| 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 |
|
|
