theorem Lec23AdaptedResolution.prop43_canonical_map {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasInjectiveResolutions C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℕ) {M : C} (K : Lec23.Resolution M) : Nonempty ((HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).obj ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj K.cocomplex) ⟶ (F.rightDerived n).obj M)

Proposition 43 (canonical comparison map): For any resolution K of M, there is a canonical map from Hⁿ(F(K)) to the right derived functor (RⁿF)(M).

theorem Lec23AdaptedResolution.prop43_adapted_resolution_computes_derived {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasInjectiveResolutions C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℕ) {M : C} (K : Lec23.Resolution M) (hadj : ∀ (i : ℕ), Lec23.IsAdjustedToFunctor F (K.cocomplex.X i)) : Nonempty ((HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).obj ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj K.cocomplex) ≅ (F.rightDerived n).obj M)

Proposition 43 (adapted resolution computes derived functor): When every term of a resolution K of M is adjusted to F, the comparison map is an isomorphism, so RⁿF(M) can be computed from Hⁿ(F(K)).

theorem Lec23AdaptedResolution.injective_is_adjusted {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasInjectiveResolutions C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [F.Additive] (M : C) [CategoryTheory.Injective M] : Lec23.IsAdjustedToFunctor F M

Every injective object is adjusted to any additive functor: RⁿF vanishes on injectives for n > 0.

noncomputable def Lec23AdaptedResolution.prop43_injective_resolution_iso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasInjectiveResolutions C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℕ) {X : C} (I : CategoryTheory.InjectiveResolution X) : (F.rightDerived n).obj X ≅ (HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).obj ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)

The standard identification of the right derived functor as the cohomology of F applied to an injective resolution.

noncomputable def Lec23AdaptedResolution.prop43_resolution_uniqueness {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.HasInjectiveResolutions C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℕ) {X : C} (I J : CategoryTheory.InjectiveResolution X) : (HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).obj ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≅ (HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).obj ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj J.cocomplex)

Independence of injective resolution: the cohomology of F applied to two different injective resolutions of X is canonically isomorphic.