| import OAI.Algebra.Finitistic.Rectification.TensorArrowComparison |
| import OAI.Algebra.Finitistic.Rectification.CanonicalArrowCompatibility |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace LittleFinitistic.FullTensorRectification |
| open ExternalTensor.Section4 QuadraticEncoding |
| 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) |
| variable (N : Type u) [AddCommGroup N] [Module k N] [Module (B P) N] |
|
|
| /-- The actual left B-module complex P tensor_B N. -/ |
| abbrev tensor := ((BimoduleBalancedTensor.functor k (B P) N).mapHomologicalComplex (.up ℤ)).obj |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H) |
|
|
| abbrev source (v : Fin 3) := |
| ((BalancedTensor.tensorFunctor k (B P) N).mapHomologicalComplex (.up ℤ)).obj |
| (VertexHomotopyEquivalence.source P D₀ D₁ D₂ v) |
| abbrev target (v : Fin 3) := |
| ((LeftVertexHomology.vertex k (B P) (ThreeVertexAlgebra.vertex (composition P) v)).mapHomologicalComplex |
| (.up ℤ)).obj (tensor P D₀ D₁ D₂ f g H N) |
|
|
| /-- The original vertex insertion after balanced tensoring and the canonical comparison. -/ |
| def iota (v : Fin 3) : source P D₀ D₁ D₂ N v ⟶ target P D₀ D₁ D₂ f g H N v := |
| ((BalancedTensor.tensorFunctor k (B P) N).mapHomologicalComplex (.up ℤ)).map |
| (VertexHomotopyEquivalence.iota P D₀ D₁ D₂ f g H v) ≫ |
| (TensorArrowComparison.complexIso k (B P) N (ThreeVertexAlgebra.vertex (composition P) v) |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem v) |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H)).hom |
|
|
| /-- The equivalences retain the full left B-valued tensor target and the exact original maps. -/ |
| theorem equivalence (v : Fin 3) (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)) : |
| ∃ E : HomotopyEquiv (source P D₀ D₁ D₂ N v) (target P D₀ D₁ D₂ f g H N v), |
| E.hom = iota P D₀ D₁ D₂ f g H N v := by |
| obtain ⟨j,⟨h₁⟩,⟨h₂⟩⟩ := VertexHomotopyEquivalence.homotopy_inverse P D₀ D₁ D₂ f g H |
| v lo hi b₀ b₁ b₂ p₀ p₁ p₂ |
| let E : HomotopyEquiv (VertexHomotopyEquivalence.source P D₀ D₁ D₂ v) |
| (VertexHomotopyEquivalence.target P D₀ D₁ D₂ f g H v) := |
| ⟨VertexHomotopyEquivalence.iota P D₀ D₁ D₂ f g H v,j,h₁,h₂⟩ |
| exact ⟨((BalancedTensor.tensorFunctor k (B P) N).mapHomotopyEquiv E).trans |
| (HomotopyEquiv.ofIso (TensorArrowComparison.complexIso k (B P) N |
| (ThreeVertexAlgebra.vertex (composition P) v) |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem v) |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H))),rfl⟩ |
|
|
| abbrev arrow01Map (a : Fin (d+1)) := |
| ((LeftVertexHomology.arrow k (B P) (ThreeVertexAlgebra.vertex (composition P) 0) |
| (ThreeVertexAlgebra.vertex (composition P) 1) (arrow01 P a) |
| (CanonicalArrowCompatibility.arrow01_target P a)).mapHomologicalComplex (.up ℤ)).app |
| (tensor P D₀ D₁ D₂ f g H N) |
| abbrev arrow12Map (b : Fin (d+1)) := |
| ((LeftVertexHomology.arrow k (B P) (ThreeVertexAlgebra.vertex (composition P) 1) |
| (ThreeVertexAlgebra.vertex (composition P) 2) (arrow12 P b) |
| (CanonicalArrowCompatibility.arrow12_target P b)).mapHomologicalComplex (.up ℤ)).app |
| (tensor P D₀ D₁ D₂ f g H N) |
|
|
| lemma original_iota0 : VertexHomotopyEquivalence.iota P D₀ D₁ D₂ f g H 0 = |
| FirstVertex.iota P D₀ D₁ D₂ f g H := by rfl |
| lemma original_iota1 : VertexHomotopyEquivalence.iota P D₀ D₁ D₂ f g H 1 = |
| SecondVertex.iota P D₀ D₁ D₂ f g H := by rfl |
| lemma original_iota2 : VertexHomotopyEquivalence.iota P D₀ D₁ D₂ f g H 2 = |
| ThirdVertex.iota P D₀ D₁ D₂ f g H := by rfl |
|
|
| /-- Actual arrow compatibility on the vertex diagram of the full B-valued tensor. -/ |
| def compatibility01 (a : Fin (d+1)) : Homotopy |
| (iota P D₀ D₁ D₂ f g H N 0 ≫ arrow01Map P D₀ D₁ D₂ f g H N a) |
| (((BalancedTensor.tensorFunctor k (B P) N).mapHomologicalComplex (.up ℤ)).map (f a) ≫ |
| iota P D₀ D₁ D₂ f g H N 1) := by |
| have h := ((BalancedTensor.tensorFunctor k (B P) N).mapHomotopy |
| (CanonicalArrowCompatibility.compatibility01 P D₀ D₁ D₂ f g H a)).compRight |
| (TensorArrowComparison.complexIso k (B P) N (ThreeVertexAlgebra.vertex (composition P) 1) |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem 1) |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H)).hom |
| erw [Functor.map_comp_assoc _ (FirstVertex.iota P D₀ D₁ D₂ f g H) |
| (CanonicalArrowCompatibility.arrow01Map P D₀ D₁ D₂ f g H a), |
| Functor.map_comp_assoc _ (f a) (SecondVertex.iota P D₀ D₁ D₂ f g H)] at h |
| have hs := TensorArrowComparison.complex_arrow_square k (B P) N |
| (ThreeVertexAlgebra.vertex (composition P) 0) (ThreeVertexAlgebra.vertex (composition P) 1) |
| (arrow01 P a) ((ThreeVertexAlgebra.directedSystem (composition P)).idem 0) |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem 1) |
| (CanonicalArrowCompatibility.arrow01_target P a) |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H) |
| erw [hs] at h |
| dsimp only [iota] |
| erw [original_iota0, original_iota1, Category.assoc] |
| exact h |
|
|
| /-- The second arrow family retains its original maps as well. -/ |
| def compatibility12 (b : Fin (d+1)) : Homotopy |
| (iota P D₀ D₁ D₂ f g H N 1 ≫ arrow12Map P D₀ D₁ D₂ f g H N b) |
| (((BalancedTensor.tensorFunctor k (B P) N).mapHomologicalComplex (.up ℤ)).map (g b) ≫ |
| iota P D₀ D₁ D₂ f g H N 2) := by |
| have h := ((BalancedTensor.tensorFunctor k (B P) N).mapHomotopy |
| (CanonicalArrowCompatibility.compatibility12 P D₀ D₁ D₂ f g H b)).compRight |
| (TensorArrowComparison.complexIso k (B P) N (ThreeVertexAlgebra.vertex (composition P) 2) |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem 2) |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H)).hom |
| erw [Functor.map_comp_assoc _ (SecondVertex.iota P D₀ D₁ D₂ f g H) |
| (CanonicalArrowCompatibility.arrow12Map P D₀ D₁ D₂ f g H b), |
| Functor.map_comp_assoc _ (g b) (ThirdVertex.iota P D₀ D₁ D₂ f g H)] at h |
| have hs := TensorArrowComparison.complex_arrow_square k (B P) N |
| (ThreeVertexAlgebra.vertex (composition P) 1) (ThreeVertexAlgebra.vertex (composition P) 2) |
| (arrow12 P b) ((ThreeVertexAlgebra.directedSystem (composition P)).idem 1) |
| ((ThreeVertexAlgebra.directedSystem (composition P)).idem 2) |
| (CanonicalArrowCompatibility.arrow12_target P b) |
| (InducedRectification.total (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g H) |
| erw [hs] at h |
| dsimp only [iota] |
| erw [original_iota1, original_iota2, Category.assoc] |
| exact h |
| end LittleFinitistic.FullTensorRectification |
|
|
| end |
|
|
| end OAI |
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