openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /LabelledStringEquiv.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.TrajectoryStages | |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.TripleBarMaps | |
| namespace OAI | |
| section | |
| section | |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory MonoidalCategory | |
| namespace SimpleAmenable.PolygonObject.Labelled | |
| variable {a n : ℕ} | |
| noncomputable def initialCore : (toStrings (a:=a) (n:=n) ⋙ stringEval).CoreMonoidal := | |
| Functor.Monoidal.coreMonoidalTransport initialIso | |
| lemma initialCore_unit : (initialCore (a:=a) (n:=n)).εIso.hom=initial (𝟙_ _) := by | |
| change 𝟙 _ ≫ initial (𝟙_ _) = _ | |
| exact Category.id_comp _ | |
| lemma initialCore_tensor (U V : Labelled a n) : | |
| (initialCore.μIso U V).hom=tensorInitial U V := by | |
| change (inv (initial U) ⊗ₘ inv (initial V)) ≫ 𝟙 _ ≫ initial (U⊗V) = | |
| inv (sumArrow (initial U) (initial V)) ≫ initial (U⊗V) | |
| simp only [Category.id_comp] | |
| congr 1 | |
| apply (cancel_epi (sumArrow (initial U) (initial V))).mp | |
| change (initial U ⊗ₘ initial V) ≫ (inv (initial U) ⊗ₘ inv (initial V)) = _ | |
| rw [tensorHom_comp_tensorHom] | |
| simp | |
| rfl | |
| noncomputable def stringsCore : (toStrings (a:=a) (n:=n)).CoreMonoidal where | |
| εIso := @asIso (StringGroupoid a n) _ _ _ unitLadder | |
| (IsIso.of_groupoid (C := StringGroupoid a n) _) | |
| μIso U V := @asIso (StringGroupoid a n) _ _ _ (tensorLadder U V) | |
| (IsIso.of_groupoid (C := StringGroupoid a n) _) | |
| μIso_hom_natural_left := by | |
| intro U V f W | |
| apply (stringEval (a:=a) (n:=n)).map_injective | |
| change sumArrow ((ladder f).hom.app 0) (𝟙 _) ≫ (tensorLadder V W).hom.app 0 = | |
| (tensorLadder U W).hom.app 0 ≫ (ladder (f ▷ W)).hom.app 0 | |
| rw [tensorLadder_zero,tensorLadder_zero,←initialCore_tensor,←initialCore_tensor] | |
| exact initialCore.μIso_hom_natural_left f W | |
| μIso_hom_natural_right := by | |
| intro U V W f | |
| apply (stringEval (a:=a) (n:=n)).map_injective | |
| change sumArrow (𝟙 _) ((ladder f).hom.app 0) ≫ (tensorLadder W V).hom.app 0 = | |
| (tensorLadder W U).hom.app 0 ≫ (ladder (W ◁ f)).hom.app 0 | |
| rw [tensorLadder_zero,tensorLadder_zero,←initialCore_tensor,←initialCore_tensor] | |
| exact initialCore.μIso_hom_natural_right W f | |
| associativity := by | |
| intro U V W | |
| apply (stringEval (a:=a) (n:=n)).map_injective | |
| change sumArrow ((tensorLadder U V).hom.app 0) (𝟙 _) ≫ | |
| (tensorLadder (U⊗V) W).hom.app 0 ≫ (ladder (α_ U V W).hom).hom.app 0 = | |
| sumAssoc _ _ _ ≫ sumArrow (𝟙 _) ((tensorLadder V W).hom.app 0) ≫ | |
| (tensorLadder U (V⊗W)).hom.app 0 | |
| simp only [tensorLadder_zero,←initialCore_tensor] | |
| exact initialCore.associativity U V W | |
| left_unitality := by | |
| intro U | |
| apply (stringEval (a:=a) (n:=n)).map_injective | |
| change leftUnit _ = sumArrow (unitLadder.hom.app 0) (𝟙 _) ≫ | |
| (tensorLadder (𝟙_ _) U).hom.app 0 ≫ (ladder (λ_ U).hom).hom.app 0 | |
| rw [unitLadder_zero,tensorLadder_zero,←initialCore_unit,←initialCore_tensor] | |
| exact initialCore.left_unitality U | |
| right_unitality := by | |
| intro U | |
| apply (stringEval (a:=a) (n:=n)).map_injective | |
| change rightUnit _ = sumArrow (𝟙 _) (unitLadder.hom.app 0) ≫ | |
| (tensorLadder U (𝟙_ _)).hom.app 0 ≫ (ladder (ρ_ U).hom).hom.app 0 | |
| rw [unitLadder_zero,tensorLadder_zero,←initialCore_unit,←initialCore_tensor] | |
| exact initialCore.right_unitality U | |
| noncomputable instance : (toStrings (a:=a) (n:=n)).Monoidal := stringsCore.toMonoidal | |
| noncomputable instance : (toStrings (a:=a) (n:=n)).Braided where | |
| braided U V := by | |
| apply (stringEval (a:=a) (n:=n)).map_injective | |
| change (tensorLadder U V).hom.app 0 ≫ (ladder (β_ U V).hom).hom.app 0 = | |
| sumSwap _ _ ≫ (tensorLadder V U).hom.app 0 | |
| rw [tensorLadder_zero,tensorLadder_zero,ladder_zero] | |
| change (inv (sumArrow (initial U) (initial V)) ≫ initial (U⊗V)) ≫ | |
| (inv (initial (U⊗V)) ≫ sumSwap _ _ ≫ initial (V⊗U)) = | |
| sumSwap _ _ ≫ inv (sumArrow (initial V) (initial U)) ≫ initial (V⊗U) | |
| simp only [Category.assoc,IsIso.hom_inv_id_assoc] | |
| apply (cancel_epi (sumArrow (initial U) (initial V))).mp | |
| simp only [IsIso.hom_inv_id_assoc] | |
| have h := BraidedCategory.braiding_naturality (initial U) (initial V) | |
| change sumArrow (initial U) (initial V) ≫ sumSwap _ _ = | |
| sumSwap _ _ ≫ sumArrow (initial V) (initial U) at h | |
| rw [←Category.assoc (sumArrow (initial U) (initial V)),h] | |
| simp | |
| lemma tripleBar_homology_isIso (j : ℕ) : | |
| IsIso (SSet.homologyMap (IntervalBar.Diagram.bar₃Map (toStrings (a:=a) (n:=n))) | |
| DiagonalResolution.Z j) := | |
| IntervalBar.Diagram.bar₃Map_homology_isIso toStrings j | |
| end SimpleAmenable.PolygonObject.Labelled | |
| end | |
| section | |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Classical | |
| namespace SimpleAmenable.PolygonObject.LabelledStage | |
| variable {a n : ℕ} | |
| variable {I : Type} [Category.{0} I] [Finite I] [∀i j:I,Finite (i ⟶ j)] | |
| theorem exists_common_stage (F : I ⥤ Labelled a n) : | |
| ∃ (P : BooleanPartition a) (s : ℕ) (L : Fin s → Fin n → CutRing × CutRing), | |
| (∀j i,orbitRepresentative (L j i)=L j i) ∧ Function.Injective L ∧ | |
| ∃G : I ⥤ UniformObject P L, G ⋙ UniformObject.forget=F := by | |
| let : Fintype I := Fintype.ofFinite I | |
| let (i j : I) : Fintype (i ⟶ j) := Fintype.ofFinite (i ⟶ j) | |
| let Ar := Σi:I,Σj:I,i ⟶ j | |
| let u : Ar → I := fun t => t.1 | |
| let v : Ar → I := fun t => t.2.1 | |
| let f : ∀t:Ar,F.obj (u t) ⟶ F.obj (v t) := fun t => F.map t.2.2 | |
| obtain ⟨P,hP,hf⟩ := Labelled.exists_uniform_partition F.obj u v f | |
| let label : (Σi:I,Fin (F.obj i).polygon.tracks) → Fin n → CutRing × CutRing := | |
| fun t => (F.obj t.1).label t.2 | |
| let S := Set.range label | |
| have : Fintype S := (Set.finite_range label).fintype | |
| let L : Fin (Fintype.card S) → Fin n → CutRing × CutRing := | |
| fun j => ((Fintype.equivFin S).symm j).val | |
| have hL : ∀j i,orbitRepresentative (L j i)=L j i := by | |
| intro j i | |
| obtain ⟨⟨k,t⟩,hk⟩ := ((Fintype.equivFin S).symm j).property | |
| change orbitRepresentative (((Fintype.equivFin S).symm j).val i)=(((Fintype.equivFin S).symm j).val i) | |
| rw [←hk] | |
| exact (F.obj k).reduced t i | |
| have hLi : Function.Injective L := by | |
| intro i j h | |
| exact (Fintype.equivFin S).symm.injective (Subtype.ext h) | |
| let obj (i : I) : UniformObject P L := { | |
| obj := F.obj i | |
| uniform := hP i | |
| supported := fun j => ⟨Fintype.equivFin S ⟨label ⟨i,j⟩,⟨⟨i,j⟩,rfl⟩⟩,by | |
| simp only [L,Equiv.symm_apply_apply]; rfl⟩ } | |
| let G : I ⥤ UniformObject P L := { | |
| obj := obj | |
| map {i j} g := ⟨F.map g,hf ⟨i,j,g⟩⟩ | |
| map_id i := UniformObject.Hom.ext _ _ (F.map_id i) | |
| map_comp g h := UniformObject.Hom.ext _ _ (F.map_comp g h) } | |
| refine ⟨P,Fintype.card S,L,hL,hLi,G,?_⟩ | |
| rfl | |
| end SimpleAmenable.PolygonObject.LabelledStage | |
| end | |
| end | |
| end OAI | |