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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 | import OAI.Algebra.Finitistic.Localization.ChainRepresentatives
namespace OAI
namespace LittleFinitistic.RectificationTotal
open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex HomComplex
variable {C : Type*} [Category C] [Preadditive C] [HasBinaryBiproducts C]
variable {L₀ L₁ L₂ : CochainComplex C ℤ} (p : L₁ ⟶ L₀) (q : L₂ ⟶ L₁)
(h : Homotopy (q ≫ p) 0)
/-- The relation null-homotopy extends p over the first cone. -/
noncomputable def rectifyingMap : mappingCone q ⟶ L₀ :=
mappingCone.desc q (Cochain.ofHomotopy h) p (by simp [δ_ofHomotopy])
/-- A genuine cochain complex rectifying a three-column homotopy diagram.
The cone convention fixes all differential signs; the three-column term
isomorphism is given below. -/
noncomputable def total : CochainComplex C ℤ := mappingCone (rectifyingMap p q h)
/-- The total complex has exactly the three finite columns. -/
noncomputable def termIso (i : ℤ) :
(total p q h).X i ≅ (L₂.X (i+2) ⊞ L₁.X (i+1)) ⊞ L₀.X i :=
homotopyCofiber.XIsoBiprod (rectifyingMap p q h) i (i+1) (by simp) ≪≫
biprod.mapIso (homotopyCofiber.XIsoBiprod q (i+1) (i+2) (by simp; omega)) (Iso.refl _)
@[simp] theorem total_isZero (i : ℤ) :
IsZero ((total p q h).X i) ↔
(IsZero (L₂.X (i+2)) ∧ IsZero (L₁.X (i+1))) ∧ IsZero (L₀.X i) := by
rw [(termIso p q h i).isZero_iff,biprod_isZero_iff,biprod_isZero_iff]
/-- Bounded columns give a bounded actual total complex. -/
theorem total_bounded (lo hi : ℤ)
(h₀ : ∀ i, i < lo ∨ hi < i → IsZero (L₀.X i))
(h₁ : ∀ i, i < lo ∨ hi < i → IsZero (L₁.X i))
(h₂ : ∀ i, i < lo ∨ hi < i → IsZero (L₂.X i)) :
∀ i, i < lo-2 ∨ hi < i → IsZero ((total p q h).X i) := by
intro i hi'
rw [total_isZero]
exact ⟨⟨h₂ _ (by omega),h₁ _ (by omega)⟩,h₀ _ (by omega)⟩
/-- Termwise projectivity is preserved by the concrete three-column construction. -/
theorem total_projective (h₀ : ∀ i, Projective (L₀.X i))
(h₁ : ∀ i, Projective (L₁.X i)) (h₂ : ∀ i, Projective (L₂.X i)) :
∀ i, Projective ((total p q h).X i) := by
intro i
let := h₀ i
let := h₁ (i+1)
let := h₂ (i+2)
apply Projective.of_iso (termIso p q h i).symm
infer_instance
/-- The original zero column includes by an actual chain map. -/
noncomputable def inclusion : L₀ ⟶ total p q h := mappingCone.inr (rectifyingMap p q h)
/-- The first-column boundary vanishes by a specified chain homotopy in the total.
For the induced B-arrow columns this is exactly the arrow compatibility homotopy. -/
noncomputable def boundaryHomotopy : Homotopy (p ≫ inclusion p q h) 0 := by
let f := rectifyingMap p q h
let H : Homotopy (f ≫ mappingCone.inr f) 0 :=
(Cochain.equivHomotopy _ _).symm ⟨mappingCone.inl f,by simp⟩
have H' := H.compLeft (mappingCone.inr q)
have heq : mappingCone.inr q ≫ f = p := mappingCone.inr_desc _ _ _ _
rw [← Category.assoc,heq,CategoryTheory.Limits.comp_zero] at H'
exact H'
variable {D : Type*} [Category D] [Preadditive D] [HasBinaryBiproducts D]
/-- Projectivity after an additive forgetful functor, needed for right projectivity
of a total built in the category of bimodules. -/
theorem total_projective_after (F : C ⥤ D) [F.Additive]
(h₀ : ∀ i, Projective (F.obj (L₀.X i)))
(h₁ : ∀ i, Projective (F.obj (L₁.X i)))
(h₂ : ∀ i, Projective (F.obj (L₂.X i))) :
∀ i, Projective (F.obj ((total p q h).X i)) := by
intro i
let := h₀ i
let := h₁ (i+1)
let := h₂ (i+2)
let := preservesBinaryBiproducts_of_preservesBiproducts F
let e := F.mapIso (termIso p q h i) ≪≫ F.mapBiprod _ _ ≪≫
biprod.mapIso (F.mapBiprod _ _) (Iso.refl _)
apply Projective.of_iso e.symm
infer_instance
end LittleFinitistic.RectificationTotal
end OAI
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