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ae00908 | 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 | 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
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