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0dd3af4 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Simplicial.LabelledMonoidal
namespace OAI
open CategoryTheory MonoidalCategory
namespace SimpleAmenable.PolygonObject.Labelled
variable {a n : ℕ} (d : (Fin n → CutRing × CutRing) → CutRing × CutRing)
noncomputable def translateObj (U : Labelled a n) : Labelled a n where
polygon := shifted U.polygon (fun i => d (U.label i))
label := U.label
reduced := U.reduced
noncomputable abbrev translateArrow (U : Labelled a n) : U.polygon ⟶ (translateObj d U).polygon :=
shiftArrow U.polygon (fun i => d (U.label i))
noncomputable def translateHom {U V : Labelled a n} (f : U ⟶ V) : translateObj d U ⟶ translateObj d V where
arrow := inv (translateArrow d U) ≫ f.arrow ≫ translateArrow d V
positional x := by
obtain ⟨y,rfl⟩ := (translateArrow d U).toEquiv.surjective x
simp only [arrow_comp_apply,inv_arrow_apply,Equiv.symm_apply_apply]
change translate a (d (V.label (f.arrow.toEquiv y).val.1)) (f.arrow.toEquiv y).val.2 =
translate a (d (U.label y.val.1)) y.val.2
rw [f.labelled,f.positional]
labelled x := by
obtain ⟨y,rfl⟩ := (translateArrow d U).toEquiv.surjective x
simp only [arrow_comp_apply,inv_arrow_apply,Equiv.symm_apply_apply]
exact f.labelled y
noncomputable def translateFunctor : Labelled a n ⥤ Labelled a n where
obj := translateObj d
map := translateHom d
map_id U := by
apply Hom.ext
change inv _ ≫ 𝟙 _ ≫ _ = _
simp
rfl
map_comp f g := by
apply Hom.ext
change inv _ ≫ (f.arrow ≫ g.arrow) ≫ _ = (inv _ ≫ f.arrow ≫ _) ≫ (inv _ ≫ g.arrow ≫ _)
simp only [Category.assoc,IsIso.hom_inv_id_assoc]
noncomputable def translationIso : forget (a:=a) (n:=n) ≅ translateFunctor d ⋙ forget :=
NatIso.ofComponents (fun U => by
let e := asIso (translateArrow d U)
exact e) (by
intro U V f
change f.arrow ≫ translateArrow d V = translateArrow d U ≫
(inv (translateArrow d U) ≫ f.arrow ≫ translateArrow d V)
simp)
noncomputable def translationCore : (translateFunctor (a:=a) d ⋙ forget).CoreMonoidal :=
Functor.Monoidal.coreMonoidalTransport (translationIso d)
lemma translationCore_unit : (translationCore (a:=a) d).εIso.hom=translateArrow d (𝟙_ _) := by
exact Category.id_comp (translateArrow d (𝟙_ _))
lemma translationCore_tensor (U V:Labelled a n) :
((translationCore d).μIso U V).hom =
inv (sumArrow (translateArrow d U) (translateArrow d V)) ≫ translateArrow d (U⊗V) := by
change (inv (translateArrow d U) ⊗ₘ inv (translateArrow d V)) ≫ 𝟙 _ ≫ _ = _
erw [Category.id_comp]
congr 1
apply (cancel_epi (sumArrow (translateArrow d U) (translateArrow d V))).mp
change (_ ⊗ₘ _) ≫ (_ ⊗ₘ _) = _
rw [tensorHom_comp_tensorHom]
simp
rfl
noncomputable def translationUnit : 𝟙_ (Labelled a n) ⟶ (translateFunctor d).obj (𝟙_ _) where
arrow := translateArrow d (𝟙_ _)
positional x := by change Point PolygonObject.empty at x; exact isEmptyElim x
labelled x := by change Point PolygonObject.empty at x; exact isEmptyElim x
noncomputable def translationTensor (U V:Labelled a n) :
(translateFunctor d).obj U ⊗ (translateFunctor d).obj V ⟶ (translateFunctor d).obj (U⊗V) where
arrow := inv (sumArrow (translateArrow d U) (translateArrow d V)) ≫ translateArrow d (U⊗V)
positional x := by
erw [arrow_comp_apply,inv_arrow_apply]
change ((translateArrow d (sum U V)).toEquiv
((sumArrow (translateArrow d U) (translateArrow d V)).toEquiv.symm x)).val.2 = x.val.2
obtain ⟨y,rfl⟩ := (sumArrow (translateArrow d U) (translateArrow d V)).toEquiv.surjective x
rw [Equiv.symm_apply_apply]
change translate a (d ((sum U V).label y.val.1)) y.val.2 =
((sumArrow (translateArrow d U) (translateArrow d V)).toEquiv y).val.2
obtain ⟨z,rfl⟩ := (sumPointEquiv U.polygon V.polygon).surjective y
cases z <;> (first | erw [sumArrow_inl] | erw [sumArrow_inr]) <;>
simp only [sumPointEquiv_inl,sumPointEquiv_inr,
translateArrow,Fin.addCases_left,Fin.addCases_right]
all_goals rfl
labelled x := by
erw [arrow_comp_apply,inv_arrow_apply]
change (sum U V).label ((translateArrow d (sum U V)).toEquiv
((sumArrow (translateArrow d U) (translateArrow d V)).toEquiv.symm x)).val.1 =
(sum (translateObj d U) (translateObj d V)).label x.val.1
obtain ⟨y,rfl⟩ := (sumArrow (translateArrow d U) (translateArrow d V)).toEquiv.surjective x
rw [Equiv.symm_apply_apply]
change (sum U V).label y.val.1 =
(sum (translateObj d U) (translateObj d V)).label
((sumArrow (translateArrow d U) (translateArrow d V)).toEquiv y).val.1
obtain ⟨z,rfl⟩ := (sumPointEquiv U.polygon V.polygon).surjective y
cases z <;> (first | erw [sumArrow_inl] | erw [sumArrow_inr]) <;> rfl
lemma translationTensor_arrow (U V : Labelled a n) :
(translationTensor d U V).arrow=((translationCore d).μIso U V).hom := by
rw [translationCore_tensor]; rfl
lemma translationUnit_arrow : (translationUnit (a:=a) d).arrow=(translationCore d).εIso.hom := by
rw [translationCore_unit]; rfl
noncomputable instance translationMonoidal : (translateFunctor (a:=a) d).Monoidal :=
Functor.CoreMonoidal.toMonoidal {
εIso := asIso (translationUnit d)
μIso U V := asIso (translationTensor d U V)
μIso_hom_natural_left := by
intro U V f W
apply Hom.ext
change sumArrow (translateHom d f).arrow (𝟙 _) ≫ (translationTensor d V W).arrow =
(translationTensor d U W).arrow ≫ (translateHom d (f ▷ W)).arrow
rw [translationTensor_arrow,translationTensor_arrow]
exact (translationCore d).μIso_hom_natural_left f W
μIso_hom_natural_right := by
intro U V W f
apply Hom.ext
change sumArrow (𝟙 _) (translateHom d f).arrow ≫ (translationTensor d W V).arrow =
(translationTensor d W U).arrow ≫ (translateHom d (W ◁ f)).arrow
rw [translationTensor_arrow,translationTensor_arrow]
exact (translationCore d).μIso_hom_natural_right W f
associativity := by
intro U V W
apply Hom.ext
change sumArrow (translationTensor d U V).arrow (𝟙 _) ≫
(translationTensor d (U⊗V) W).arrow ≫ (translateHom d (α_ U V W).hom).arrow =
sumAssoc _ _ _ ≫ sumArrow (𝟙 _) (translationTensor d V W).arrow ≫
(translationTensor d U (V⊗W)).arrow
simp only [translationTensor_arrow]
exact (translationCore d).associativity U V W
left_unitality := by
intro U
apply Hom.ext
change leftUnit _ = sumArrow (translationUnit d).arrow (𝟙 _) ≫
(translationTensor d (𝟙_ _) U).arrow ≫ (translateHom d (λ_ U).hom).arrow
rw [translationUnit_arrow,translationTensor_arrow]
exact (translationCore d).left_unitality U
right_unitality := by
intro U
apply Hom.ext
change rightUnit _ = sumArrow (𝟙 _) (translationUnit d).arrow ≫
(translationTensor d U (𝟙_ _)).arrow ≫ (translateHom d (ρ_ U).hom).arrow
rw [translationUnit_arrow,translationTensor_arrow]
exact (translationCore d).right_unitality U }
noncomputable instance translationBraided : (translateFunctor (a:=a) d).Braided where
braided U V := by
apply Hom.ext
change (translationTensor d U V).arrow ≫ (translateHom d (β_ U V).hom).arrow =
sumSwap _ _ ≫ (translationTensor d V U).arrow
change (inv (sumArrow (translateArrow d U) (translateArrow d V)) ≫ translateArrow d (U⊗V)) ≫
(inv (translateArrow d (U⊗V)) ≫ sumSwap _ _ ≫ translateArrow d (V⊗U)) =
sumSwap _ _ ≫ inv (sumArrow (translateArrow d V) (translateArrow d U)) ≫ translateArrow d (V⊗U)
simp only [Category.assoc,IsIso.hom_inv_id_assoc]
apply (cancel_epi (sumArrow (translateArrow d U) (translateArrow d V))).mp
simp only [IsIso.hom_inv_id_assoc]
have h := BraidedCategory.braiding_naturality (translateArrow d U) (translateArrow d V)
change sumArrow (translateArrow d U) (translateArrow d V) ≫ sumSwap _ _ =
sumSwap _ _ ≫ sumArrow (translateArrow d V) (translateArrow d U) at h
rw [←Category.assoc (sumArrow (translateArrow d U) (translateArrow d V)),h]
simp
end SimpleAmenable.PolygonObject.Labelled
end OAI
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