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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 | import OAI.Algebra.Finitistic.Encoding.FiniteBasisIso
import OAI.Algebra.Finitistic.Encoding.RelationPathCoordinates
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
open scoped TensorProduct
namespace LittleFinitistic.RelationMatrixSequence
open QuadraticEncoding RelationBasisCoefficients
universe u
variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t)
abbrev relationIso := FiniteBasisIso.iso (k := k) (V := relationSpace P) (RelationSequence.relationBasis P)
abbrev pathIso := PathBasisIso.iso k d
def inclusion := (relationIso P).hom ≫ (RelationSequence.sequence P).f ≫ (pathIso (k := k) (d := d)).inv
def quotient := (pathIso (k := k) (d := d)).hom ≫ (RelationSequence.sequence P).g
private theorem transported_zero {C : Type*} [Category C] [HasZeroMorphisms C]
{X₁ X₂ X₃ Y₁ Y₂ : C} (e₁ : Y₁ ≅ X₁) (e₂ : Y₂ ≅ X₂)
(f : X₁ ⟶ X₂) (g : X₂ ⟶ X₃) (h : f ≫ g = 0) :
(e₁.hom ≫ f ≫ e₂.inv) ≫ (e₂.hom ≫ g) = 0 := by
simp only [Category.assoc, Iso.inv_hom_id_assoc, h, comp_zero]
private theorem transported_comp_hom {C : Type*} [Category C]
{X₁ X₂ Y₁ Y₂ : C} (e₁ : Y₁ ≅ X₁) (e₂ : Y₂ ≅ X₂) (f : X₁ ⟶ X₂) :
e₁.hom ≫ f = (e₁.hom ≫ f ≫ e₂.inv) ≫ e₂.hom := by
simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id]
def sequence : ShortComplex (ModuleCat.{u} k) :=
.mk (inclusion P) (quotient P)
(transported_zero (relationIso P) pathIso _ _ (RelationSequence.sequence P).zero)
/-- The actual independent relation columns and first-arrow path columns are the canonical exact sequence. -/
def sequenceIso : sequence P ≅ RelationSequence.sequence P :=
ShortComplex.isoMk (relationIso P) (pathIso (k := k) (d := d)) (Iso.refl _)
(transported_comp_hom (relationIso P) pathIso _) (by simp [sequence,quotient]; rfl)
theorem shortExact : (sequence P).ShortExact :=
(ShortComplex.shortExact_iff_of_iso (sequenceIso P)).2 (RelationSequence.sequence_shortExact P)
/-- The concrete matrix sequence splits over the base field. -/
def splitting : (sequence P).Splitting := (shortExact P).splittingOfProjective
theorem realized_total_acyclic {C : Type*} [Category C] [Abelian C]
(F : ModuleCat.{u} k ⥤ CochainComplex C ℤ) [F.Additive] :
(RectificationTotal.total ((sequence P).map F).g ((sequence P).map F).f
(Homotopy.ofEq ((sequence P).map F).zero)).Acyclic :=
RectificationAcyclic.short_exact_total_acyclic _ ((splitting P).map F).shortExact
/-- Each middle column maps by the original length-two path quotient. -/
theorem quotient_injection (a : Fin (d+1)) (w : Arrows k d) :
(biproduct.ι (PathBasisIso.family k d) a ≫ quotient P).hom w =
(relationSpace P).mkQ (w ⊗ₜ[k] basis a) := by
change (relationSpace P).mkQ ((biproduct.ι (PathBasisIso.family k d) a ≫
(PathBasisIso.iso k d).hom).hom w) = _
rw [PathBasisIso.injection_apply]
/-- An independent relation column has exactly the source coefficient matrix. -/
theorem inclusion_coordinate (r : RelationIndex P) (a : Fin (d+1)) (c : k) :
(biproduct.ι (FiniteBasisIso.family (k := k) : RelationIndex P → _) r ≫
inclusion P ≫ biproduct.π (PathBasisIso.family k d) a).hom c =
c • ∑ b, basisCoefficient P r a b • basis b := by
have hp : (pathIso (k := k) (d := d)).inv ≫ biproduct.π (PathBasisIso.family k d) a =
ModuleCat.ofHom ((LinearMap.proj a).comp RelationPathCoordinates.coordinates.toLinearMap) := by
simp only [pathIso,PathBasisIso.iso,Iso.trans_inv,Category.assoc,
ModuleCat.biproductIsoPi_inv_comp_π]
rfl
simp only [inclusion, Category.assoc]
erw [Category.assoc (RelationSequence.sequence P).f, hp]
change RelationPathCoordinates.coordinates
(((biproduct.ι (FiniteBasisIso.family (k := k) : RelationIndex P → _) r ≫
(relationIso P).hom).hom c : relationSpace P) : Paths k d) a = _
rw [FiniteBasisIso.injection_apply]
change RelationPathCoordinates.coordinates
(c • ((RelationSequence.relationBasis P r : relationSpace P) : Paths k d)) a = _
rw [map_smul]
exact congrArg (c • ·) (RelationPathCoordinates.basis_coordinate P r a)
end LittleFinitistic.RelationMatrixSequence
end
end OAI
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