Derivators

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.

Nine lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Diagrams up to weak equivalence
2. Kan extensions and the derivator axioms
3. Homotopy exact squares
4. The derivator of a ring
5. Pointed and stable derivators
6. Recognizing cocartesian squares
7. Stable derivators are additive
8. The canonical triangulated structure
9. The triangles of the derivator of a ring

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Simplicial sets, nerves and Kan complexes · Algebraic spaces and stacks
• Test categories and Grothendieck's homotopy programme · Algebraic spaces and stacks
• Complexes, cones and localization (Derived categories of sheaves)
• Flat modules and K-flat resolutions (Derived categories of sheaves)
• K-injective resolutions in Grothendieck abelian categories (Derived categories of sheaves)
• Complexes, cones, derived categories and resolutions (course Derived categories and sheaf operations: Complexes, cones and localization; Flat modules and K-flat resolutions; K-injective resolutions in Grothendieck abelian categories). The homotopy programme of Pursuing Stacks (course Algebraic spaces and stacks, Lessons 11 and 12).