Results on derivators and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• A morphism its components do not see — Proposition 2.1. In the derived category of arrows of complexes of abelian groups: an explicit nonzero morphism whose two components are zero; its cone induces -1 on cohomology in degree -1.
• No functorial cones — Corollary 2.2. No functor on the arrow category of D(Ab) is naturally isomorphic to the cone of arrows of complexes; the underlying diagram functor of arrows is not faithful.
• The prederivator of a category with weak equivalences — Theorem 4.2. Any category with a class of morphisms containing the identities, with chosen localizations of its diagram categories: exactly one strict 2-functor with these values compatible with restriction.
• Represented derivators — Theorem 4.2. The represented prederivator of a category is a derivator exactly when the category has all small limits and colimits.
• Fully faithful extensions — Proposition 6.1. In every derivator, homotopy Kan extensions along a fully faithful functor are fully faithful.
• Pasting — Lemma 1.1. Base change morphisms of squares pasted side by side are composites of the base change morphisms; Lemma 1.2 for squares pasted on top of each other.
• Right adjoints are homotopy final — Proposition 2.3. In every derivator, restriction along a right adjoint preserves homotopy colimits; dually for left adjoints and homotopy limits.
• Extension along sieves and cosieves — Proposition 3.3. In every derivator, left Kan extension along a cosieve extends by initial objects and right Kan extension along a sieve extends by terminal objects.
• The derivator of a ring — Theorem 3.1. For every ring R, the assignment of D(Mod_R^I) to small categories I is a derivator, with derived Kan extensions computed from semifree and K-injective resolutions.
• Strongness — Theorem 4.1. For every ring R and small category J, the partial underlying diagram functor of shape [1] x J is full and essentially surjective.
• Suspension and loops — Proposition 4.2. In a stable derivator, suspension and loops are inverse equivalences.
• Stability of the derivator of a ring — Theorem 5.3. For every ring R, a square of complexes is cocartesian in the derivator of R exactly when its total complex is acyclic, exactly when it is cartesian.
• Recognition lemma — Lemma 2.1. In every derivator, a square in a diagram is cocartesian exactly when the counit of the left Kan extension from the other objects is invertible at its corner, when the induced functor to the comma category has a left adjoint.
• Composition and cancellation — Proposition 2.3. In every derivator, if the left square of a diagram of shape [2] x [1] is cocartesian, the right square is cocartesian exactly when the composite square is; dually for cartesian squares.
• Shifted derivators — Proposition 3.1. For a derivator D and a small category M, D^M is a derivator, pointed, stable or strong when D is, and its squares are cocartesian objectwise (Corollaries 3.2 and 3.3).
• The cofibre sequence of an arrow — Proposition 4.1. In a pointed derivator, every arrow has a diagram of shape [2] x [1] with cocartesian squares containing the arrow, its cofibre and the suspension of its source.
• Biproducts — Theorem 1.3. In a stable derivator, finite coproducts and finite products agree in every value.
• The flip of a suspension — Theorem 4.1. In a stable derivator, the automorphism of a suspension that exchanges the two legs of its span is the negative of the identity.
• Additivity — Corollary 4.2. In a stable derivator, every value is additive, and restrictions, homotopy Kan extensions, suspension and loops are additive functors.
• The canonical triangulated structure — Theorem 3.1. In a strong stable derivator, every value is triangulated, with the suspension and the triangles of cofibre diagrams; rotation carries the sign of the flip.
• The octahedron — Proposition 3.5. In a strong stable derivator, composable morphisms have triangles on both morphisms, their composite and the cofibres that form an octahedron, read off one diagram.
• Exact functors — Proposition 1.2. In a strong stable derivator, restriction and homotopy Kan extensions are exact for the canonical triangulated structures.
• Dold-Puppe sequences — Proposition 2.1. In a strong stable derivator, the rotations of a distinguished triangle give long exact sequences under Hom(T, -) and Hom(-, T).
• Comparison with the classical triangles — Theorem 3.2. For every ring R, the canonical triangles of the derivator of R are the classical cone triangles of D(R) once the suspension is identified with the shift through the horizontal leg of the suspension square; through the vertical leg every connecting map changes sign.

Reading order

Nine lessons, to be read in order. The first five construct derivators and show that the derivator of a ring is strong and stable; the last four prove that every strong stable derivator has additive, triangulated values, and compare the result with the classical triangles of the derived category of a ring. Complexes, cones, localization and resolutions are taken from the course Derived categories and sheaf operations.