| 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 | |