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695bd70 | 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 | import OAI.Geometry.SingularHomology.HomologyCycle
import OAI.Geometry.SingularHomology.HomologyZero
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits Simplicial
namespace Problem340
def realSingularZeroClass (X : Type) [TopologicalSpace X] :
(realSingularChainComplex X).X 0 →ₗ[ℝ] RealSingularHomology 0 X :=
(((realSingularChainComplex X).liftCycles (𝟙 _) 0 (by simp) (by simp)) ≫
(realSingularChainComplex X).homologyπ 0).hom
theorem realSingularZeroClass_eq_moduleHomologyClass
(X : Type) [TopologicalSpace X] (z : (realSingularChainComplex X).X 0) :
realSingularZeroClass X z =
moduleHomologyClass ((realSingularChainComplex X).sc 0) z (by
change ((realSingularChainComplex X).d 0 ((ComplexShape.down ℕ).next 0)).hom z = 0
rw [ChainComplex.next_nat_zero, HomologicalComplex.shape _ 0 0 (by simp)]
rfl) := by
let C := realSingularChainComplex X
erw [moduleHomologyClass_eq_pi]
change (C.homologyπ 0).hom ((C.liftCycles (𝟙 _) 0 (by simp) (by simp)).hom z) = _
apply congrArg (C.homologyπ 0).hom
apply (ModuleCat.mono_iff_injective (C.iCycles 0)).mp inferInstance
have h := congrArg (fun k : C.X 0 ⟶ C.X 0 => k.hom z)
(C.liftCycles_i (𝟙 _) 0 (by simp) (by simp))
change (C.iCycles 0).hom _ = (C.iCycles 0).hom _
exact h.trans (((C.sc 0).moduleCatCyclesIso_inv_iCycles_apply
⟨z, by
change (C.d 0 ((ComplexShape.down ℕ).next 0)).hom z = 0
rw [ChainComplex.next_nat_zero, HomologicalComplex.shape _ 0 0 (by simp)]
rfl⟩).symm)
def realSingularAugmentation (X : Type) [TopologicalSpace X] :
(realSingularChainComplex X).X 0 →ₗ[ℝ] ℝ :=
((TopCat.of X).singularHomology₀ε (ModuleCat.of ℝ ℝ)).hom.comp
(realSingularZeroClass X)
theorem realSingularAugmentation_eq_sum (X : Type) [TopologicalSpace X] :
realSingularAugmentation X =
(Sigma.desc (fun _ : (TopCat.toSSet.obj (TopCat.of X)) _⦋0⦌ =>
𝟙 (ModuleCat.of ℝ ℝ))).hom := by
let K := TopCat.toSSet.obj (TopCat.of X)
let R := ModuleCat.of ℝ ℝ
let C := K.chainComplex R
change (((C.liftCycles (𝟙 _) 0 (by simp) (by simp)) ≫
C.homologyπ 0) ≫ K.homology₀ε R).hom = _
apply congrArg (fun k : C.X 0 ⟶ R => k.hom)
apply SSet.chainComplex_hom_ext
intro x
rw [Category.assoc, HomologicalComplex.comp_liftCycles_assoc]
change C.liftCycles (K.ιChainComplex x) 0 _ _ ≫
C.homologyπ 0 ≫ K.homology₀ε R = _
rw [SSet.liftCycles_ιChainComplex_homologyπ_homology₀ε]
simp only [SSet.ιChainComplex, Sigma.ι_comp_desc]
rfl
theorem realSingularZeroClass_eq_zero_iff
(X : Type) [TopologicalSpace X] (z : (realSingularChainComplex X).X 0) :
realSingularZeroClass X z = 0 ↔
∃ w : (realSingularChainComplex X).X 1,
((realSingularChainComplex X).d 1 0).hom w = z := by
rw [realSingularZeroClass_eq_moduleHomologyClass]
have h := moduleHomologyClass_eq_zero_iff ((realSingularChainComplex X).sc 0)
(show ((realSingularChainComplex X).sc 0).X₂ from z) (by
change ((realSingularChainComplex X).d 0 ((ComplexShape.down ℕ).next 0)).hom z = 0
rw [ChainComplex.next_nat_zero, HomologicalComplex.shape _ 0 0 (by simp)]
rfl)
change _ ↔ ∃ w : (realSingularChainComplex X).X ((ComplexShape.down ℕ).prev 0),
((realSingularChainComplex X).d ((ComplexShape.down ℕ).prev 0) 0).hom w = z at h
rw [ChainComplex.prev ℕ 0] at h
exact h
theorem realSingularAugmentation_eq_zero_iff
(X : Type) [TopologicalSpace X] [PathConnectedSpace X]
(z : (realSingularChainComplex X).X 0) :
realSingularAugmentation X z = 0 ↔
∃ w : (realSingularChainComplex X).X 1,
((realSingularChainComplex X).d 1 0).hom w = z := by
change (realSingularHomologyZeroEquiv X) (realSingularZeroClass X z) = 0 ↔ _
rw [map_eq_zero_iff _ (realSingularHomologyZeroEquiv X).injective,
realSingularZeroClass_eq_zero_iff]
@[simp] theorem realSingularAugmentation_boundary
(X : Type) [TopologicalSpace X] (w : (realSingularChainComplex X).X 1) :
realSingularAugmentation X (((realSingularChainComplex X).d 1 0).hom w) = 0 := by
have h := (realSingularZeroClass_eq_zero_iff X _).mpr ⟨w, rfl⟩
simp only [realSingularAugmentation, LinearMap.comp_apply, h, map_zero]
@[simp] theorem realSingularAugmentation_generator
(X : Type) [TopologicalSpace X]
(σ : (TopCat.toSSet.obj (TopCat.of X)) _⦋0⦌) (r : ℝ) :
realSingularAugmentation X
(((TopCat.toSSet.obj (TopCat.of X)).ιChainComplex (R := ModuleCat.of ℝ ℝ) σ).hom r) = r := by
rw [realSingularAugmentation_eq_sum]
exact congrArg (fun k : ModuleCat.of ℝ ℝ ⟶ ModuleCat.of ℝ ℝ => k.hom r)
(Sigma.ι_comp_desc (fun _ : (TopCat.toSSet.obj (TopCat.of X)) _⦋0⦌ =>
𝟙 (ModuleCat.of ℝ ℝ)) σ)
theorem realSingularAugmentation_naturality
{X Y : Type} [TopologicalSpace X] [TopologicalSpace Y]
(f : X → Y) (hf : Continuous f) :
(realSingularAugmentation Y).comp ((realSingularChainMap f hf).f 0).hom =
realSingularAugmentation X := by
rw [realSingularAugmentation_eq_sum, realSingularAugmentation_eq_sum]
let K := TopCat.toSSet.obj (TopCat.of X)
let L := TopCat.toSSet.obj (TopCat.of Y)
let R := ModuleCat.of ℝ ℝ
let φ : K ⟶ L := TopCat.toSSet.map (TopCat.ofHom (continuousMapOf f hf))
change ((SSet.chainComplexMap φ R).f 0 ≫
Sigma.desc (fun _ : L _⦋0⦌ => 𝟙 R)).hom = _
apply congrArg (fun k : (K.chainComplex R).X 0 ⟶ R => k.hom)
apply SSet.chainComplex_hom_ext
intro σ
rw [← Category.assoc, SSet.ι_chainComplexMap_f]
simp only [SSet.ιChainComplex, Sigma.ι_comp_desc]
rfl
def realSingularAugmentedBoundary (X : Type) [TopologicalSpace X] :
(realSingularChainComplex X).X 1 →ₗ[ℝ] LinearMap.ker (realSingularAugmentation X) :=
((realSingularChainComplex X).d 1 0).hom.codRestrict
(LinearMap.ker (realSingularAugmentation X))
(fun w => realSingularAugmentation_boundary X w)
theorem realSingularAugmentedBoundary_surjective
(X : Type) [TopologicalSpace X] [PathConnectedSpace X] :
Function.Surjective (realSingularAugmentedBoundary X) := by
intro z
obtain ⟨w, hw⟩ := (realSingularAugmentation_eq_zero_iff X z.val).mp z.property
exact ⟨w, Subtype.ext hw⟩
def realSingularZeroFill (X : Type) [TopologicalSpace X] [PathConnectedSpace X] :
LinearMap.ker (realSingularAugmentation X) →ₗ[ℝ] (realSingularChainComplex X).X 1 :=
Classical.choose ((realSingularAugmentedBoundary X).exists_rightInverse_of_surjective
(LinearMap.range_eq_top.mpr (realSingularAugmentedBoundary_surjective X)))
theorem realSingularZeroFill_boundary
(X : Type) [TopologicalSpace X] [PathConnectedSpace X]
(z : LinearMap.ker (realSingularAugmentation X)) :
((realSingularChainComplex X).d 1 0).hom (realSingularZeroFill X z) = z.val := by
have h := LinearMap.congr_fun
(Classical.choose_spec ((realSingularAugmentedBoundary X).exists_rightInverse_of_surjective
(LinearMap.range_eq_top.mpr (realSingularAugmentedBoundary_surjective X)))) z
exact congrArg Subtype.val h
end Problem340
end
end OAI
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