| import OAI.Algebra.Finitistic.Rectification.VertexZeroTotal |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace LittleFinitistic.FirstVertex |
| open ExternalTensor.Section4 |
| section General |
| variable {C D : Type*} [Category C] [Category D] [Preadditive C] [Preadditive D] |
| [HasBinaryBiproducts C] [HasBinaryBiproducts D] |
| def biprodInlIso {X Y : D} (hY : IsZero Y) : X ≅ X ⊞ Y where |
| hom := biprod.inl |
| inv := biprod.fst |
| hom_inv_id := by simp |
| inv_hom_id := by |
| apply biprod.hom_ext |
| · simp |
| · exact hY.eq_of_tgt _ _ |
|
|
| theorem mapped_inl_isIso (F : C ⥤ D) [F.Additive] (X Y : C) (hY : IsZero (F.obj Y)) : |
| IsIso (F.map (biprod.inl : X ⟶ X ⊞ Y)) := by |
| have := preservesBinaryBiproducts_of_preservesBiproducts F |
| rw [← isIso_comp_right_iff _ (F.mapBiprod X Y).hom] |
| have he : F.map (biprod.inl : X ⟶ X ⊞ Y) ≫ (F.mapBiprod X Y).hom = biprod.inl := by |
| apply biprod.hom_ext |
| · erw [Functor.mapBiprod_hom, Category.assoc, biprod.lift_fst, |
| ← Functor.map_comp, biprod.inl_fst, F.map_id, biprod.inl_fst] |
| · erw [Functor.mapBiprod_hom, Category.assoc, biprod.lift_snd, |
| ← Functor.map_comp, biprod.inl_snd, F.map_zero, biprod.inl_snd] |
| rw [he] |
| exact (biprodInlIso hY).isIso_hom |
| end General |
|
|
| universe u |
| variable {k : Type u} [Field k] {d t : ℕ} (P : QuadraticEncoding.Relations k d t) |
| local notation "B" => QuadraticEncoding.B P |
| local notation "s" => ConcreteQuadraticDiagram.quadratic P |
| abbrev vertexComplex := (VertexZeroTotal.vertexFunctor P).mapHomologicalComplex (.up ℤ) |
| variable (D₀ D₁ D₂ : RightComplex (QuadraticEncoding.B P)) |
| variable (f : Fin (d+1) → (D₀ ⟶ D₁)) (g : Fin (d+1) → (D₁ ⟶ D₂)) |
| variable (H : ∀ r, Homotopy (relationMap (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g r) 0) |
|
|
| /-- The original first induced column includes into the actual zero column. -/ |
| def columnInclusion : (columnComplex k B (s).e₀).obj D₀ ⟶ L0Prime s D₀ D₁ D₂ f g := biprod.inl |
|
|
| theorem columnInclusion_isIso : |
| IsIso ((vertexComplex P).map (columnInclusion P D₀ D₁ D₂ f g)) := by |
| have := preservesBinaryBiproducts_of_preservesBiproducts (vertexComplex P) |
| apply mapped_inl_isIso |
| apply ((vertexComplex P).mapBiprod _ _).isZero_iff.mpr |
| rw [biprod_isZero_iff] |
| exact ⟨VertexColumnComplex.reverse_isZero (QuadraticEncoding.composition P) 0 1 (by decide) D₁, |
| VertexColumnComplex.reverse_isZero (QuadraticEncoding.composition P) 0 2 (by decide) D₂⟩ |
|
|
| /-- The manuscript's original first-vertex insertion into the constructed total. -/ |
| def iota : D₀ ⟶ (vertexComplex P).obj (InducedRectification.total s D₀ D₁ D₂ f g H) := |
| (VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 0 D₀).hom ≫ |
| (vertexComplex P).map (columnInclusion P D₀ D₁ D₂ f g) ≫ |
| (vertexComplex P).map (InducedRectification.inclusion s D₀ D₁ D₂ f g H) |
|
|
| /-- At the first vertex the original insertion is already a chain isomorphism. -/ |
| theorem iota_isIso : IsIso (iota P D₀ D₁ D₂ f g H) := by |
| have := columnInclusion_isIso P D₀ D₁ D₂ f g |
| have := VertexZeroTotal.inclusion_isIso P D₀ D₁ D₂ f g H |
| let e : D₀ ≅ (vertexComplex P).obj ((columnComplex k B (s).e₀).obj D₀) := |
| VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 0 D₀ |
| change IsIso (e.hom ≫ (vertexComplex P).map (columnInclusion P D₀ D₁ D₂ f g) ≫ |
| (vertexComplex P).map (InducedRectification.inclusion s D₀ D₁ D₂ f g H)) |
| infer_instance |
| end LittleFinitistic.FirstVertex |
|
|
| end |
|
|
| end OAI |
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|
