| import OAI.Algebra.Finitistic.Rectification.SourceFiltrationReal |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex |
| open LittleFinitistic.RectificationTotal |
| noncomputable section |
| namespace ExternalTensor.Section4.SourceFiltration |
| universe u |
| variable {k B : Type u} [Field k] [Ring B] [Algebra k B] |
|
|
| /-- An isomorphism of the two input columns gives an isomorphism of actual cones. -/ |
| def coneIso {X Y X' Y' : BiComplex k B} (p : X ⟶ Y) (p' : X' ⟶ Y') |
| (a : X ≅ X') (b : Y ≅ Y') (w : a.hom ≫ p' = p ≫ b.hom) : |
| mappingCone p ≅ mappingCone p' where |
| hom := coneMap p p' a.hom b.hom w |
| inv := coneMap p' p a.inv b.inv (by |
| apply (cancel_epi a.hom).1 |
| simp only [← Category.assoc, Iso.hom_inv_id, Category.id_comp] |
| rw [w, Category.assoc, Iso.hom_inv_id, Category.comp_id]) |
| hom_inv_id := by rw [← coneMap_comp]; simp |
| inv_hom_id := by rw [← coneMap_comp]; simp |
| |
| variable (X : BiComplex k B) |
|
|
| theorem zeroCone_inclusion_product (n : ℤ) : |
| (mappingCone.inr (0 : Z ⟶ X)).f n ≫ |
| ((homotopyCofiber.XIsoBiprod (0 : Z ⟶ X) n (n+1) rfl) ≪≫ |
| ModuleCat.biprodIsoProd _ _).hom = |
| ModuleCat.ofHom (LinearMap.inr (Enveloping k B) (Z.X (n+1)) (X.X n)) := by |
| erw [Iso.trans_hom, ← Category.assoc, cone_inr_termIso] |
| exact inr_prod _ _ |
|
|
| theorem zeroCone_inclusion_bijective (n : ℤ) : |
| Function.Bijective ((mappingCone.inr (0 : Z ⟶ X)).f n) := by |
| let e := ((homotopyCofiber.XIsoBiprod (0 : Z ⟶ X) n (n+1) rfl) ≪≫ |
| ModuleCat.biprodIsoProd _ _).toLinearEquiv |
| have he (x : X.X n) : e ((mappingCone.inr (0 : Z ⟶ X)).f n x) = (0,x) := |
| congrArg (fun a => a x) (zeroCone_inclusion_product X n) |
| refine ⟨?_, ?_⟩ |
| · intro x y hxy |
| have := congrArg e hxy |
| simpa only [he] using congrArg Prod.snd this |
| · intro y |
| refine ⟨(e y).2, e.injective ?_⟩ |
| rw [he] |
| exact Prod.ext (Subsingleton.elim _ _) rfl |
|
|
| instance zeroCone_inclusion_isIso : IsIso (mappingCone.inr (0 : Z ⟶ X)) := by |
| have (n : ℤ) : IsIso ((mappingCone.inr (0 : Z ⟶ X)).f n) := |
| (ConcreteCategory.isIso_iff_bijective _).2 (zeroCone_inclusion_bijective X n) |
| exact HomologicalComplex.Hom.isIso_of_components _ |
|
|
| /-- The ordinary two-column total identifies with the three-column API having |
| zero relation column; its zero-column inclusion is unchanged. -/ |
| def twoColumnIso {Y : BiComplex k B} (p : X ⟶ Y) |
| (h : Homotopy ((0 : Z ⟶ X) ≫ p) 0) : mappingCone p ≅ total p 0 h := |
| coneIso p (rectifyingMap p 0 h) (asIso (mappingCone.inr (0 : Z ⟶ X))) |
| (Iso.refl _) (by simp [rectifyingMap]) |
|
|
| theorem twoColumnIso_inclusion {Y : BiComplex k B} (p : X ⟶ Y) |
| (h : Homotopy ((0 : Z ⟶ X) ≫ p) 0) : |
| mappingCone.inr p ≫ (twoColumnIso X p h).hom = inclusion p 0 h := by |
| apply HomologicalComplex.hom_ext |
| intro n |
| change (mappingCone.inr p).f n ≫ |
| (coneMap p (rectifyingMap p 0 h) (mappingCone.inr (0 : Z ⟶ X)) (𝟙 Y) _).f n = |
| (mappingCone.inr (rectifyingMap p 0 h)).f n |
| erw [coneMap_inr, id_f, Category.id_comp] |
|
|
| variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R] |
| variable (t : AssemblyData (C := BiComplex k B) (A := A) (J := J) (R := R)) |
|
|
| def f1ConeIso : mappingCone (p1 t) ≅ total (p1 t) 0 (h1 t) := |
| twoColumnIso _ _ _ |
| def grade1ConeIso : mappingCone (pGrade1 t) ≅ total (pGrade1 t) 0 (hGrade1 t) := |
| twoColumnIso _ _ _ |
| end ExternalTensor.Section4.SourceFiltration |
|
|
| end |
|
|
| end OAI |
|
|
