{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "C0KFX2CMRR"
    }
  ],
  "body": "\n## Summary\n\nA stable ∞-category is an [[(∞,1)-category|YFY1FWA1MY]] with a [[zero object|AQET9YHH6F]] and with fibers and cofibers of all morphisms.\nIn it, fiber sequences and cofiber sequences coincide.\nIts homotopy category is a [[triangulated category|T2V7GJJ2EN]] in a natural way.\nStable ∞-categories give a well-behaved replacement for triangulated categories: cones are functorial, and all finite limits and finite colimits exist.\nMain examples are the ∞-category of [[spectra|1GHF4EKBXS]] and the derived ∞-category of a ring.\n\n## Explanations\n\n### L1 human\n\nIn a stable ∞-category, every map has a kernel-like object, the fiber, and a cokernel-like object, the cofiber.\nStability says that these two constructions fit together perfectly: a sequence is exact from the left exactly when it is exact from the right.\nSo the loop and suspension operations are inverse to each other, as in the world of chain complexes.\nFor chain complexes, the cofiber of a map is its mapping cone, and the fiber is the cone shifted down by one.\nPointed topological spaces are not stable: the loop space of the suspension of a space is in general not the space itself.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18N60",
      "18G80",
      "55P42"
    ],
    "name": "Stable ∞-category",
    "same_as": {
      "wikidata": "Q17073594",
      "wikipedia": "Stable ∞-category"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stable_%E2%88%9E-category"
      }
    ],
    "specializes": [
      "YFY1FWA1MY"
    ],
    "tag": "X2AHW5VJZR"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/X2AHW5VJZR/",
  "links": [
    {
      "field": "specializes",
      "target": "YFY1FWA1MY"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/X2AHW5VJZR.md",
  "msc": [
    "18N60",
    "18G80",
    "55P42"
  ],
  "name": "Stable ∞-category",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/X2A/X2AHW5VJZR.md",
  "status": null,
  "tag": "X2AHW5VJZR",
  "uri": "m2://X2AHW5VJZR",
  "url": "https://tamnd.github.io/math2/wiki/stable-category/",
  "writing": {
    "tag": "X2AHW5VJZR",
    "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
      }
    ]
  }
}