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216fb31 | 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 | 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
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