{
  "id": "derivators",
  "title": "Derivators",
  "description": "Grothendieck's proposal in Pursuing Stacks to keep all derived categories of diagram categories at once, with restriction functors that have adjoints on both sides. Why cones in a derived category cannot be chosen functorially; localization, prederivators, Kan's formulas and the derivator axioms; homotopy exact squares; the derivator of a ring, its strongness and its stability; recognition of cocartesian squares and shifted derivators; additivity of stable derivators, with the flip of a suspension equal to -1; the canonical triangulated structure, exact functors, the Dold-Puppe sequences, and the comparison with the classical triangles of the derived category of a ring, sign included.",
  "units": [
    {
      "id": "diagrams-up-to-weak-equivalence",
      "title": "Diagrams up to weak equivalence",
      "reader": "diagrams-up-to-weak-equivalence.html",
      "source": "src/diagrams-up-to-weak-equivalence.md",
      "source_sha256": "f5ee8de4758edf160e340fda9808c1f0e5f27852963908874d335af5283731bb",
      "status": "author_self_checked"
    },
    {
      "id": "kan-extensions-and-the-derivator-axioms",
      "title": "Kan extensions and the derivator axioms",
      "reader": "kan-extensions-and-the-derivator-axioms.html",
      "source": "src/kan-extensions-and-the-derivator-axioms.md",
      "source_sha256": "fd42f9a05af9162b6b674f0f2b4de5642b73822e9e2f11bf2d1f77ba1379fec4",
      "status": "author_self_checked"
    },
    {
      "id": "homotopy-exact-squares",
      "title": "Homotopy exact squares",
      "reader": "homotopy-exact-squares.html",
      "source": "src/homotopy-exact-squares.md",
      "source_sha256": "11daab9c2ef8386596b34e7077fb27a0fa6a7df2acefd354c2d0ce1908992755",
      "status": "author_self_checked"
    },
    {
      "id": "the-derivator-of-a-ring",
      "title": "The derivator of a ring",
      "reader": "the-derivator-of-a-ring.html",
      "source": "src/the-derivator-of-a-ring.md",
      "source_sha256": "dd6db29a01c1f6599703483bbb1ff317eabb3fdefdec313ce6dc82a52df87847",
      "status": "author_self_checked"
    },
    {
      "id": "pointed-and-stable-derivators",
      "title": "Pointed and stable derivators",
      "reader": "pointed-and-stable-derivators.html",
      "source": "src/pointed-and-stable-derivators.md",
      "source_sha256": "ff4d1f8d7c2df5f225f0f5b3ada992639a87c20effcc2128837a2fa8ecfc1c65",
      "status": "author_self_checked"
    },
    {
      "id": "recognizing-cocartesian-squares",
      "title": "Recognizing cocartesian squares",
      "reader": "recognizing-cocartesian-squares.html",
      "source": "src/recognizing-cocartesian-squares.md",
      "source_sha256": "27ec9792e072c849714000fc3d2a51775d4ff837306599900c3b7d530ddf913f",
      "status": "author_self_checked"
    },
    {
      "id": "stable-derivators-are-additive",
      "title": "Stable derivators are additive",
      "reader": "stable-derivators-are-additive.html",
      "source": "src/stable-derivators-are-additive.md",
      "source_sha256": "77d018da74884e1cd01a48eb193d3f65a47cd879c2afb92e39d2a6ff683f3ea3",
      "status": "author_self_checked"
    },
    {
      "id": "the-canonical-triangulated-structure",
      "title": "The canonical triangulated structure",
      "reader": "the-canonical-triangulated-structure.html",
      "source": "src/the-canonical-triangulated-structure.md",
      "source_sha256": "115bc3df252c7eda1967a799acdd17cc6c1ad14996ee8573316877a977adb583",
      "status": "author_self_checked"
    },
    {
      "id": "the-triangles-of-the-derivator-of-a-ring",
      "title": "The triangles of the derivator of a ring",
      "reader": "the-triangles-of-the-derivator-of-a-ring.html",
      "source": "src/the-triangles-of-the-derivator-of-a-ring.md",
      "source_sha256": "e6f84520c4107e15ad9640427be9da4ba1397a397c98cfc6d99842570fc61f4a",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "categories, functors, natural transformations, limits, colimits and adjoint functors",
    "complexes, cones and derived categories (programme lesson Complexes, cones and localization, Lemmas 1.1-1.2, Lemma 2.1, Theorems 4.2 and 5.1, Proposition 5.2)",
    "semifree and K-projective resolutions over a ring (programme lesson Flat modules and K-flat resolutions, Theorem 3.4)",
    "K-injective resolutions in Grothendieck abelian categories (programme lesson of that name, Theorems 4.1 and 5.1, equation (5.1), Proposition 5.2)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}