| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Configurations.StageFiberSum |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.TripleComposition |
| import OAI.GroupTheory.SimpleAmenable.Configurations.EmptyStage |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PointFiberEvaluation |
| import OAI.GroupTheory.SimpleAmenable.Configurations.CellFunctor |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.CellMonoidal |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.PointFiberMonoidal |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.StageMonoidal |
|
|
| namespace OAI |
|
|
| section |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Classical MonoidalCategory |
| namespace SimpleAmenable.PolygonObject.Labelled.Cell |
| open BarFinitePower |
|
|
| variable {a n : ℕ} (P : polygonAlgebra a) (l : LabelledStage.ReducedLabel n) |
| variable (c : GenericSquare a × (Fin n → CutRing × CutRing)) |
| variable (hp : c.1∈P.val) (hl : l.val=c.2) |
| noncomputable def fiber (U : FiniteSetGroupoid) : |
| Fin U.size ≃ PointFiber.Fiber (object P l U) c where |
| toFun i := ⟨⟨(i,c.1),hp⟩,rfl,hl⟩ |
| invFun y := y.val.val.1 |
| left_inv _ := rfl |
| right_inv y := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl y.property.1.symm |
| lemma fiber_natural {U V : FiniteSetGroupoid} (f : U ⟶ V) (x : Fin U.size) : |
| fiber P l c hp hl V (f x)= |
| PointFiber.fiberMap (arrow P l f) c (fiber P l c hp hl U x) := rfl |
| noncomputable def evaluationIso : |
| 𝟭 FiniteSetGroupoid ≅ functor P l ⋙ PointFiber.evaluation ⋙ project _ c := |
| NatIso.ofComponents (fun U => asIso (X := U) (Y := PointFiber.E.obj (object P l U) c) |
| ((fiber P l c hp hl U).trans (PointFiber.enumerate (object P l U) c))) (by |
| intro U V f |
| apply Equiv.ext; intro x |
| change PointFiber.enumerate (object P l V) c (fiber P l c hp hl V (f x)) = |
| PointFiber.E.map (arrow P l f) c (PointFiber.enumerate (object P l U) c |
| (fiber P l c hp hl U x)) |
| rw [PointFiber.E_map_enum,fiber_natural]) |
| lemma evaluationIso_apply (U : FiniteSetGroupoid) (x : Fin U.size) : |
| (evaluationIso P l c hp hl).hom.app U x = |
| PointFiber.enumerate (object P l U) c (fiber P l c hp hl U x) := rfl |
| lemma fiber_sum_left (U V : FiniteSetGroupoid) (x : Fin U.size) : |
| PointFiber.fiberMap (tensorArrow P l U V) c |
| (PointFiber.fiberSum (object P l U) (object P l V) c (.inl (fiber P l c hp hl U x))) = |
| fiber P l c hp hl (U⊗V) (FiniteSetGroupoid.sumEquiv U V (.inl x)) := rfl |
| lemma fiber_sum_right (U V : FiniteSetGroupoid) (x : Fin V.size) : |
| PointFiber.fiberMap (tensorArrow P l U V) c |
| (PointFiber.fiberSum (object P l U) (object P l V) c (.inr (fiber P l c hp hl V x))) = |
| fiber P l c hp hl (U⊗V) (FiniteSetGroupoid.sumEquiv U V (.inr x)) := rfl |
| noncomputable instance evaluationIsoMonoidal : NatTrans.IsMonoidal (evaluationIso P l c hp hl).hom where |
| unit := by apply Equiv.ext; intro x; exact Fin.elim0 x |
| tensor U V := by |
| apply Equiv.ext; intro x |
| change (evaluationIso P l c hp hl).hom.app (U⊗V) x = |
| PointFiber.E.map (tensorArrow P l U V) c |
| (PointFiber.mu (object P l U) (object P l V) c |
| (FiniteSetGroupoid.sumHom ((evaluationIso P l c hp hl).hom.app U) |
| ((evaluationIso P l c hp hl).hom.app V) x)) |
| obtain ⟨x,rfl⟩ := (FiniteSetGroupoid.sumEquiv U V).surjective x |
| cases x with |
| | inl x => erw [FiniteSetGroupoid.sumHom_inl,evaluationIso_apply,PointFiber.mu_inl, |
| PointFiber.E_map_enum,fiber_sum_left,evaluationIso_apply] |
| | inr x => erw [FiniteSetGroupoid.sumHom_inr,evaluationIso_apply,PointFiber.mu_inr, |
| PointFiber.E_map_enum,fiber_sum_right,evaluationIso_apply] |
|
|
| variable {hp hl} |
| lemma fiber_empty (h : ¬(c.1∈P.val ∧ l.val=c.2)) (U : FiniteSetGroupoid) : |
| IsEmpty (PointFiber.Fiber (object P l U) c) := ⟨fun x => h |
| ⟨x.property.1 ▸ x.val.property,x.property.2⟩⟩ |
| lemma empty_size (h : ¬(c.1∈P.val ∧ l.val=c.2)) (U : FiniteSetGroupoid) : |
| (PointFiber.E.obj (object P l U) c).size=0 := by |
| have := fiber_empty P l c h U |
| exact @Fintype.card_eq_zero _ (PointFiber.instFintypeFiber (object P l U) c) _ |
| noncomputable def emptyIso (h : ¬(c.1∈P.val ∧ l.val=c.2)) : |
| FiniteSetGroupoid.terminal ⋙ FiniteSetGroupoid.emptyFunctor ≅ |
| functor P l ⋙ PointFiber.evaluation ⋙ project _ c := |
| NatIso.ofComponents (fun U => asIso (X := FiniteSetGroupoid.empty) |
| (Y := PointFiber.E.obj (object P l U) c) (finCongr (empty_size P l c h U).symm)) (by |
| intros; apply Equiv.ext; intro x; exact Fin.elim0 x) |
| noncomputable instance emptyIsoMonoidal (h : ¬(c.1∈P.val ∧ l.val=c.2)) : |
| NatTrans.IsMonoidal (emptyIso P l c h).hom where |
| unit := by apply Equiv.ext; intro x; exact Fin.elim0 x |
| tensor _ _ := by apply Equiv.ext; intro x; exact Fin.elim0 x |
| end SimpleAmenable.PolygonObject.Labelled.Cell |
|
|
| end |
|
|
| section |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Classical MonoidalCategory |
| namespace SimpleAmenable.PolygonObject.LabelledStage.Stage |
| open UniformObject BarFinitePower |
|
|
| variable {a n : ℕ} (S : Stage a n) |
| variable (b : Fin S.partition.size × Fin S.support.card) |
| (c : GenericSquare a × (Fin n → CutRing × CutRing)) |
| (hp : S.partition.color c.1=b.1) (hl : S.L b.2=c.2) |
|
|
| noncomputable def pointFiber (U : S.Obj) : |
| UniformObject.Fiber U b ≃ Labelled.PointFiber.Fiber U.obj c where |
| toFun x := ⟨⟨(x.val.val.1,c.1), |
| (U.uniform x.val.val.1 x.val.val.2 c.1 (by |
| rw [x.property.1,S.partition.color_point,hp])).mp x.val.property⟩,rfl, |
| x.property.2.trans hl⟩ |
| invFun y := ⟨⟨(y.val.val.1,S.partition.point b.1), |
| (U.uniform y.val.val.1 y.val.val.2 (S.partition.point b.1) (by |
| rw [y.property.1,hp,S.partition.color_point])).mp y.val.property⟩,rfl, |
| y.property.2.trans hl.symm⟩ |
| left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl x.property.1.symm |
| right_inv y := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext rfl y.property.1.symm |
|
|
| lemma pointFiber_natural {U V : S.Obj} (f : U ⟶ V) (x : Fiber U b) : |
| pointFiber S b c hp hl V (fiberMap f b x)= |
| Labelled.PointFiber.fiberMap f.arrow c (pointFiber S b c hp hl U x) := by |
| apply Subtype.ext |
| apply Subtype.ext |
| apply Prod.ext |
| · exact f.uniform x.val (pointFiber S b c hp hl U x).val rfl (by |
| change S.partition.color x.val.val.2=S.partition.color c.1 |
| rw [x.property.1,S.partition.color_point,hp]) |
| · exact (f.arrow.positional (pointFiber S b c hp hl U x).val).symm |
|
|
| noncomputable def pointEvaluationIso : |
| evaluation (P:=S.partition) (L:=S.L) ⋙ project _ b ≅ |
| S.forget ⋙ Labelled.PointFiber.evaluation ⋙ project _ c := |
| NatIso.ofComponents (fun U => asIso (X := UniformObject.E.obj U b) |
| (Y := Labelled.PointFiber.E.obj U.obj c) |
| ((enumerate U b).symm.trans ((pointFiber S b c hp hl U).trans |
| (Labelled.PointFiber.enumerate U.obj c)))) (by |
| intro U V f |
| apply Equiv.ext |
| intro x |
| obtain ⟨x,rfl⟩ := (enumerate U b).surjective x |
| dsimp only [Functor.comp_map,Pi.eval,project,asIso_hom,Equiv.trans_apply,Equiv.symm_apply_apply] |
| change Labelled.PointFiber.enumerate V.obj c (pointFiber S b c hp hl V |
| ((enumerate V b).symm (UniformObject.E.map f b (enumerate U b x)))) = |
| Labelled.PointFiber.E.map f.arrow c (Labelled.PointFiber.enumerate U.obj c |
| (pointFiber S b c hp hl U ((enumerate U b).symm (enumerate U b x)))) |
| rw [E_map_enum,Equiv.symm_apply_apply,Equiv.symm_apply_apply, |
| Labelled.PointFiber.E_map_enum,pointFiber_natural]) |
|
|
| lemma pointFiber_sum_inl (U V : S.Obj) (x : Fiber U b) : |
| pointFiber S b c hp hl (UniformObject.sum U V) (fiberSum U V b (.inl x)) = |
| Labelled.PointFiber.fiberSum U.obj V.obj c (.inl (pointFiber S b c hp hl U x)) := rfl |
| lemma pointFiber_sum_inr (U V : S.Obj) (x : Fiber V b) : |
| pointFiber S b c hp hl (UniformObject.sum U V) (fiberSum U V b (.inr x)) = |
| Labelled.PointFiber.fiberSum U.obj V.obj c (.inr (pointFiber S b c hp hl V x)) := rfl |
| lemma pointEvaluationIso_enum (U : S.Obj) (x : Fiber U b) : |
| (pointEvaluationIso S b c hp hl).hom.app U (enumerate U b x) = |
| Labelled.PointFiber.enumerate U.obj c (pointFiber S b c hp hl U x) := by |
| change Labelled.PointFiber.enumerate U.obj c (pointFiber S b c hp hl U |
| ((enumerate U b).symm (enumerate U b x)))=_ |
| rw [Equiv.symm_apply_apply] |
|
|
| instance pointEvaluationIso_monoidal : NatTrans.IsMonoidal (pointEvaluationIso S b c hp hl).hom where |
| unit := by apply Equiv.ext; intro x; exact Fin.elim0 x |
| tensor U V := by |
| change ((𝟙 _ ≫ mu U V b) ≫ _) = (_ ≫ (𝟙 _ ≫ Labelled.PointFiber.mu U.obj V.obj c) ≫ |
| (Labelled.PointFiber.E.map (𝟙 (U.obj ⊗ V.obj))) c) |
| rw [CategoryTheory.Functor.map_id] |
| simp only [Category.id_comp] |
| apply Equiv.ext |
| intro x |
| change (pointEvaluationIso S b c hp hl).hom.app (UniformObject.sum U V) (mu U V b x) = |
| Labelled.PointFiber.mu U.obj V.obj c |
| (FiniteSetGroupoid.sumHom ((pointEvaluationIso S b c hp hl).hom.app U) |
| ((pointEvaluationIso S b c hp hl).hom.app V) x) |
| obtain ⟨x,rfl⟩ := (FiniteSetGroupoid.sumEquiv (UniformObject.E.obj U b) (UniformObject.E.obj V b)).surjective x |
| cases x with |
| | inl x => |
| obtain ⟨x,rfl⟩ := (enumerate U b).surjective x |
| erw [mu_inl,pointEvaluationIso_enum,pointFiber_sum_inl,FiniteSetGroupoid.sumHom_inl, |
| pointEvaluationIso_enum,Labelled.PointFiber.mu_inl] |
| | inr x => |
| obtain ⟨x,rfl⟩ := (enumerate V b).surjective x |
| erw [mu_inr,pointEvaluationIso_enum,pointFiber_sum_inr,FiniteSetGroupoid.sumHom_inr, |
| pointEvaluationIso_enum,Labelled.PointFiber.mu_inr] |
|
|
| variable {S b hp hl} |
| lemma pointFiber_empty_of_new_label (hc : ∀j,S.L j ≠ c.2) (U : S.Obj) : |
| IsEmpty (Labelled.PointFiber.Fiber U.obj c) := ⟨by |
| intro x |
| obtain ⟨j,hj⟩ := U.supported x.val.val.1 |
| exact hc j (hj.trans x.property.2)⟩ |
| lemma point_new_label_size (hc : ∀j,S.L j ≠ c.2) (U : S.Obj) : |
| (Labelled.PointFiber.E.obj U.obj c).size=0 := by |
| have := pointFiber_empty_of_new_label c hc U |
| exact Fintype.card_eq_zero |
| noncomputable def emptyPointIso (hc : ∀j,S.L j ≠ c.2) : |
| FiniteSetGroupoid.terminal ⋙ FiniteSetGroupoid.emptyFunctor ≅ |
| S.forget ⋙ Labelled.PointFiber.evaluation ⋙ project _ c := |
| NatIso.ofComponents (fun U => asIso (X := FiniteSetGroupoid.empty) |
| (Y := Labelled.PointFiber.E.obj U.obj c) (finCongr (point_new_label_size c hc U).symm)) (by |
| intros; apply Equiv.ext; intro x; exact Fin.elim0 x) |
| noncomputable instance emptyPointIsoMonoidal (hc : ∀j,S.L j ≠ c.2) : |
| NatTrans.IsMonoidal (emptyPointIso c hc).hom where |
| unit := by apply Equiv.ext; intro x; exact Fin.elim0 x |
| tensor _ _ := by apply Equiv.ext; intro x; exact Fin.elim0 x |
|
|
| end SimpleAmenable.PolygonObject.LabelledStage.Stage |
|
|
| end |
|
|
| end OAI |
|
|
