| import OAI.Algebra.Finitistic.Bimodule.BimoduleVertex |
| import OAI.Algebra.Finitistic.Rectification.InducedRectification |
| import OAI.Algebra.Finitistic.Rectification.ArrowCompatibility |
|
|
| namespace OAI |
| open CategoryTheory HomologicalComplex |
| open scoped TensorProduct |
| noncomputable section |
| namespace ExternalTensor.Section4.ArrowVertex.NativeBridge |
| universe u |
| variable (k B : Type u) [Field k] [Ring B] [Algebra k B] |
|
|
| /-- The native and generic functors retain definitionally the same original fixed submodule. -/ |
| theorem vertex_eq (e : B) : vertex k B e = LittleFinitistic.BimoduleVertex.functor k B e := rfl |
|
|
| variable {k B} |
| variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R] |
| variable (s : Quadratic k B (A := A) (J := J) (R := R)) |
| variable (D₀ D₁ D₂ : RightComplex B) |
| variable (f : A → (D₀ ⟶ D₁)) (g : J → (D₁ ⟶ D₂)) |
| variable (H : ∀ r, Homotopy (relationMap s D₀ D₁ D₂ f g r) 0) |
|
|
| /-- The native P is exactly the total already used for actual arrow compatibility. -/ |
| theorem total_eq : actualP s D₀ D₁ D₂ f g H = |
| LittleFinitistic.InducedRectification.total s D₀ D₁ D₂ f g H := rfl |
|
|
| /-- The native zero-column inclusion is exactly the one in the proved endpoints. -/ |
| theorem inclusion_eq : LittleFinitistic.RectificationTotal.inclusion |
| (p s D₀ D₁ D₂ f g) (q s D₀ D₁ D₂ f g) (nullHomotopy s D₀ D₁ D₂ f g H) = |
| LittleFinitistic.InducedRectification.inclusion s D₀ D₁ D₂ f g H := rfl |
|
|
| abbrev nativeVertex (e : B) := |
| (LittleFinitistic.BimoduleVertex.functor k B e).mapHomologicalComplex (ComplexShape.up ℤ) |
|
|
| /-- The direct original insertion in the native P and native fixed-submodule functor. -/ |
| def originalIota (e : B) (he : e * e = e) (D : RightComplex B) |
| (t : (columnComplex k B e).obj D ⟶ L0Prime s D₀ D₁ D₂ f g) : |
| D ⟶ (nativeVertex e).obj (LittleFinitistic.InducedRectification.total s D₀ D₁ D₂ f g H) := |
| diagonalComplex k B e he D ≫ (nativeVertex e).map t ≫ |
| (nativeVertex e).map (LittleFinitistic.InducedRectification.inclusion s D₀ D₁ D₂ f g H) |
|
|
| /-- No comparison hypothesis is needed to identify the proved insertion with native maps. -/ |
| theorem iota_eq_original (e : B) (he : e * e = e) (D : RightComplex B) |
| (t : (columnComplex k B e).obj D ⟶ L0Prime s D₀ D₁ D₂ f g) : |
| iota (p s D₀ D₁ D₂ f g) (q s D₀ D₁ D₂ f g) |
| (nullHomotopy s D₀ D₁ D₂ f g H) e he D t = |
| originalIota s D₀ D₁ D₂ f g H e he D t := by |
| simp only [iota, originalIota, Functor.map_comp] |
| rfl |
|
|
| /-- The native original diagonal map is determined by its tensor formula. -/ |
| theorem diagonal_eq_of_original (e : B) (he : e * e = e) (D : RightComplex B) |
| (j : D ⟶ (nativeVertex e).obj ((columnComplex k B e).obj D)) |
| (hj : ∀ n x, ((j.f n) x).val = generator B e ⊗ₜ[k] x) : |
| j = diagonalComplex k B e he D := by |
| ext n x |
| apply Subtype.ext |
| exact hj n x |
|
|
| /-- Actual arrow-01 compatibility with the native P, vertex functors and inclusions. -/ |
| def nativeCompatibility01 (h₀ : s.e₀ * s.e₀ = s.e₀) (h₁ : s.e₁ * s.e₁ = s.e₁) |
| (a : A) (ha : s.e₁ * s.a a = s.a a) : |
| Homotopy |
| (originalIota s D₀ D₁ D₂ f g H s.e₀ h₀ D₀ (inducedData s D₀ D₁ D₂ f g).in₀ ≫ |
| arrowComplex k B s.e₀ s.e₁ (s.a a) ha |
| (LittleFinitistic.InducedRectification.total s D₀ D₁ D₂ f g H)) |
| (f a ≫ originalIota s D₀ D₁ D₂ f g H s.e₁ h₁ D₁ |
| (inducedData s D₀ D₁ D₂ f g).in₁) := by |
| have ht := compatibility01 s D₀ D₁ D₂ f g H h₀ h₁ a ha |
| dsimp only [iota0, iota1] at ht |
| erw [iota_eq_original s D₀ D₁ D₂ f g H, |
| iota_eq_original s D₀ D₁ D₂ f g H] at ht |
| exact ht |
|
|
| /-- Actual arrow-12 compatibility with the native P, vertex functors and inclusions. -/ |
| def nativeCompatibility12 (h₁ : s.e₁ * s.e₁ = s.e₁) (h₂ : s.e₂ * s.e₂ = s.e₂) |
| (b : J) (hb : s.e₂ * s.b b = s.b b) : |
| Homotopy |
| (originalIota s D₀ D₁ D₂ f g H s.e₁ h₁ D₁ (inducedData s D₀ D₁ D₂ f g).in₁ ≫ |
| arrowComplex k B s.e₁ s.e₂ (s.b b) hb |
| (LittleFinitistic.InducedRectification.total s D₀ D₁ D₂ f g H)) |
| (g b ≫ originalIota s D₀ D₁ D₂ f g H s.e₂ h₂ D₂ |
| (inducedData s D₀ D₁ D₂ f g).in₂) := by |
| have ht := compatibility12 s D₀ D₁ D₂ f g H h₁ h₂ b hb |
| dsimp only [iota1, iota2] at ht |
| erw [iota_eq_original s D₀ D₁ D₂ f g H, |
| iota_eq_original s D₀ D₁ D₂ f g H] at ht |
| exact ht |
| end ExternalTensor.Section4.ArrowVertex.NativeBridge |
|
|
| end |
|
|
| end OAI |
|
|
