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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 | import OAI.Algebra.Finitistic.Rectification.VertexTensorLinear
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
open scoped TensorProduct ModuleCat.Algebra
namespace LittleFinitistic.RelationVertexCoordinates
open QuadraticEncoding RelationBasisCoefficients ExternalTensor.Section4
universe u
variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t)
variable (D : RightComplex (QuadraticEncoding.B P))
local notation "T" => QuadraticEncoding.B P
local notation "s" => ConcreteQuadraticDiagram.quadratic P
abbrev vertex := AdjacentArrowSum.vertexComplex (composition P) 2
abbrev tensor := TensorComplex.tensorFunctor (k := k) D
def zeroIso := CornerPathTransport.coordinateIso (composition P) D 2 0
(CornerSpaces.corner20Equiv (composition P))
def oneIso := CornerPathTransport.coordinateIso (composition P) D 2 1
(CornerSpaces.corner21Equiv (composition P))
def twoIso := CornerPathTransport.coordinateIso (composition P) D 2 2
(CornerSpaces.diagonalEquiv (composition P) 2)
def quotientEntry (a : Fin (d+1)) :=
biproduct.ι (PathBasisIso.family k d) a ≫ RelationMatrixSequence.quotient P
def relationEntry (r : RelationIndex P) (a : Fin (d+1)) :=
biproduct.ι (FiniteBasisIso.family (k := k) : RelationIndex P → _) r ≫
RelationMatrixSequence.inclusion P ≫ biproduct.π (PathBasisIso.family k d) a
theorem relationEntry_eq (r : RelationIndex P) (a : Fin (d+1)) :
relationEntry P r a = ∑ b, basisCoefficient P r a b • TensorArrowSum.insertion k d b := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro c
change (biproduct.ι _ r ≫ RelationMatrixSequence.inclusion P ≫ biproduct.π _ a).hom c = _
rw [RelationMatrixSequence.inclusion_coordinate]
simp only [Finset.smul_sum,ModuleCat.hom_sum,ModuleCat.hom_smul,
LinearMap.sum_apply,LinearMap.smul_apply]
apply Finset.sum_congr rfl
intro b hb
rw [TensorArrowSum.insertion_apply]
simp only [smul_smul,mul_comm]
/-- The source-zero p entry becomes the actual length-two path quotient. -/
theorem p_square (a : Fin (d+1)) :
(oneIso P D).hom ≫ (tensor P D).map (quotientEntry P a) =
(vertex P).map (path k T (s).e₀ (s).e₁ (arrow01 P a) ((s).ha a) D) ≫
(zeroIso P D).hom := by
apply CornerPathTransport.path_square
intro x
change (CornerSpaces.corner20Equiv (composition P))
(CornerColumnPaths.cornerMap (composition P) 2 0 1 (arrow01 P a) ((s).ha a) x) =
(quotientEntry P a).hom ((CornerSpaces.corner21Equiv (composition P)) x)
rw [show (quotientEntry P a).hom ((CornerSpaces.corner21Equiv (composition P)) x) =
(relationSpace P).mkQ (((CornerSpaces.corner21Equiv (composition P)) x) ⊗ₜ[k] basis a)
from RelationMatrixSequence.quotient_injection P a _]
simpa only [LinearEquiv.symm_apply_apply] using
CornerArrowCoordinates.lengthTwo_coordinate P a ((CornerSpaces.corner21Equiv (composition P)) x)
/-- The source-zero q entry uses the original coefficients and actual b-arrow maps. -/
theorem q_square (r : RelationIndex P) (a : Fin (d+1)) :
(twoIso P D).hom ≫ (tensor P D).map (relationEntry P r a) =
(vertex P).map (∑ b, basisCoefficient P r a b •
path k T (s).e₁ (s).e₂ (arrow12 P b) ((s).hb b) D) ≫ (oneIso P D).hom := by
have hb (b : Fin (d+1)) :
(twoIso P D).hom ≫ (tensor P D).map (TensorArrowSum.insertion k d b) =
(vertex P).map (path k T (s).e₁ (s).e₂ (arrow12 P b) ((s).hb b) D) ≫
(oneIso P D).hom := by
refine CornerPathTransport.path_square (composition P) D 2 1 2
(CornerSpaces.diagonalEquiv (composition P) 2) (CornerSpaces.corner21Equiv (composition P))
(arrow12 P b) ((s).hb b) (TensorArrowSum.insertion k d b).hom ?_
intro x
change (CornerSpaces.corner21Equiv (composition P))
(CornerColumnPaths.cornerMap (composition P) 2 1 2 (arrow12 P b) ((s).hb b) x) =
(TensorArrowSum.insertion k d b).hom ((CornerSpaces.diagonalEquiv (composition P) 2) x)
rw [TensorArrowSum.insertion_apply]
simpa only [LinearEquiv.symm_apply_apply] using
CornerArrowCoordinates.arrow12_coordinate P b ((CornerSpaces.diagonalEquiv (composition P) 2) x)
rw [relationEntry_eq]
simp only [Functor.map_sum,Functor.map_smul,Preadditive.comp_sum,Linear.comp_smul]
refine (Finset.sum_congr rfl
(fun b _ => congrArg (basisCoefficient P r a b • ·) (hb b))).trans ?_
refine (Finset.sum_congr rfl (fun b _ =>
(Linear.smul_comp _ _ _ (basisCoefficient P r a b)
((vertex P).map (path k T (s).e₁ (s).e₂ (arrow12 P b) ((s).hb b) D))
(oneIso P D).hom).symm)).trans ?_
exact (Preadditive.sum_comp Finset.univ
(fun b => basisCoefficient P r a b •
(vertex P).map (path k T (s).e₁ (s).e₂ (arrow12 P b) ((s).hb b) D))
(oneIso P D).hom).symm
end LittleFinitistic.RelationVertexCoordinates
end
end OAI
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