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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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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