| import OAI.Algebra.Finitistic.Bar.FlatSupportedAssembly |
| import OAI.Algebra.Finitistic.Bar.BlockCoordinateGenerators |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt |
| open scoped TensorProduct DirectSum |
| noncomputable section |
| namespace DerivedBar.ParityBridge |
| open DerivedBar.Multibar DerivedBar.RecursiveTotal |
| |
| private theorem assembly_eq {C : Type*} [Category C] {A B V W U T : C} |
| {l : A ⟶ B} {e : B ⟶ V} {f : V ⟶ W} {i : A ⟶ U} |
| {s : U ⟶ V} {a : U ⟶ T} {b : T ⟶ W} {t : A ⟶ W} {u : B ⟶ W} |
| (h₁ : l ≫ e = i ≫ s) (h₂ : s ≫ f = a ≫ b) |
| (h₃ : i ≫ a ≫ b = t) (h₄ : l ≫ u = t) : l ≫ e ≫ f = l ≫ u := by |
| rw [← Category.assoc, h₁, Category.assoc, h₂, h₃, h₄] |
|
|
| universe u |
| variable (k D X : Type u) [Field k] [Ring D] [Algebra k D] |
| [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| local instance (V : ModuleCat.{u} D) : Module k V := BimoduleTensor.scalarModule k D V |
| local instance (V : ModuleCat.{u} D) : IsScalarTower k D V := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| local instance indexedAssemblyComparisonIndexDecidableEq (r n : ℕ) : |
| DecidableEq (Index r n) := Classical.decEq _ |
| variable (N : ModuleCat.{u} D) |
| local notation "A" => TrivSqZeroExt D X |
| local notation "XB" => BimoduleTensor.twoSidedBarFunctor k D X |
| local notation "Z" => Functor.obj (ChainComplex.single₀ (ModuleCat D)) N |
| local notation "B" => (fun _ : ℕ => XB) |
| local notation "c" => ComplexShape.down ℕ |
| local instance : Module A N := inflatedModule D X N |
| local instance : IsScalarTower k A N := IsScalarTower.of_algebraMap_smul |
| (fun a v => IsScalarTower.algebraMap_smul D a v) |
|
|
| local instance indexedAssemblyComparisonShapeDecidableEq (n r : ℕ) : |
| DecidableEq (Multibar.Indexed.Shape n r) := Classical.decEq _ |
|
|
| def shapeAssembly (r n : ℕ) : indexedModule k D X N r n ⟶ |
| ModuleCat.of D (D ⊗[k] term k A N (r+n)) := |
| ModuleCat.ofHom ((SquareZero.twoSidedWeight k D X N (r+n) r).subtype.comp |
| (Multibar.Indexed.decomposition k D X N (r+n) r).toLinearMap) |
|
|
| theorem shapeAssembly_injective (r n : ℕ) : Function.Injective (shapeAssembly k D X N r n).hom := by |
| intro v w h |
| exact (Multibar.Indexed.decomposition k D X N (r+n) r).injective (Subtype.ext h) |
|
|
| @[reassoc] theorem shapeAssembly_lof (r n : ℕ) (p : Multibar.Indexed.Shape (r+n) r) : |
| ModuleCat.ofHom (DirectSum.lof D _ (Multibar.Indexed.outerTensor k D X N) p) ≫ |
| shapeAssembly k D X N r n = ModuleCat.ofHom (Multibar.Indexed.outerInclusion k D X N p) := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro v |
| change (Multibar.Indexed.assemble k D X N (r+n) r |
| (DirectSum.lof D _ _ p v)).val = _ |
| simp only [Multibar.Indexed.assemble, DirectSum.toModule_lof, LinearMap.codRestrict_apply] |
| rfl |
|
|
| theorem outerInclusion_eqToHom {n r : ℕ} (p q : Multibar.Indexed.Shape n r) (h : p=q) : |
| eqToHom (congrArg (fun t => ModuleCat.of D (Multibar.Indexed.outerTensor k D X N t)) h) ≫ |
| ModuleCat.ofHom (Multibar.Indexed.outerInclusion k D X N q) = |
| ModuleCat.ofHom (Multibar.Indexed.outerInclusion k D X N p) := by |
| subst q |
| simp |
|
|
| theorem reindexedModuleSummand_assembly (r n : ℕ) (p : Multibar.Indexed.Shape (r+n) r) : |
| (reindexedModuleSummandIso k D X N r n p).inv ≫ |
| (moduleSummandIso k D X N r n ((moduleShapeEquiv r n).symm p)).hom ≫ |
| ModuleCat.ofHom (Multibar.Indexed.outerInclusion k D X N |
| (moduleShapeEquiv r n ((moduleShapeEquiv r n).symm p))) = |
| ModuleCat.ofHom (Multibar.Indexed.outerInclusion k D X N p) := by |
| simp only [reindexedModuleSummandIso, Iso.trans_inv, Category.assoc, Iso.inv_hom_id_assoc] |
| exact outerInclusion_eqToHom k D X N _ _ ((moduleShapeEquiv r n).apply_symm_apply p).symm |
|
|
| /-- The new embedding is exactly the old Shape assembly after the old inverse |
| coordinates. No new differential or replacement coordinate system is used. -/ |
| theorem moduleFlatCoordinates_inv_flatEmbedding (r n : ℕ) : |
| (moduleFlatCoordinates k D X N r n).inv ≫ flatEmbedding k D X N r n = |
| shapeAssembly k D X N r n := by |
| apply ModuleCat.hom_ext |
| apply DirectSum.linearMap_ext |
| intro p |
| change (ModuleCat.ofHom (DirectSum.lof D _ (Multibar.Indexed.outerTensor k D X N) p) ≫ |
| (moduleFlatCoordinates k D X N r n).inv ≫ flatEmbedding k D X N r n).hom = |
| (ModuleCat.ofHom (DirectSum.lof D _ (Multibar.Indexed.outerTensor k D X N) p) ≫ |
| shapeAssembly k D X N r n).hom |
| apply congrArg ModuleCat.Hom.hom |
| exact assembly_eq (moduleFlatCoordinates_inv_lof k D X N r n p) |
| (flatEmbedding_supported k D X N r n ((moduleShapeEquiv r n).symm p)) |
| (reindexedModuleSummand_assembly k D X N r n p) (shapeAssembly_lof k D X N r n p) |
|
|
| theorem flatEmbedding_coordinates (r n : ℕ) : flatEmbedding k D X N r n = |
| (moduleFlatCoordinates k D X N r n).hom ≫ shapeAssembly k D X N r n := by |
| apply (cancel_epi (moduleFlatCoordinates k D X N r n).inv).mp |
| rw [moduleFlatCoordinates_inv_flatEmbedding, Iso.inv_hom_id_assoc] |
| end DerivedBar.ParityBridge |
|
|
| end |
|
|
| end OAI |
|
|
