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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 | import OAI.Algebra.Finitistic.Rectification.RectificationTotal
namespace OAI
open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex
open CochainComplex.HomComplex
noncomputable section
namespace LittleFinitistic.RectificationTotal
variable {C : Type*} [Category C] [Preadditive C] [HasBinaryBiproducts C]
theorem biprodMap_comp {U V W X Y Z : C} (f : U ⟶ W) (g : V ⟶ X)
(f' : W ⟶ Y) (g' : X ⟶ Z) :
biprod.map f g ≫ biprod.map f' g' = biprod.map (f ≫ f') (g ≫ g') := by
apply biprod.hom_ext <;> simp
variable {A B A' B' A'' B'' : CochainComplex C ℤ}
/-- A strictly commuting square induces the ordinary diagonal map of cones. -/
def coneMap (q : A ⟶ B) (q' : A' ⟶ B') (u : A ⟶ A') (v : B ⟶ B')
(w : u ≫ q' = q ≫ v) : mappingCone q ⟶ mappingCone q' :=
homotopyCofiber.mapArrowHom q q' (fun j => ⟨j-1, by simp⟩)
(Arrow.homMk u v w)
@[reassoc (attr := simp)]
theorem coneMap_inl (q : A ⟶ B) (q' : A' ⟶ B') (u : A ⟶ A') (v : B ⟶ B')
(w : u ≫ q' = q ≫ v) (i j : ℤ) (hij : i + (-1) = j) :
(mappingCone.inl q).v i j hij ≫ (coneMap q q' u v w).f j =
u.f i ≫ (mappingCone.inl q').v i j hij := by
dsimp [coneMap, homotopyCofiber.mapArrowHom, mappingCone.inl]
erw [homotopyCofiber.inlX_desc_f]
simp
congr 1
exact homotopyCofiber.inrCompHomotopy_hom q' _ i j (by change j+1=i; omega)
@[reassoc (attr := simp)]
theorem coneMap_inr (q : A ⟶ B) (q' : A' ⟶ B') (u : A ⟶ A') (v : B ⟶ B')
(w : u ≫ q' = q ≫ v) (i : ℤ) :
(mappingCone.inr q).f i ≫ (coneMap q q' u v w).f i =
v.f i ≫ (mappingCone.inr q').f i := by
dsimp [coneMap, homotopyCofiber.mapArrowHom, mappingCone.inr]
exact homotopyCofiber.inrX_desc_f _ _ _ _
@[reassoc (attr := simp)]
theorem cone_inl_termIso (q : A ⟶ B) (i j : ℤ) (hij : i+1=j) :
(mappingCone.inl q).v j i (by omega) ≫ (homotopyCofiber.XIsoBiprod q i j hij).hom =
biprod.inl :=
homotopyCofiber.inlX_XIsoBiprod_hom _ _ _ _
@[reassoc (attr := simp)]
theorem cone_inr_termIso (q : A ⟶ B) (i j : ℤ) (hij : i+1=j) :
(mappingCone.inr q).f i ≫ (homotopyCofiber.XIsoBiprod q i j hij).hom =
biprod.inr :=
homotopyCofiber.inrX_XIsoBiprod_hom _ _ _ _
/-- On terms the cone map is exactly the sum of the two supplied maps. -/
theorem coneMap_term (q : A ⟶ B) (q' : A' ⟶ B') (u : A ⟶ A') (v : B ⟶ B')
(w : u ≫ q' = q ≫ v) (i j : ℤ) (hij : i+1=j) :
(coneMap q q' u v w).f i ≫ (homotopyCofiber.XIsoBiprod q' i j hij).hom =
(homotopyCofiber.XIsoBiprod q i j hij).hom ≫ biprod.map (u.f j) (v.f i) := by
apply mappingCone.ext_from q j i hij
· erw [coneMap_inl_assoc, cone_inl_termIso, ← Category.assoc, cone_inl_termIso,
biprod.inl_map]
· erw [coneMap_inr_assoc, cone_inr_termIso, ← Category.assoc, cone_inr_termIso,
biprod.inr_map]
@[simp]
theorem coneMap_id (q : A ⟶ B) : coneMap q q (𝟙 A) (𝟙 B) (by simp) = 𝟙 _ := by
change homotopyCofiber.mapArrowHom q q _ (𝟙 _) = 𝟙 _
exact homotopyCofiber.mapArrowHom_id _ _
theorem coneMap_comp (q : A ⟶ B) (q' : A' ⟶ B') (q'' : A'' ⟶ B'')
(u : A ⟶ A') (v : B ⟶ B') (w : u ≫ q' = q ≫ v)
(u' : A' ⟶ A'') (v' : B' ⟶ B'') (w' : u' ≫ q'' = q' ≫ v') :
coneMap q q'' (u ≫ u') (v ≫ v') (by rw [Category.assoc, w', ← Category.assoc, w, Category.assoc]) =
coneMap q q' u v w ≫ coneMap q' q'' u' v' w' := by
exact homotopyCofiber.mapArrowHom_comp _ _ _ _ (Arrow.homMk u v w) (Arrow.homMk u' v' w')
variable {L₀ L₁ L₂ M₀ M₁ M₂ N₀ N₁ N₂ : CochainComplex C ℤ}
variable {p : L₁ ⟶ L₀} {q : L₂ ⟶ L₁} {h : Homotopy (q ≫ p) 0}
variable {p' : M₁ ⟶ M₀} {q' : M₂ ⟶ M₁} {h' : Homotopy (q' ≫ p') 0}
/-- Morphisms preserve both degree-zero maps and the specified relation
homotopy, which is part of the three-column input data. -/
structure MapData (p : L₁ ⟶ L₀) (q : L₂ ⟶ L₁) (h : Homotopy (q ≫ p) 0)
(p' : M₁ ⟶ M₀) (q' : M₂ ⟶ M₁) (h' : Homotopy (q' ≫ p') 0) where
f₀ : L₀ ⟶ M₀
f₁ : L₁ ⟶ M₁
f₂ : L₂ ⟶ M₂
commP : f₁ ≫ p' = p ≫ f₀
commQ : f₂ ≫ q' = q ≫ f₁
commH : ∀ i j, f₂.f i ≫ h'.hom i j = h.hom i j ≫ f₀.f j
namespace MapData
variable (a : MapData p q h p' q' h')
def inner : mappingCone q ⟶ mappingCone q' := coneMap q q' a.f₂ a.f₁ a.commQ
theorem rectifyingMap_naturality : a.inner ≫ rectifyingMap p' q' h' =
rectifyingMap p q h ≫ a.f₀ := by
ext n
apply mappingCone.ext_from q (n+1) n rfl
· simp only [comp_f, inner, coneMap_inl_assoc, rectifyingMap, mappingCone.inl_v_desc_f_assoc,
mappingCone.inl_v_desc_f]
exact a.commH (n+1) n
· simp only [comp_f, inner, coneMap_inr_assoc, rectifyingMap,
mappingCone.inr_f_desc_f_assoc, mappingCone.inr_f_desc_f]
exact congrArg (fun f => f.f n) a.commP
/-- The induced chain map of the native primary's actual iterated-cone total. -/
def totalMap : total p q h ⟶ total p' q' h' :=
coneMap (rectifyingMap p q h) (rectifyingMap p' q' h') a.inner a.f₀
a.rectifyingMap_naturality
/-- The zero-column inclusion is preserved by the actual induced total map. -/
@[reassoc]
theorem inclusion_totalMap : inclusion p q h ≫ a.totalMap =
a.f₀ ≫ inclusion p' q' h' := by
ext n
exact coneMap_inr _ _ _ _ _ n
/-- The induced total map is the actual diagonal triple under the native
term decomposition, including both cohomological shifts. -/
theorem totalMap_term (n : ℤ) :
a.totalMap.f n ≫ (termIso p' q' h' n).hom =
(termIso p q h n).hom ≫
biprod.map (biprod.map (a.f₂.f (n+2)) (a.f₁.f (n+1))) (a.f₀.f n) := by
unfold totalMap termIso
erw [Iso.trans_hom, Iso.trans_hom, biprod.mapIso_hom, biprod.mapIso_hom, Iso.refl_hom, Iso.refl_hom]
erw [← Category.assoc, coneMap_term]
erw [Category.assoc, Category.assoc, biprodMap_comp, biprodMap_comp,
Category.comp_id, Category.id_comp]
congr 2
exact coneMap_term q q' a.f₂ a.f₁ a.commQ (n+1) (n+2) (by omega)
def id (p : L₁ ⟶ L₀) (q : L₂ ⟶ L₁) (h : Homotopy (q ≫ p) 0) :
MapData p q h p q h where
f₀ := 𝟙 _
f₁ := 𝟙 _
f₂ := 𝟙 _
commP := by simp
commQ := by simp
commH := by intros; simp
variable {p'' : N₁ ⟶ N₀} {q'' : N₂ ⟶ N₁} {h'' : Homotopy (q'' ≫ p'') 0}
def comp (b : MapData p' q' h' p'' q'' h'') : MapData p q h p'' q'' h'' where
f₀ := a.f₀ ≫ b.f₀
f₁ := a.f₁ ≫ b.f₁
f₂ := a.f₂ ≫ b.f₂
commP := by rw [Category.assoc, b.commP, ← Category.assoc, a.commP, Category.assoc]
commQ := by rw [Category.assoc, b.commQ, ← Category.assoc, a.commQ, Category.assoc]
commH i j := by
simp only [comp_f]
rw [Category.assoc, b.commH, ← Category.assoc, a.commH, Category.assoc]
@[simp] theorem totalMap_id : (id p q h).totalMap = 𝟙 _ := by
simp [totalMap, inner, id, coneMap_id, total]
theorem totalMap_comp (b : MapData p' q' h' p'' q'' h'') :
(a.comp b).totalMap = a.totalMap ≫ b.totalMap := by
have hi : (a.comp b).inner = a.inner ≫ b.inner := coneMap_comp _ _ _ _ _ _ _ _ _
refine Eq.trans ?_ (coneMap_comp _ _ _ a.inner a.f₀ a.rectifyingMap_naturality
b.inner b.f₀ b.rectifyingMap_naturality)
change coneMap _ _ (a.comp b).inner (a.f₀ ≫ b.f₀) _ = _
congr 1
end MapData
end LittleFinitistic.RectificationTotal
end
end OAI
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