| import OAI.Algebra.Finitistic.Rectification.ColumnComplex |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| noncomputable section |
| namespace ExternalTensor.Section4 |
| universe u |
| variable {k : Type u} [Field k] |
| variable {C : Type u} [Category C] [Preadditive C] [Linear k C] [HasFiniteBiproducts C] [HasBinaryBiproducts C] |
| variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R] |
|
|
| /-- The concrete column/map interface. No composite or null-homotopy is assumed. |
| The three equations are path naturality, the algebra relations, and the chosen |
| relation homotopies after column induction. -/ |
| structure AssemblyData where |
| V₀ : C |
| V₁ : C |
| V₂ : C |
| U : A → C |
| V : J → C |
| W : R → C |
| pa₀ : ∀ a, U a ⟶ V₀ |
| pa₁ : ∀ a, U a ⟶ V₁ |
| pb₁ : ∀ b, V b ⟶ V₁ |
| pb₂ : ∀ b, V b ⟶ V₂ |
| qa : ∀ relation arrow, J → (W relation ⟶ U arrow) |
| qb : ∀ relation, A → ∀ arrow, W relation ⟶ V arrow |
| rel : ∀ r, W r ⟶ V₂ |
| middle : ∀ r a b, qa r a b ≫ pa₁ a = qb r a b ≫ pb₁ b |
| first : ∀ r, ∑ a, ∑ b, qa r a b ≫ pa₀ a = 0 |
| last : ∀ r, ∑ a, ∑ b, qb r a b ≫ pb₂ b = rel r |
|
|
| namespace AssemblyData |
| variable (s : AssemblyData (C := C) (A := A) (J := J) (R := R)) |
| abbrev L₀ := s.V₀ ⊞ (s.V₁ ⊞ s.V₂) |
| abbrev L₁ := (⨁ s.U) ⊞ (⨁ s.V) |
| abbrev L₂ := ⨁ s.W |
| abbrev in₀ : s.V₀ ⟶ s.L₀ := biprod.inl |
| abbrev in₁ : s.V₁ ⟶ s.L₀ := biprod.inl ≫ biprod.inr |
| abbrev in₂ : s.V₂ ⟶ s.L₀ := biprod.inr ≫ biprod.inr |
|
|
| def p : s.L₁ ⟶ s.L₀ := |
| biprod.desc |
| (biproduct.desc fun a => s.pa₀ a ≫ s.in₀ - s.pa₁ a ≫ s.in₁) |
| (biproduct.desc fun b => s.pb₁ b ≫ s.in₁ - s.pb₂ b ≫ s.in₂) |
|
|
| def q : s.L₂ ⟶ s.L₁ := |
| biproduct.desc fun r => ∑ a, ∑ b, |
| (s.qa r a b ≫ biproduct.ι s.U a ≫ biprod.inl + |
| s.qb r a b ≫ biproduct.ι s.V b ≫ biprod.inr) |
|
|
| /-- The complete cancellation in Section 4, including the minus sign. -/ |
| theorem composite : s.q ≫ s.p = |
| -(biproduct.desc fun r => s.rel r ≫ s.in₂) := by |
| apply biproduct.hom_ext' |
| intro r |
| simp only [q, biproduct.ι_desc_assoc, Preadditive.comp_neg, |
| biproduct.ι_desc] |
| simp only [Preadditive.sum_comp, Preadditive.add_comp, Category.assoc, p, |
| biprod.inl_desc, biprod.inr_desc, biproduct.ι_desc, Preadditive.comp_sub, |
| ] |
| simp only [← Category.assoc, s.middle] |
| simp only [Category.assoc] |
| have cancel (a : A) (b : J) : |
| s.qa r a b ≫ s.pa₀ a ≫ s.in₀ - s.qb r a b ≫ s.pb₁ b ≫ s.in₁ + |
| (s.qb r a b ≫ s.pb₁ b ≫ s.in₁ - s.qb r a b ≫ s.pb₂ b ≫ s.in₂) = |
| s.qa r a b ≫ s.pa₀ a ≫ s.in₀ - s.qb r a b ≫ s.pb₂ b ≫ s.in₂ := by abel |
| simp_rw [cancel, Finset.sum_sub_distrib] |
| have hf := congrArg (fun z => z ≫ s.in₀) (s.first r) |
| have hl := congrArg (fun z => z ≫ s.in₂) (s.last r) |
| simp only [Preadditive.sum_comp, Category.assoc, zero_comp] at hf hl |
| rw [hf, hl] |
| simp |
|
|
| /-- A componentwise specified finite sum of homotopies. -/ |
| def sumHomotopy {E : Type*} [Category E] [Preadditive E] |
| {X Y : CochainComplex E ℤ} {I : Type} [Fintype I] |
| (f : I → (X ⟶ Y)) (H : ∀ i, Homotopy (f i) 0) : |
| Homotopy (∑ i, f i) 0 where |
| hom n m := ∑ i, (H i).hom n m |
| zero n m h := by simp [fun i => (H i).zero n m h] |
| comm n := by |
| change Hom.fAddMonoidHom n (∑ i, f i) = _ |
| rw [map_sum] |
| change (∑ i, (f i).f n) = _ |
| calc |
| _ = ∑ i, (dNext n (H i).hom + prevD n (H i).hom + (0 : X ⟶ Y).f n) := |
| Finset.sum_congr rfl (fun i _ => (H i).comm n) |
| _ = _ := by |
| simp [dNext, prevD, Preadditive.comp_sum, Preadditive.sum_comp, |
| Finset.sum_add_distrib] |
| end AssemblyData |
| end ExternalTensor.Section4 |
|
|
| end |
|
|
| end OAI |
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|
