File size: 4,903 Bytes
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 | import OAI.Geometry.SingularHomology.HomologyCycle
namespace OAI
noncomputable section
open CategoryTheory
namespace Problem340
section ChainMaps
variable {R : Type} [Ring R]
variable {C D : ChainComplex (ModuleCat R) ℕ}
variable {A B : ℕ → ModuleCat R}
def chainMapOfLinearModels
(e : ∀ n, A n ≃ₗ[R] C.X n) (f : ∀ n, B n ≃ₗ[R] D.X n)
(dA : ∀ n, A (n + 1) →ₗ[R] A n) (dB : ∀ n, B (n + 1) →ₗ[R] B n)
(he : ∀ n x, (C.d (n + 1) n).hom (e (n + 1) x) = e n (dA n x))
(hf : ∀ n x, (D.d (n + 1) n).hom (f (n + 1) x) = f n (dB n x))
(T : ∀ n, A n →ₗ[R] B n)
(hT : ∀ n x, dB n (T (n + 1) x) = T n (dA n x)) : C ⟶ D where
f n := ModuleCat.ofHom ((f n).toLinearMap.comp ((T n).comp (e n).symm.toLinearMap))
comm' i j hij := by
change j + 1 = i at hij
subst i
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
obtain ⟨y, rfl⟩ := (e (j + 1)).surjective x
change (D.d (j + 1) j).hom
(f (j + 1) (T (j + 1) ((e (j + 1)).symm (e (j + 1) y)))) =
f j (T j ((e j).symm ((C.d (j + 1) j).hom (e (j + 1) y))))
simp only [LinearEquiv.symm_apply_apply, hf, he, hT]
@[simp]
theorem chainMapOfLinearModels_apply
(e : ∀ n, A n ≃ₗ[R] C.X n) (f : ∀ n, B n ≃ₗ[R] D.X n)
(dA : ∀ n, A (n + 1) →ₗ[R] A n) (dB : ∀ n, B (n + 1) →ₗ[R] B n)
(he : ∀ n x, (C.d (n + 1) n).hom (e (n + 1) x) = e n (dA n x))
(hf : ∀ n x, (D.d (n + 1) n).hom (f (n + 1) x) = f n (dB n x))
(T : ∀ n, A n →ₗ[R] B n)
(hT : ∀ n x, dB n (T (n + 1) x) = T n (dA n x))
(n : ℕ) (x : A n) :
((chainMapOfLinearModels e f dA dB he hf T hT).f n).hom (e n x) = f n (T n x) := by
simp [chainMapOfLinearModels]
end ChainMaps
section Homotopies
variable {V : Type*} [Category V] [Preadditive V]
variable {C D : ChainComplex V ℕ} (f g : C ⟶ D)
def chainHomotopyMatrix (h : ∀ n, C.X n ⟶ D.X (n + 1)) (i j : ℕ) : C.X i ⟶ D.X j :=
if hij : i + 1 = j then h i ≫ eqToHom (congrArg D.X hij) else 0
@[simp]
theorem chainHomotopyMatrix_succ (h : ∀ n, C.X n ⟶ D.X (n + 1)) (i : ℕ) :
chainHomotopyMatrix h i (i + 1) = h i := by
simp [chainHomotopyMatrix]
def chainHomotopyOfComponents (h : ∀ n, C.X n ⟶ D.X (n + 1))
(hzero : f.f 0 = h 0 ≫ D.d 1 0 + g.f 0)
(hsucc : ∀ n, f.f (n + 1) =
C.d (n + 1) n ≫ h n + h (n + 1) ≫ D.d (n + 2) (n + 1) + g.f (n + 1)) :
Homotopy f g where
hom := chainHomotopyMatrix h
zero i j hij := by
change ¬ i + 1 = j at hij
simp [chainHomotopyMatrix, hij]
comm i := by
cases i with
| zero =>
rw [Homotopy.dNext_zero_chainComplex, Homotopy.prevD_chainComplex]
simpa using hzero
| succ n =>
rw [Homotopy.dNext_succ_chainComplex, Homotopy.prevD_chainComplex]
simpa using hsucc n
end Homotopies
section LinearHomotopies
variable {R : Type} [Ring R]
variable {C D : ChainComplex (ModuleCat R) ℕ}
variable {A B : ℕ → ModuleCat R}
variable (e : ∀ n, A n ≃ₗ[R] C.X n) (f : ∀ n, B n ≃ₗ[R] D.X n)
variable (dA : ∀ n, A (n + 1) →ₗ[R] A n) (dB : ∀ n, B (n + 1) →ₗ[R] B n)
variable (he : ∀ n x, (C.d (n + 1) n).hom (e (n + 1) x) = e n (dA n x))
variable (hf : ∀ n x, (D.d (n + 1) n).hom (f (n + 1) x) = f n (dB n x))
variable (S T : ∀ n, A n →ₗ[R] B n)
variable (hS : ∀ n x, dB n (S (n + 1) x) = S n (dA n x))
variable (hT : ∀ n x, dB n (T (n + 1) x) = T n (dA n x))
def chainHomotopyOfLinearModels
(H : ∀ n, A n →ₗ[R] B (n + 1))
(hzero : ∀ x, S 0 x = dB 0 (H 0 x) + T 0 x)
(hsucc : ∀ n x, S (n + 1) x =
H n (dA n x) + dB (n + 1) (H (n + 1) x) + T (n + 1) x) :
Homotopy (chainMapOfLinearModels e f dA dB he hf S hS)
(chainMapOfLinearModels e f dA dB he hf T hT) := by
let h : ∀ n, C.X n ⟶ D.X (n + 1) := fun n =>
ModuleCat.ofHom ((f (n + 1)).toLinearMap.comp
((H n).comp (e n).symm.toLinearMap))
apply chainHomotopyOfComponents _ _ h
· apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
obtain ⟨y, rfl⟩ := (e 0).surjective x
simp [chainMapOfLinearModels, h, hf, hzero, map_add]
· intro n
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
obtain ⟨y, rfl⟩ := (e (n + 1)).surjective x
simp [chainMapOfLinearModels, h, he, hf, hsucc, map_add]
theorem homologyMap_eq_of_linearModel_homotopy
(H : ∀ n, A n →ₗ[R] B (n + 1))
(hzero : ∀ x, S 0 x = dB 0 (H 0 x) + T 0 x)
(hsucc : ∀ n x, S (n + 1) x =
H n (dA n x) + dB (n + 1) (H (n + 1) x) + T (n + 1) x)
(n : ℕ) :
HomologicalComplex.homologyMap (chainMapOfLinearModels e f dA dB he hf S hS) n =
HomologicalComplex.homologyMap (chainMapOfLinearModels e f dA dB he hf T hT) n :=
(chainHomotopyOfLinearModels e f dA dB he hf S T hS hT H hzero hsucc).homologyMap_eq n
end LinearHomotopies
end Problem340
end
end OAI
|