structure Lec23DeltaFunctorDefs.DeltaFunctorDef43 (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)

Definition 43 (δ-functor): A sequence of additive functors (Tⁿ : A → B)_{n ≥ 0} with T⁰ left exact, equipped with connecting maps δⁿ : Tⁿ(X₃) → T^{n+1}(X₁) for every short exact sequence in A, fitting into long exact sequences that are natural in the SES.

• 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 Lec23DeltaFunctorDefs.IsUniversalDeltaFunctorDef44 {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 : DeltaFunctorDef43 A B) : Prop

Definition 44 (universal δ-functor): A δ-functor T is universal if every natural transformation S⁰ → T⁰ from another δ-functor extends uniquely to a morphism of δ-functors S → T compatible with all connecting homomorphisms.

• exists_hom (S : DeltaFunctorDef43 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 : DeltaFunctorDef43 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 Lec23DeltaFunctorDefs.alternatingSum (d : ℕ) (dims : ℕ → ℕ) : ℤ

The truncated Euler characteristic ∑_{i=0}^{d} (-1)^i · dims i, used to compute sheaf Euler characteristics from cohomological dimensions.

structure Lec23DeltaFunctorDefs.SheafCohomologyData : Type (max (u_1 + 1) (u_2 + 1))

Axiomatic data of a (truncated) sheaf cohomology theory: a class of "sheaves" with finite cohomological dimensions and an additivity axiom for the Euler characteristic along short exact sequences.

• k : Type u_1
• field : Field self.k
• CohSheaf : Type u_2
• d : ℕ
• cohomDim : self.CohSheaf → ℕ → ℕ
• cohom_vanishing (F : self.CohSheaf) (i : ℕ) : self.d < i → self.cohomDim F i = 0
• ShortExactSeq : self.CohSheaf → self.CohSheaf → self.CohSheaf → Prop
• eulerChar_additive (F' F F'' : self.CohSheaf) : self.ShortExactSeq F' F F'' → alternatingSum self.d (self.cohomDim F) = alternatingSum self.d (self.cohomDim F') + alternatingSum self.d (self.cohomDim F'')

def Lec23DeltaFunctorDefs.SheafCohomologyData.eulerChar (data : SheafCohomologyData) (F : data.CohSheaf) : ℤ

The Euler characteristic of a sheaf F, the alternating sum of its cohomology dimensions truncated at the cohomological dimension d.

theorem Lec23DeltaFunctorDefs.SheafCohomologyData.eulerChar_add (data : SheafCohomologyData) (F' F F'' : data.CohSheaf) (hses : data.ShortExactSeq F' F F'') : data.eulerChar F = data.eulerChar F' + data.eulerChar F''

Additivity of the Euler characteristic on short exact sequences: χ(F) = χ(F') + χ(F'').