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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 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 | import Mathlib.CategoryTheory.Localization.CalculusOfFractions
import Mathlib.CategoryTheory.Linear.LinearFunctor
namespace OAI
namespace LittleFinitistic.DiagramLifting
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
variable {C D : Type*} [Category C] [Category D]
variable (L : C ⥤ D) (W : MorphismProperty C) [L.IsLocalization W]
variable [W.HasRightCalculusOfFractions]
/-- Finitely many localized arrows admit one common source denominator. -/
theorem finite_roof (n : ℕ) (X : C) (Y : Fin n → C)
(f : ∀ i, L.obj X ⟶ L.obj (Y i)) :
∃ (Z : C) (u : Z ⟶ X), W u ∧
∃ (g : ∀ i, Z ⟶ Y i), ∀ i, L.map u ≫ f i = L.map (g i) := by
induction n generalizing X with
| zero => exact ⟨X,𝟙 X,W.id_mem X,(fun i => Fin.elim0 i),fun i => Fin.elim0 i⟩
| succ n ih =>
obtain ⟨φ,hφ⟩ := Localization.exists_rightFraction L W (f 0)
obtain ⟨Z,u,hu,g,hg⟩ := ih φ.X' (fun i => Y i.succ)
(fun i => L.map φ.s ≫ f i.succ)
refine ⟨Z,u ≫ φ.s,W.comp_mem _ _ hu φ.hs,Fin.cases (u ≫ φ.f) g,?_⟩
intro i
refine Fin.cases ?_ (fun j => ?_) i
· simp only [Fin.cases_zero,Functor.map_comp,assoc,hφ,φ.map_s_comp_map]
· simpa only [Fin.cases_succ,Functor.map_comp,assoc] using hg j
section Zero
variable [HasZeroMorphisms C] [HasZeroMorphisms D] [L.PreservesZeroMorphisms]
/-- A finite family vanishing after localization is annihilated by one source denominator. -/
theorem finite_zero_refinement (n : ℕ) (X : C) (Y : Fin n → C)
(f : ∀ i, X ⟶ Y i) (hf : ∀ i, L.map (f i) = 0) :
∃ (Z : C) (u : Z ⟶ X), W u ∧ ∀ i, u ≫ f i = 0 := by
induction n generalizing X with
| zero => exact ⟨X,𝟙 X,W.id_mem X,fun i => Fin.elim0 i⟩
| succ n ih =>
have h0 : L.map (f 0) = L.map (0 : X ⟶ Y 0) := by simpa using hf 0
obtain ⟨X',u,hu,he⟩ := (MorphismProperty.map_eq_iff_precomp L W (f 0) 0).mp h0
have he0 : u ≫ f 0 = 0 := by simpa using he
obtain ⟨Z,v,hv,hg⟩ := ih X' (fun i => Y i.succ) (fun i => u ≫ f i.succ)
(fun i => by simp [hf])
refine ⟨Z,v ≫ u,W.comp_mem _ _ hv hu,?_⟩
intro i
refine Fin.cases ?_ (fun j => ?_) i
· simp [assoc,he0]
· simpa only [assoc] using hg j
end Zero
open scoped BigOperators
variable {k : Type*} [CommRing k] [Preadditive C] [Preadditive D]
variable [Linear k C] [Linear k D] [L.Additive] [L.Linear k]
/-- A finite three-vertex diagram with arbitrary homogeneous quadratic relations
lifts along a right-fraction localization. The displayed denominator maps give
an isomorphism from the localized lift to the original entire diagram. -/
theorem quadratic_diagram_lift (na nb nr : ℕ) (X₀ X₁ X₂ : C)
(a : Fin na → (L.obj X₀ ⟶ L.obj X₁))
(b : Fin nb → (L.obj X₁ ⟶ L.obj X₂))
(r : Fin nr → Fin na → Fin nb → k)
(hr : ∀ ρ, ∑ i, ∑ j, r ρ i j • (a i ≫ b j) = 0) :
∃ (D₀ D₁ : C) (u₀ : D₀ ⟶ X₀) (u₁ : D₁ ⟶ X₁), W u₀ ∧ W u₁ ∧
∃ (f : Fin na → (D₀ ⟶ D₁)) (g : Fin nb → (D₁ ⟶ X₂)),
(∀ i, L.map u₀ ≫ a i = L.map (f i) ≫ L.map u₁) ∧
(∀ j, L.map u₁ ≫ b j = L.map (g j)) ∧
(∀ ρ, ∑ i, ∑ j, r ρ i j • (f i ≫ g j) = 0) := by
classical
obtain ⟨D₁,u₁,hu₁,g,hg⟩ := finite_roof L W nb X₁ (fun _ => X₂) b
have : IsIso (L.map u₁) := Localization.inverts L W _ hu₁
obtain ⟨D₀,u₀,hu₀,f,hf⟩ := finite_roof L W na X₀ (fun _ => D₁)
(fun i => a i ≫ inv (L.map u₁))
have hpath (i : Fin na) (j : Fin nb) :
L.map (f i ≫ g j) = L.map u₀ ≫ (a i ≫ b j) := by
rw [Functor.map_comp,← hf i,← hg j]
simp only [assoc,IsIso.inv_hom_id_assoc]
let relations : Fin nr → (D₀ ⟶ X₂) := fun ρ => ∑ i, ∑ j, r ρ i j • (f i ≫ g j)
have hrel (ρ : Fin nr) : L.map (relations ρ) = 0 := by
dsimp [relations]
simp only [Functor.map_sum,Functor.map_smul,hpath]
have h := congrArg (fun t => L.map u₀ ≫ t) (hr ρ)
simpa only [Preadditive.comp_sum,Linear.comp_smul,comp_zero] using h
obtain ⟨D₀',v,hv,hvrel⟩ := finite_zero_refinement L W nr D₀ (fun _ => X₂) relations hrel
refine ⟨D₀',D₁,v ≫ u₀,u₁,W.comp_mem _ _ hv hu₀,hu₁,
(fun i => v ≫ f i),g,?_,hg,?_⟩
· intro i
have hi := congrArg (fun t => t ≫ L.map u₁) (hf i)
simp only [assoc,IsIso.inv_hom_id,comp_id] at hi
simp only [Functor.map_comp,assoc]
rw [← hi]
· intro ρ
have h := hvrel ρ
dsimp [relations] at h
simpa only [Preadditive.comp_sum,Linear.comp_smul,assoc] using h
include W in
/-- Every finite three-level quadratic diagram in the localized category,
including objects not literally presented as functor images, has an isomorphic lift. -/
theorem finite_quadratic_diagram_lifting (na nb nr : ℕ) (Y₀ Y₁ Y₂ : D)
(a : Fin na → (Y₀ ⟶ Y₁)) (b : Fin nb → (Y₁ ⟶ Y₂))
(r : Fin nr → Fin na → Fin nb → k)
(hr : ∀ ρ, ∑ i, ∑ j, r ρ i j • (a i ≫ b j) = 0) :
∃ (X₀ X₁ X₂ : C) (e₀ : L.obj X₀ ≅ Y₀) (e₁ : L.obj X₁ ≅ Y₁)
(e₂ : L.obj X₂ ≅ Y₂) (f : Fin na → (X₀ ⟶ X₁)) (g : Fin nb → (X₁ ⟶ X₂)),
(∀ i, L.map (f i) ≫ e₁.hom = e₀.hom ≫ a i) ∧
(∀ j, L.map (g j) ≫ e₂.hom = e₁.hom ≫ b j) ∧
(∀ ρ, ∑ i, ∑ j, r ρ i j • (f i ≫ g j) = 0) := by
classical
have := Localization.essSurj L W
let X₀ := L.objPreimage Y₀
let X₁ := L.objPreimage Y₁
let X₂ := L.objPreimage Y₂
let e₀ : L.obj X₀ ≅ Y₀ := L.objObjPreimageIso Y₀
let e₁ : L.obj X₁ ≅ Y₁ := L.objObjPreimageIso Y₁
let e₂ : L.obj X₂ ≅ Y₂ := L.objObjPreimageIso Y₂
let a' := fun i => e₀.hom ≫ a i ≫ e₁.inv
let b' := fun j => e₁.hom ≫ b j ≫ e₂.inv
have hr' (ρ : Fin nr) : ∑ i, ∑ j, r ρ i j • (a' i ≫ b' j) = 0 := by
dsimp [a',b']
simp only [assoc,Iso.inv_hom_id_assoc]
have h := congrArg (fun t => e₀.hom ≫ t ≫ e₂.inv) (hr ρ)
simpa only [Preadditive.comp_sum,Preadditive.sum_comp,Linear.comp_smul,
Linear.smul_comp,assoc,comp_zero,zero_comp] using h
obtain ⟨D₀,D₁,u₀,u₁,hu₀,hu₁,f,g,hf,hg,hrel⟩ :=
quadratic_diagram_lift L W na nb nr X₀ X₁ X₂ a' b' r hr'
refine ⟨D₀,D₁,X₂,(Localization.isoOfHom L W u₀ hu₀) ≪≫ e₀,
(Localization.isoOfHom L W u₁ hu₁) ≪≫ e₁,e₂,f,g,?_,?_,hrel⟩
· intro i
have h := congrArg (fun t => t ≫ e₁.hom) (hf i)
simpa only [a',Iso.trans_hom,Localization.isoOfHom_hom,assoc,
Iso.inv_hom_id,comp_id] using h.symm
· intro j
have h := congrArg (fun t => t ≫ e₂.hom) (hg j)
simpa only [b',Iso.trans_hom,Localization.isoOfHom_hom,assoc,
Iso.inv_hom_id,comp_id] using h.symm
end LittleFinitistic.DiagramLifting
end OAI
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