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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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | 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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