| 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 |
|
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| end OAI |
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