| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.LabelledMonoidal |
|
|
| namespace OAI |
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|
| section |
| open CategoryTheory MonoidalCategory |
| namespace SimpleAmenable.PolygonObject.Labelled |
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|
| variable {a n m : ℕ} |
| abbrev Reduced (g : Fin n → CutRing × CutRing) : Prop := |
| ∀ i, orbitRepresentative (g i)=g i |
| variable (f : (Fin n → CutRing × CutRing) → (Fin m → CutRing × CutRing)) |
| (hf : ∀g, Reduced g → Reduced (f g)) |
| noncomputable def relabel : Labelled a n ⥤ Labelled a m where |
| obj U := ⟨U.polygon,fun k => f (U.label k),fun k => hf _ (U.reduced k)⟩ |
| map g := ⟨g.arrow,g.positional,fun x => congrArg f (g.labelled x)⟩ |
| map_id _ := Hom.ext _ _ rfl |
| map_comp _ _ := Hom.ext _ _ rfl |
| noncomputable def relabelUnit : 𝟙_ (Labelled a m) ⟶ (relabel f hf).obj (𝟙_ (Labelled a n)) where |
| arrow := 𝟙 _ |
| positional _ := rfl |
| labelled x := by |
| change Point PolygonObject.empty at x |
| exact isEmptyElim x |
| noncomputable def relabelTensor (U V : Labelled a n) : |
| (relabel f hf).obj U ⊗ (relabel f hf).obj V ⟶ (relabel f hf).obj (U ⊗ V) where |
| arrow := 𝟙 _ |
| positional _ := rfl |
| labelled x := by |
| change f ((sum U V).label x.val.1) = |
| (sum ((relabel f hf).obj U) ((relabel f hf).obj V)).label x.val.1 |
| obtain ⟨y,rfl⟩ := (sumPointEquiv U.polygon V.polygon).surjective x |
| cases y <;> simp only [relabel,sumPointEquiv_inl,sumPointEquiv_inr,Fin.addCases_left,Fin.addCases_right] |
| noncomputable instance relabelMonoidal : (relabel (a:=a) f hf).Monoidal := |
| Functor.CoreMonoidal.toMonoidal { |
| εIso := asIso (relabelUnit f hf) |
| μIso U V := asIso (relabelTensor f hf U V) |
| μIso_hom_natural_left := by |
| intros; apply Hom.ext |
| change _ ≫ 𝟙 _ = 𝟙 _ ≫ _ |
| simp only [Category.comp_id] |
| rfl |
| μIso_hom_natural_right := by |
| intros; apply Hom.ext |
| change _ ≫ 𝟙 _ = 𝟙 _ ≫ _ |
| simp only [Category.comp_id] |
| rfl |
| associativity := by |
| intros; apply Hom.ext |
| change sumArrow (𝟙 _) (𝟙 _) ≫ 𝟙 _ ≫ _ = _ ≫ sumArrow (𝟙 _) (𝟙 _) ≫ 𝟙 _ |
| simp only [sumArrow_id,Category.id_comp,Category.comp_id] |
| rfl |
| left_unitality := by |
| intros; apply Hom.ext |
| change _ = sumArrow (𝟙 _) (𝟙 _) ≫ 𝟙 _ ≫ _ |
| simp only [sumArrow_id,Category.id_comp] |
| rfl |
| right_unitality := by |
| intros; apply Hom.ext |
| change _ = sumArrow (𝟙 _) (𝟙 _) ≫ 𝟙 _ ≫ _ |
| simp only [sumArrow_id,Category.id_comp] |
| rfl } |
| noncomputable instance relabelBraided : (relabel (a:=a) f hf).Braided where |
| braided U V := by |
| apply Hom.ext |
| change 𝟙 _ ≫ sumSwap U.polygon V.polygon = sumSwap U.polygon V.polygon ≫ 𝟙 _ |
| simp only [Category.id_comp,Category.comp_id] |
| @[simp] lemma relabel_map_arrow {U V : Labelled a n} (g:U⟶V) : |
| ((relabel f hf).map g).arrow=g.arrow := rfl |
| @[simp] lemma relabel_ε_arrow : (Functor.LaxMonoidal.ε (relabel (a:=a) f hf)).arrow=𝟙 _ := rfl |
| @[simp] lemma relabel_μ_arrow (U V : Labelled a n) : |
| (Functor.LaxMonoidal.μ (relabel f hf) U V).arrow=𝟙 _ := rfl |
| end SimpleAmenable.PolygonObject.Labelled |
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| end |
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| end OAI |
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