\nA triangulated category is an [[additive category|B2VP96NJSK]] with a shift functor and a class of distinguished triangles $X \\to Y \\to Z \\to X[1]$ that satisfy a list of axioms.\nDistinguished triangles play the role that short exact sequences play in an abelian category.\nThe [[homotopy category of chain complexes|9Y5Y92R4D5]], the [[derived category|5ST7NVJFSM]] and the [[stable homotopy category|26GTNMAF7B]] are triangulated.\n\n## Explanations\n\n### L1 human\n\nIn the derived category, maps need not have kernels and cokernels, so short exact sequences are not available.\nTheir replacement is the triangle: each map $f \\colon X \\to Y$ has a cone $Z$, and the sequence $X \\to Y \\to Z \\to X[1]$ behaves like a short exact sequence that continues forever by shifting.\nApplying a functor such as $\\Hom(W, -)$ to a triangle gives a long exact sequence.\nThe axioms list the rules that these triangles obey in the examples.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18G80"
    ],
    "name": "Triangulated category",
    "same_as": {
      "wikidata": "Q7840150",
      "wikipedia": "Triangulated category"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Triangulated_category"
      }
    ],
    "specializes": [
      "B2VP96NJSK"
    ],
    "tag": "T2V7GJJ2EN"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/T2V7GJJ2EN/",
  "links": [
    {
      "field": "specializes",
      "target": "B2VP96NJSK"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/T2V7GJJ2EN.md",
  "msc": [
    "18G80"
  ],
  "name": "Triangulated category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T2V/T2V7GJJ2EN.md",
  "status": null,
  "tag": "T2V7GJJ2EN",
  "uri": "m2://T2V7GJJ2EN",
  "url": "https://tamnd.github.io/math2/wiki/triangulated-category/",
  "writing": {
    "tag": "T2V7GJJ2EN",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}