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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 | import OAI.Algebra.Finitistic.Encoding.RelationVertexCoordinates
import OAI.Algebra.Finitistic.Localization.FiniteSumTransport
import OAI.Algebra.Finitistic.Rectification.SourceFiltrationComplex
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
open scoped TensorProduct ModuleCat.Algebra
namespace LittleFinitistic.RelationVertexSequence
open QuadraticEncoding RelationBasisCoefficients ExternalTensor.Section4
open ExternalTensor.Section4.SourceFiltration RelationVertexCoordinates
universe u
variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t)
variable (D₀ D₁ D₂ : RightComplex (QuadraticEncoding.B P))
variable (f : Fin (d+1) → (D₀ ⟶ D₁)) (g : Fin (d+1) → (D₁ ⟶ D₂))
local notation "T" => QuadraticEncoding.B P
local notation "s" => ConcreteQuadraticDiagram.quadratic P
local notation "data" => Exactness.data s D₀ D₁ D₂ f g
local instance : HasFiniteBiproducts (RightComplex T) := HasFiniteBiproducts.of_hasFiniteProducts
local instance : HasFiniteBiproducts (BiComplex k T) := HasFiniteBiproducts.of_hasFiniteProducts
abbrev arrowFamily : Fin (d+1) → BiComplex k T := fun _ => (columnComplex k T (s).e₁).obj D₀
abbrev relationFamily : RelationIndex P → BiComplex k T := fun _ => (columnComplex k T (s).e₂).obj D₀
def oneSumIso := FiniteSumTransport.iso (vertex P) (tensor P D₀)
(arrowFamily P D₀) (PathBasisIso.family k d) (fun _ => oneIso P D₀)
def twoSumIso := FiniteSumTransport.iso (vertex P) (tensor P D₀)
(relationFamily P D₀) (FiniteBasisIso.family (k := k) : RelationIndex P → _) (fun _ => twoIso P D₀)
theorem p_square : (oneSumIso P D₀).hom ≫ (tensor P D₀).map (RelationMatrixSequence.quotient P) =
(vertex P).map (p0 data) ≫ (zeroIso P D₀).hom := by
have hg : biproduct.desc (quotientEntry P) = RelationMatrixSequence.quotient P := by
apply biproduct.hom_ext'
intro a
simp [quotientEntry]
rw [← hg]
exact FiniteSumTransport.desc_square (vertex P) (tensor P D₀)
(arrowFamily P D₀) (PathBasisIso.family k d) (fun _ => oneIso P D₀)
(zeroIso P D₀) (data).pa₀ (quotientEntry P) (RelationVertexCoordinates.p_square P D₀)
theorem q_entry (r : RelationIndex P) (a : Fin (d+1)) :
biproduct.ι (relationFamily P D₀) r ≫ q0 data ≫ biproduct.π (arrowFamily P D₀) a =
∑ b, basisCoefficient P r a b •
path k T (s).e₁ (s).e₂ (arrow12 P b) ((s).hb b) D₀ := by
classical
change biproduct.ι (data).W r ≫ q0 data ≫ biproduct.π (data).U a = _
simp only [q0,biproduct.ι_desc_assoc,Preadditive.sum_comp,Category.assoc]
simp only [biproduct.ι_π]
rw [Finset.sum_eq_single a]
· simp [Exactness.data,inducedData,ConcreteQuadraticDiagram.quadratic]
rfl
· intro b hb hba
simp [hba]
· simp
theorem q_square : (twoSumIso P D₀).hom ≫ (tensor P D₀).map (RelationMatrixSequence.inclusion P) =
(vertex P).map (q0 data) ≫ (oneSumIso P D₀).hom := by
apply FiniteSumTransport.matrix_square
intro r a
have he := q_entry P D₀ D₁ D₂ f g r a
change (twoIso P D₀).hom ≫ (tensor P D₀).map (relationEntry P r a) =
(vertex P).map (biproduct.ι (relationFamily P D₀) r ≫ q0 data ≫
biproduct.π (arrowFamily P D₀) a) ≫ (oneIso P D₀).hom
rw [he]
exact RelationVertexCoordinates.q_square P D₀ r a
def originalSequence : ShortComplex (BiComplex k T) := .mk (q0 data) (p0 data) (q0_p0 data)
def sequenceIso : (originalSequence P D₀ D₁ D₂ f g).map (vertex P) ≅
(RelationMatrixSequence.sequence P).map (tensor P D₀) :=
ShortComplex.isoMk (twoSumIso P D₀) (oneSumIso P D₀) (zeroIso P D₀)
(q_square P D₀ D₁ D₂ f g) (p_square P D₀ D₁ D₂ f g)
/-- The original source-zero p/q maps are a short exact relation sequence at the final vertex. -/
theorem shortExact : ((originalSequence P D₀ D₁ D₂ f g).map (vertex P)).ShortExact :=
(ShortComplex.shortExact_iff_of_iso (sequenceIso P D₀ D₁ D₂ f g)).2
(((RelationMatrixSequence.splitting P).map (tensor P D₀)).shortExact)
end LittleFinitistic.RelationVertexSequence
end
end OAI
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