| import Mathlib.Algebra.Homology.HomotopyCategory.Plus | |
| import Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory Limits ZeroObject Pretriangulated | |
| namespace LittleFinitistic.BoundedHomotopy | |
| variable (C : Type*) [Category* C] [Preadditive C] | |
| /-- Strict boundedness of the underlying complex, at both ends. -/ | |
| def boundedComplex : ObjectProperty (CochainComplex C ℤ) := | |
| fun X => ∃ a b : ℤ, X.IsStrictlyGE a ∧ X.IsStrictlyLE b | |
| /-- The actual category of bounded cochain complexes, with chain maps. -/ | |
| abbrev Complex := (boundedComplex C).FullSubcategory | |
| /-- Bounded representatives, with homotopy classes of chain maps. -/ | |
| def bounded : ObjectProperty (HomotopyCategory C (.up ℤ)) := | |
| (boundedComplex C).strictMap (HomotopyCategory.quotient _ _) | |
| @[simp] theorem bounded_quotient_obj_iff (X : CochainComplex C ℤ) : | |
| bounded C ((HomotopyCategory.quotient _ _).obj X) ↔ boundedComplex C X := by | |
| refine ⟨?_, fun h => ⟨_, h⟩⟩ | |
| simp only [bounded, ObjectProperty.strictMap_iff] | |
| rintro ⟨Y, h, he⟩ | |
| obtain rfl : Y = X := congr_arg CategoryTheory.Quotient.as he | |
| exact h | |
| instance [HasZeroObject C] : (bounded C).ContainsZero where | |
| exists_zero := ⟨(HomotopyCategory.quotient _ _).obj 0, | |
| Functor.map_isZero _ (isZero_zero _), by | |
| rw [bounded_quotient_obj_iff] | |
| exact ⟨0, 0, inferInstance, inferInstance⟩⟩ | |
| instance : (bounded C).IsStableUnderShift ℤ where | |
| isStableUnderShiftBy n := | |
| { le_shift := by | |
| intro X hX | |
| obtain ⟨X : CochainComplex _ _, rfl⟩ := X.quotient_obj_surjective | |
| rw [bounded_quotient_obj_iff] at hX | |
| obtain ⟨a, b, ha, hb⟩ := hX | |
| rw [ObjectProperty.prop_shift_iff, HomotopyCategory.shift_quotient_obj, | |
| bounded_quotient_obj_iff] | |
| exact ⟨a-n, b-n, X.isStrictlyGE_shift a n (a-n) (by omega), | |
| X.isStrictlyLE_shift b n (b-n) (by omega)⟩ } | |
| variable {C} | |
| theorem bounded_mappingCone [HasBinaryBiproducts C] | |
| {X Y : CochainComplex C ℤ} (f : X ⟶ Y) | |
| (hX : boundedComplex C X) (hY : boundedComplex C Y) : | |
| boundedComplex C (CochainComplex.mappingCone f) := by | |
| obtain ⟨a,b,ha,hb⟩ := hX | |
| obtain ⟨c,d,hc,hd⟩ := hY | |
| refine ⟨min (a-1) c, max b d, CochainComplex.isStrictlyGE_mappingCone f a c _ | |
| (by omega) (by omega), ?_⟩ | |
| rw [CochainComplex.isStrictlyLE_iff] | |
| intro i hi | |
| rw [CochainComplex.mappingCone.isZero_X_iff] | |
| exact ⟨X.isZero_of_isStrictlyLE b _ (by omega), | |
| Y.isZero_of_isStrictlyLE d _ (by omega)⟩ | |
| variable (C) | |
| instance [HasZeroObject C] [HasBinaryBiproducts C] : | |
| (bounded C).IsTriangulatedClosed₃ where | |
| ext₃' T hT h₁ h₂ := by | |
| have h₁' : boundedComplex C T.obj₁.as := (bounded_quotient_obj_iff _ _).mp h₁ | |
| have h₂' : boundedComplex C T.obj₂.as := (bounded_quotient_obj_iff _ _).mp h₂ | |
| obtain ⟨f : T.obj₁.as ⟶ T.obj₂.as, hf⟩ := | |
| (HomotopyCategory.quotient _ _).map_surjective T.mor₁ | |
| refine ⟨_, ?_, ⟨Triangle.π₃.mapIso (isoTriangleOfIso₁₂ T _ hT | |
| (HomotopyCategory.mappingCone_triangleh_distinguished f) | |
| (Iso.refl _) (Iso.refl _) ?_)⟩⟩ | |
| · exact (bounded_quotient_obj_iff _ _).mpr (bounded_mappingCone f h₁' h₂') | |
| · exact (Category.comp_id _).trans (hf.symm.trans (Category.id_comp _).symm) | |
| instance [HasZeroObject C] [HasBinaryBiproducts C] : (bounded C).IsTriangulated where | |
| toIsTriangulatedClosed₂ := .of_isTriangulatedClosed₃ | |
| /-- The homotopy category Kᵇ(C); boundedness is part of each object. -/ | |
| abbrev K := (bounded C).FullSubcategory | |
| /-- Pass from bounded chain maps to their homotopy classes. -/ | |
| def quotient : Complex C ⥤ K C := | |
| ObjectProperty.lift _ ((boundedComplex C).ι ⋙ HomotopyCategory.quotient _ _) | |
| (fun X => (bounded_quotient_obj_iff _ _).mpr X.property) | |
| instance [HasZeroObject C] [HasBinaryBiproducts C] : IsTriangulated (K C) := inferInstance | |
| end LittleFinitistic.BoundedHomotopy | |
| end | |
| end OAI | |