File size: 5,332 Bytes
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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Algebra.Finitistic.Localization.SourceVerdier
import OAI.Algebra.Finitistic.Rectification.VertexTensorEvaluation
import OAI.Algebra.Finitistic.Localization.TensorEvaluationFunctor
namespace OAI
noncomputable section
open CategoryTheory Limits Category
namespace LittleFinitistic.ConcreteTensorEvaluation
open QuadraticEncoding
universe u
variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t}
variable {Y : Type u} [AddCommGroup Y] [Module k Y] (X : Representation P Y)
/-- Match the opposite-regular presentation of eB to the fixed-point presentation. -/
def idealEquiv (e : B P) (he : e*e=e) :
VertexProjectives.ideal e ≃ₗ[(B P)ᵐᵒᵖ] IdempotentTensor.rightIdeal e where
toFun x := ⟨x.val.unop, by
exact congrArg MulOpposite.unop (VertexProjectives.fixed e he x)⟩
invFun x := ⟨MulOpposite.op x.val, ⟨MulOpposite.op x.val, by
change MulOpposite.op x.val * MulOpposite.op e = MulOpposite.op x.val
exact congrArg MulOpposite.op x.property⟩⟩
left_inv x := by rfl
right_inv x := by rfl
map_add' x y := rfl
map_smul' a x := rfl
section TensorCongr
variable {A : Type u} [Ring A] [Algebra k A]
variable {M M' N : Type u} [AddCommGroup M] [AddCommGroup M'] [AddCommGroup N]
variable [Module k M] [Module k M'] [Module k N]
variable [Module Aᵐᵒᵖ M] [Module Aᵐᵒᵖ M'] [Module A N]
variable [IsScalarTower k Aᵐᵒᵖ M] [IsScalarTower k Aᵐᵒᵖ M']
/-- Balanced tensor respects actual right-module equivalences. -/
def tensorCongr (e : M ≃ₗ[Aᵐᵒᵖ] M') :
BalancedTensor.Tensor k A M N ≃ₗ[k] BalancedTensor.Tensor k A M' N where
__ := BalancedTensor.mapLeft (k := k) N e.toLinearMap
invFun := BalancedTensor.mapLeft (k := k) N e.symm.toLinearMap
left_inv z := by
change BalancedTensor.mapLeft (k := k) N e.symm.toLinearMap
(BalancedTensor.mapLeft (k := k) N e.toLinearMap z) = z
induction z using BalancedTensor.induction_on with
| hz => simp
| ht m n => simp
| ha a b ha hb =>
simp only [map_add]
exact congrArg₂ (· + ·) ha hb
right_inv z := by
change BalancedTensor.mapLeft (k := k) N e.toLinearMap
(BalancedTensor.mapLeft (k := k) N e.symm.toLinearMap z) = z
induction z using BalancedTensor.induction_on with
| hz => simp
| ht m n => simp
| ha a b ha hb =>
simp only [map_add]
exact congrArg₂ (· + ·) ha hb
end TensorCongr
/-- Actual tensor evaluation on projectives. -/
abbrev moduleEvaluation : SourceVerdier.Proj P ⥤ ModuleCat.{u} k :=
(ProjectiveComplexes.finiteProjective (B P)).ι ⋙
BalancedTensor.tensorFunctor k (B P) X.Evaluation
instance vertexScalarTower (i : Fin 3) :
IsScalarTower k (B P)ᵐᵒᵖ (BalancedTensor.ScalarModule k (B P) (SourceVerdier.E P i).obj) :=
BalancedTensor.scalarModuleTower k (B P) (SourceVerdier.E P i).obj
/-- The exact natural vertex identification E_i tensor_B M(Y) = Y. -/
def vertexIso (i : Fin 3) : (moduleEvaluation X).obj (SourceVerdier.E P i) ≅ ModuleCat.of k Y :=
((@tensorCongr k _ (B P) _ _
(BalancedTensor.ScalarModule k (B P) (SourceVerdier.E P i).obj)
(IdempotentTensor.rightIdeal (SourceVerdier.vertex P i)) X.Evaluation
_ _ _ (BalancedTensor.ScalarModule k (B P) (SourceVerdier.E P i).obj).isModule _ _
_ _ _ (vertexScalarTower (P := P) i) inferInstance
(idealEquiv (SourceVerdier.vertex P i) (SourceVerdier.vertex_idem P i))).trans
(X.vertexTensorEquiv i)).toModuleIso
@[simp] theorem vertexIso_mk (i : Fin 3)
(m : BalancedTensor.ScalarModule k (B P) (SourceVerdier.E P i).obj) (y : X.Evaluation) :
(vertexIso X i).hom.hom
(BalancedTensor.mk k (B P) _ X.Evaluation m y) = X.projection i (m.val.unop • y) := rfl
/-- The two actual identity-arrow tensor maps become identities on Y. -/
theorem s_naturality :
(moduleEvaluation X).map (SourceVerdier.arrow01 P 0) ≫ (vertexIso X 1).hom =
(vertexIso X 0).hom := by
apply ModuleCat.hom_ext
apply BalancedTensor.ext
intro m y
change X.projection 1 ((m.val * MulOpposite.op (QuadraticEncoding.arrow01 P 0)).unop • y) =
X.projection 0 (m.val.unop • y)
rw [MulOpposite.unop_mul, MulOpposite.unop_op, mul_smul, X.arrow01_smul]
simp [Representation.projection]
theorem s'_naturality :
(moduleEvaluation X).map (SourceVerdier.arrow12 P 0) ≫ (vertexIso X 2).hom =
(vertexIso X 1).hom := by
apply ModuleCat.hom_ext
apply BalancedTensor.ext
intro m y
change X.projection 2 ((m.val * MulOpposite.op (QuadraticEncoding.arrow12 P 0)).unop • y) =
X.projection 1 (m.val.unop • y)
rw [MulOpposite.unop_mul, MulOpposite.unop_op, mul_smul, X.arrow12_smul]
simp [Representation.projection]
instance s_inverted : IsIso ((moduleEvaluation X).map (SourceVerdier.arrow01 P 0)) := by
have : IsIso ((moduleEvaluation X).map (SourceVerdier.arrow01 P 0) ≫ (vertexIso X 1).hom) := by
rw [s_naturality]
infer_instance
exact IsIso.of_isIso_comp_right _ (vertexIso X 1).hom
instance s'_inverted : IsIso ((moduleEvaluation X).map (SourceVerdier.arrow12 P 0)) := by
have : IsIso ((moduleEvaluation X).map (SourceVerdier.arrow12 P 0) ≫ (vertexIso X 2).hom) := by
rw [s'_naturality]
infer_instance
exact IsIso.of_isIso_comp_right _ (vertexIso X 2).hom
end LittleFinitistic.ConcreteTensorEvaluation
end
end OAI
|