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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 96 97 98 99 100 101 102 | 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
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