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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 | 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
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