| import OAI.Algebra.Finitistic.Rectification.CornerArrowCoordinates |
| import OAI.Algebra.Finitistic.Rectification.TensorArrowSum |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace LittleFinitistic.AdjacentArrowSum |
| universe u |
| variable {k V W U : Type u} [Field k] [AddCommGroup V] [Module k V] |
| [AddCommGroup W] [Module k W] [AddCommGroup U] [Module k U] |
| variable (μ : W →ₗ[k] V →ₗ[k] U) |
| variable (D : CochainComplex (ModuleCat.{u} (ThreeVertexAlgebra μ)ᵐᵒᵖ) ℤ) |
| local notation "T" => ThreeVertexAlgebra μ |
| local instance : HasFiniteBiproducts (CochainComplex (ModuleCat.{u} Tᵐᵒᵖ) ℤ) := |
| HasFiniteBiproducts.of_hasFiniteProducts |
| abbrev vertexComplex (v : Fin 3) := |
| (BimoduleVertex.functor k T (ThreeVertexAlgebra.vertex μ v)).mapHomologicalComplex (.up ℤ) |
|
|
| def sourceIso (v : Fin 3) : |
| (vertexComplex μ v).obj ((ExternalTensor.Section4.columnComplex k T |
| (ThreeVertexAlgebra.vertex μ v)).obj D) ≅ |
| (TensorComplex.tensorFunctor (k := k) D).obj (ModuleCat.of k k) := |
| (CornerColumnComplex.complexIso μ v v D).symm ≪≫ |
| (TensorComplex.tensorFunctor (k := k) D).mapIso (CornerSpaces.diagonalEquiv μ v).toModuleIso |
|
|
| def targetIso (v i : Fin 3) (d : ℕ) |
| (ξ : CornerSpaces.corner μ v i ≃ₗ[k] QuadraticEncoding.Arrows k d) : |
| (vertexComplex μ v).obj ((ExternalTensor.Section4.columnComplex k T |
| (ThreeVertexAlgebra.vertex μ i)).obj D) ≅ |
| (TensorComplex.tensorFunctor (k := k) D).obj (ModuleCat.of k (QuadraticEncoding.Arrows k d)) := |
| (CornerColumnComplex.complexIso μ v i D).symm ≪≫ |
| (TensorComplex.tensorFunctor (k := k) D).mapIso ξ.toModuleIso |
|
|
| /-- An actual adjacent path family becomes the full arrow basis under corner coordinates. -/ |
| theorem desc_isIso (v i : Fin 3) (d : ℕ) |
| (ξ : CornerSpaces.corner μ v i ≃ₗ[k] QuadraticEncoding.Arrows k d) |
| (a : Fin (d+1) → ThreeVertexAlgebra μ) |
| (ha : ∀ j, a j * ThreeVertexAlgebra.vertex μ i = a j) |
| (hc : ∀ j c, ξ (CornerColumnPaths.cornerMap μ v i v (a j) (ha j) |
| ((CornerSpaces.diagonalEquiv μ v).symm c)) = c • QuadraticEncoding.basis j) : |
| IsIso (biproduct.desc (fun j => (vertexComplex μ v).map |
| (ExternalTensor.Section4.path k T (ThreeVertexAlgebra.vertex μ i) |
| (ThreeVertexAlgebra.vertex μ v) (a j) (ha j) D))) := by |
| let F := TensorComplex.tensorFunctor (k := k) D |
| apply TensorArrowSum.desc_isIso_of_coordinates k d F _ _ |
| (fun _ => sourceIso μ D v) (targetIso μ D v i d ξ) |
| intro j |
| have hm : ModuleCat.ofHom (CornerSpaces.diagonalEquiv μ v).toLinearMap ≫ |
| TensorArrowSum.insertion k d j = |
| ModuleCat.ofHom (CornerColumnPaths.cornerMap μ v i v (a j) (ha j)) ≫ |
| ModuleCat.ofHom ξ.toLinearMap := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| change (TensorArrowSum.insertion k d j).hom ((CornerSpaces.diagonalEquiv μ v) x) = |
| ξ (CornerColumnPaths.cornerMap μ v i v (a j) (ha j) x) |
| rw [TensorArrowSum.insertion_apply] |
| simpa only [LinearEquiv.symm_apply_apply] |
| using (hc j ((CornerSpaces.diagonalEquiv μ v) x)).symm |
| have hF := congrArg F.map hm |
| simp only [Functor.map_comp] at hF |
| apply (cancel_epi (CornerColumnComplex.complexIso μ v v D).hom).1 |
| calc |
| _ = F.map (ModuleCat.ofHom (CornerSpaces.diagonalEquiv μ v).toLinearMap) ≫ |
| F.map (TensorArrowSum.insertion k d j) := by |
| change (CornerColumnComplex.complexIso μ v v D).hom ≫ |
| ((CornerColumnComplex.complexIso μ v v D).inv ≫ |
| F.map (ModuleCat.ofHom (CornerSpaces.diagonalEquiv μ v).toLinearMap)) ≫ |
| F.map (TensorArrowSum.insertion k d j) = _ |
| erw [Category.assoc] |
| exact (CornerColumnComplex.complexIso μ v v D).hom_inv_id_assoc _ |
| _ = F.map (ModuleCat.ofHom (CornerColumnPaths.cornerMap μ v i v (a j) (ha j))) ≫ |
| F.map (ModuleCat.ofHom ξ.toLinearMap) := hF |
| _ = (F.map (ModuleCat.ofHom |
| (CornerColumnPaths.cornerMap μ v i v (a j) (ha j))) ≫ |
| (CornerColumnComplex.complexIso μ v i D).hom) ≫ |
| (targetIso μ D v i d ξ).hom := by simp [targetIso,F,TensorComplex.tensorFunctor] |
| _ = _ := by |
| erw [CornerColumnPaths.path_naturality] |
| exact Category.assoc _ _ _ |
| end LittleFinitistic.AdjacentArrowSum |
|
|
| end |
|
|
| end OAI |
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