| import OAI.Algebra.Finitistic.Rectification.CanonicalArrowCompatibility |
| import OAI.Algebra.Finitistic.Rectification.TotalFinite |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace LittleFinitistic.RectificationLemma |
| open ExternalTensor.Section4 QuadraticEncoding CanonicalArrowCompatibility |
| universe u |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) |
| variable (D₀ D₁ D₂ : RightComplex (B P)) |
| variable (f : Fin (d+1) → (D₀ ⟶ D₁)) (g : Fin (d+1) → (D₁ ⟶ D₂)) |
| variable (H : ∀ r, Homotopy (relationMap (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g r) 0) |
|
|
| /-- The conclusions of the paper's rectification lemma for the explicitly constructed |
| bimodule complex and original vertex insertion maps. -/ |
| structure Verified (lo hi : ℤ) : Prop where |
| bounded : ∀ n, n < lo-2 ∨ hi < n → IsZero |
| ((InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H).X n) |
| projective : ∀ n, Projective |
| ((InducedRectification.rightTotal (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H).X n) |
| finite : ∀ n, FiniteDimensional k ((forgetK k (B P)).obj |
| ((InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H).X n)) |
| vertices : ∀ v : Fin 3, ∃ j : |
| VertexHomotopyEquivalence.target P D₀ D₁ D₂ f g H v ⟶ |
| VertexHomotopyEquivalence.source P D₀ D₁ D₂ v, |
| Nonempty (Homotopy (VertexHomotopyEquivalence.iota P D₀ D₁ D₂ f g H v ≫ j) (𝟙 _)) ∧ |
| Nonempty (Homotopy (j ≫ VertexHomotopyEquivalence.iota P D₀ D₁ D₂ f g H v) (𝟙 _)) |
| arrows01 : ∀ a, Nonempty (Homotopy |
| (FirstVertex.iota P D₀ D₁ D₂ f g H ≫ arrow01Map P D₀ D₁ D₂ f g H a) |
| (f a ≫ SecondVertex.iota P D₀ D₁ D₂ f g H)) |
| arrows12 : ∀ b, Nonempty (Homotopy |
| (SecondVertex.iota P D₀ D₁ D₂ f g H ≫ arrow12Map P D₀ D₁ D₂ f g H b) |
| (g b ≫ ThirdVertex.iota P D₀ D₁ D₂ f g H)) |
|
|
| /-- Full rectification for arbitrary bounded finite projective input complexes, |
| using only the assigned relation homotopies. No conclusion about the constructed |
| P, its vertices, or its arrow homotopies is assumed. -/ |
| theorem rectification (lo hi : ℤ) |
| (b₀ : ∀ n, n < lo ∨ hi < n → IsZero (D₀.X n)) |
| (b₁ : ∀ n, n < lo ∨ hi < n → IsZero (D₁.X n)) |
| (b₂ : ∀ n, n < lo ∨ hi < n → IsZero (D₂.X n)) |
| (p₀ : ∀ n, Module.Projective (B P)ᵐᵒᵖ (D₀.X n)) |
| (p₁ : ∀ n, Module.Projective (B P)ᵐᵒᵖ (D₁.X n)) |
| (p₂ : ∀ n, Module.Projective (B P)ᵐᵒᵖ (D₂.X n)) |
| (f₀ : ∀ n, Module.Finite (B P)ᵐᵒᵖ (D₀.X n)) |
| (f₁ : ∀ n, Module.Finite (B P)ᵐᵒᵖ (D₁.X n)) |
| (f₂ : ∀ n, Module.Finite (B P)ᵐᵒᵖ (D₂.X n)) : |
| Verified P D₀ D₁ D₂ f g H lo hi := by |
| have := QuadraticEncoding.finiteDimensional P |
| exact { |
| bounded := InducedRectification.total_bounded _ _ _ _ _ _ _ lo hi b₀ b₁ b₂ |
| projective := InducedRectification.rightTotal_projective _ _ _ _ _ _ _ p₀ p₁ p₂ |
| finite := induced_total_finite _ _ _ _ _ _ _ f₀ f₁ f₂ |
| vertices := fun v => VertexHomotopyEquivalence.homotopy_inverse P D₀ D₁ D₂ f g H |
| v lo hi b₀ b₁ b₂ p₀ p₁ p₂ |
| arrows01 := fun a => ⟨CanonicalArrowCompatibility.compatibility01 P D₀ D₁ D₂ f g H a⟩ |
| arrows12 := fun b => ⟨CanonicalArrowCompatibility.compatibility12 P D₀ D₁ D₂ f g H b⟩ } |
| end LittleFinitistic.RectificationLemma |
|
|
| end |
|
|
| end OAI |
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