structure Lec23.DeltaFunctor (A : Type u) [CategoryTheory.Category.{v, u} A] [CategoryTheory.Abelian A] (B : Type u_1) [CategoryTheory.Category.{u_2, u_1} B] [CategoryTheory.Abelian B] : Type (max (max (max u u_1) u_2) v)

A δ-functor T = (Tⁿ)_{n ≥ 0} : A → B: additive functors with T⁰ left exact and natural connecting homomorphisms δⁿ fitting into long exact sequences for every short exact sequence of A.

• T : ℕ → CategoryTheory.Functor A B
• additive (n : ℕ) : (self.T n).Additive
• leftExact : CategoryTheory.Limits.PreservesFiniteLimits (self.T 0)
• δ (n : ℕ) (S : CategoryTheory.ShortComplex A) : S.ShortExact → ((self.T n).obj S.X₃ ⟶ (self.T (n + 1)).obj S.X₁)
• δ_comp (n : ℕ) (S : CategoryTheory.ShortComplex A) (hS : S.ShortExact) : CategoryTheory.CategoryStruct.comp (self.δ n S hS) ((self.T (n + 1)).map S.f) = 0
• comp_δ (n : ℕ) (S : CategoryTheory.ShortComplex A) (hS : S.ShortExact) : CategoryTheory.CategoryStruct.comp ((self.T n).map S.g) (self.δ n S hS) = 0
• exact₁ (n : ℕ) (S : CategoryTheory.ShortComplex A) (hS : S.ShortExact) : { X₁ := (self.T n).obj S.X₃, X₂ := (self.T (n + 1)).obj S.X₁, X₃ := (self.T (n + 1)).obj S.X₂, f := self.δ n S hS, g := (self.T (n + 1)).map S.f, zero := ⋯ }.Exact
• exact₂ (n : ℕ) (S : CategoryTheory.ShortComplex A) (_hS : S.ShortExact) : { X₁ := (self.T n).obj S.X₁, X₂ := (self.T n).obj S.X₂, X₃ := (self.T n).obj S.X₃, f := (self.T n).map S.f, g := (self.T n).map S.g, zero := ⋯ }.Exact
• exact₃ (n : ℕ) (S : CategoryTheory.ShortComplex A) (hS : S.ShortExact) : { X₁ := (self.T n).obj S.X₂, X₂ := (self.T n).obj S.X₃, X₃ := (self.T (n + 1)).obj S.X₁, f := (self.T n).map S.g, g := self.δ n S hS, zero := ⋯ }.Exact
• δ_natural (n : ℕ) (S S' : CategoryTheory.ShortComplex A) (hS : S.ShortExact) (hS' : S'.ShortExact) (φ : S ⟶ S') : CategoryTheory.CategoryStruct.comp ((self.T n).map φ.τ₃) (self.δ n S' hS') = CategoryTheory.CategoryStruct.comp (self.δ n S hS) ((self.T (n + 1)).map φ.τ₁)

structure Lec23.IsUniversalDeltaFunctor {A : Type u} [CategoryTheory.Category.{v, u} A] [CategoryTheory.Abelian A] {B : Type u_1} [CategoryTheory.Category.{u_2, u_1} B] [CategoryTheory.Abelian B] (T : DeltaFunctor A B) : Prop

A δ-functor T is universal if every natural transformation S⁰ → T⁰ from another δ-functor extends to a unique morphism of δ-functors S → T.

• exists_hom (S : DeltaFunctor A B) (η₀ : S.T 0 ⟶ T.T 0) : ∃ (η : (n : ℕ) → S.T n ⟶ T.T n), η 0 = η₀ ∧ ∀ (n : ℕ) (SC : CategoryTheory.ShortComplex A) (hSC : SC.ShortExact), CategoryTheory.CategoryStruct.comp ((η n).app SC.X₃) (T.δ n SC hSC) = CategoryTheory.CategoryStruct.comp (S.δ n SC hSC) ((η (n + 1)).app SC.X₁)
• unique (S : DeltaFunctor A B) (η₀ : S.T 0 ⟶ T.T 0) (η η' : (n : ℕ) → S.T n ⟶ T.T n) : η 0 = η₀ → η' 0 = η₀ → (∀ (n : ℕ) (SC : CategoryTheory.ShortComplex A) (hSC : SC.ShortExact), CategoryTheory.CategoryStruct.comp ((η n).app SC.X₃) (T.δ n SC hSC) = CategoryTheory.CategoryStruct.comp (S.δ n SC hSC) ((η (n + 1)).app SC.X₁)) → (∀ (n : ℕ) (SC : CategoryTheory.ShortComplex A) (hSC : SC.ShortExact), CategoryTheory.CategoryStruct.comp ((η' n).app SC.X₃) (T.δ n SC hSC) = CategoryTheory.CategoryStruct.comp (S.δ n SC hSC) ((η' (n + 1)).app SC.X₁)) → ∀ (n : ℕ), η n = η' n

def Lec23.DeltaFunctor.IsEffaceable {A : Type u} [CategoryTheory.Category.{v, u} A] [CategoryTheory.Abelian A] {B : Type u_1} [CategoryTheory.Category.{u_2, u_1} B] [CategoryTheory.Abelian B] (T : DeltaFunctor A B) : Prop

A δ-functor is effaceable if for every n > 0 and every object M, there is a monomorphism M ↪ N killed by Tⁿ. Grothendieck's criterion says effaceable δ-functors are universal.

theorem Lec23.effaceable_deltaFunctor_isUniversal {A : Type u} [CategoryTheory.Category.{v, u} A] [CategoryTheory.Abelian A] {B : Type u_1} [CategoryTheory.Category.{u_2, u_1} B] [CategoryTheory.Abelian B] (T : DeltaFunctor A B) (hT : T.IsEffaceable) : IsUniversalDeltaFunctor T

Proposition 41 (Grothendieck's criterion): An effaceable δ-functor is universal.

theorem Lec23.rightDerived_vanishes_on_injectives {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) [CategoryTheory.Injective X] : CategoryTheory.Limits.IsZero ((F.rightDerived (n + 1)).obj X)

The higher right derived functors R^{n+1}F vanish on injective objects.

theorem Lec23.rightDerived_effaceable {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.EnoughInjectives C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [F.Additive] (n : ℕ) (hn : 0 < n) (M : C) : ∃ (I : C) (φ : M ⟶ I), CategoryTheory.Mono φ ∧ CategoryTheory.Limits.IsZero ((F.rightDerived n).obj I)

In a category with enough injectives, the δ-functor RF = (RⁿF)_{n ≥ 0} is effaceable: every object embeds into an injective on which RⁿF vanishes for n > 0.

noncomputable def Lec23.prop42_connectingHom {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Abelian C] {ι : Type u_2} {c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (hS : S.ShortExact) (i j : ι) (hij : c.Rel i j) : S.X₃.homology i ⟶ S.X₁.homology j

Proposition 42 (snake lemma): the connecting homomorphism Hⁱ(S.X₃) → Hʲ(S.X₁) for a short exact sequence of complexes.

theorem Lec23.prop42_exactness_at_X1 {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Abelian C] {ι : Type u_2} {c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (hS : S.ShortExact) (i j : ι) (hij : c.Rel i j) : { X₁ := S.X₃.homology i, X₂ := S.X₁.homology j, X₃ := S.X₂.homology j, f := hS.δ i j hij, g := HomologicalComplex.homologyMap S.f j, zero := ⋯ }.Exact

Proposition 42 (long exact sequence): Exactness at Hʲ(X₁) in the homology long exact sequence associated with a short exact sequence of complexes.

theorem Lec23.prop42_exactness_at_X2 {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Abelian C] {ι : Type u_2} {c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (hS : S.ShortExact) (i : ι) : { X₁ := S.X₁.homology i, X₂ := S.X₂.homology i, X₃ := S.X₃.homology i, f := HomologicalComplex.homologyMap S.f i, g := HomologicalComplex.homologyMap S.g i, zero := ⋯ }.Exact

Proposition 42 (long exact sequence): Exactness at Hⁱ(X₂) in the homology long exact sequence associated with a short exact sequence of complexes.

theorem Lec23.prop42_exactness_at_X3 {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Abelian C] {ι : Type u_2} {c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (hS : S.ShortExact) (i j : ι) (hij : c.Rel i j) : { X₁ := S.X₂.homology i, X₂ := S.X₃.homology i, X₃ := S.X₁.homology j, f := HomologicalComplex.homologyMap S.g i, g := hS.δ i j hij, zero := ⋯ }.Exact

Proposition 42 (long exact sequence): Exactness at Hⁱ(X₃) in the homology long exact sequence associated with a short exact sequence of complexes.

structure Lec23.Resolution {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] (M : C) : Type (max u v)

A (right) resolution of M: a cochain complex K^• together with a quasi-isomorphism (single₀ M) → K^•.

• cocomplex : CochainComplex C ℕ
• hasHomology (i : ℕ) : HomologicalComplex.HasHomology self.cocomplex i
• ι : (CochainComplex.single₀ C).obj M ⟶ self.cocomplex
• quasiIso : QuasiIso self.ι

def Lec23.InjectiveResolution.toResolution {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {M : C} (I : CategoryTheory.InjectiveResolution M) : Resolution M

Every injective resolution gives rise to a general resolution by forgetting injectivity.

def Lec23.IsAdjustedToFunctor {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) : Prop

M is adjusted to F (also called F-acyclic) if every higher right derived functor of F vanishes on M. Such objects can replace injectives in computing RⁿF.

theorem Lec23.injective_isAdjustedToFunctor {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] : IsAdjustedToFunctor F M

Every injective object is F-acyclic for any additive functor F.

theorem Lec23.injRes_objects_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} (I : CategoryTheory.InjectiveResolution M) (n : ℕ) : IsAdjustedToFunctor F (I.cocomplex.X n)

The objects in an injective resolution are all F-acyclic.

theorem Lec23.prop43_canonical_map_exists {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 : 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 (existence of canonical map): For any resolution K of M, there is a canonical map from Hⁿ(F(K)) to (RⁿF)(M).

theorem Lec23.prop43_iso_when_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] (n : ℕ) {M : C} (K : Resolution M) (hadj : ∀ (i : ℕ), 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 (adjusted resolution): When every term of the resolution K is F-acyclic, the canonical map Hⁿ(F(K)) → (RⁿF)(M) is an isomorphism.

noncomputable def Lec23.derived_functor_via_resolution {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 canonical identification (RⁿF)(X) ≅ Hⁿ(F(I•)) for any injective resolution I• of X.

instance Lec23.enoughInjectives_implies_hasInjectiveResolutions (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.EnoughInjectives C] : CategoryTheory.HasInjectiveResolutions C

An abelian category with enough injectives admits injective resolutions.

noncomputable def Lec23.H0_iso_self {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] [CategoryTheory.Limits.PreservesFiniteLimits F] : F.rightDerived 0 ≅ F

The zeroth right derived functor of a left exact additive functor F is naturally isomorphic to F itself.