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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Simplicial.LabelledMonoidal
import OAI.GroupTheory.SimpleAmenable.Simplicial.FiniteSetMonoidal
import OAI.GroupTheory.SimpleAmenable.Configurations.StageOrder
namespace OAI
section
open CategoryTheory Classical MonoidalCategory
namespace SimpleAmenable.PolygonObject.Labelled.Cell
variable {a n : ℕ} (P : polygonAlgebra a) (l : LabelledStage.ReducedLabel n)
noncomputable abbrev object (U : FiniteSetGroupoid) : Labelled a n :=
⟨⟨U.size,fun _ => P⟩,fun _ => l.val,fun _ => l.property⟩
noncomputable def pointMap {U V : FiniteSetGroupoid} (f : U ⟶ V) :
(object P l U).polygon.Point ≃ (object P l V).polygon.Point where
toFun x := ⟨(f x.val.1,x.val.2),x.property⟩
invFun y := ⟨(f.symm y.val.1,y.val.2),y.property⟩
left_inv x := by apply Subtype.ext; exact Prod.ext (f.symm_apply_apply _) rfl
right_inv y := by apply Subtype.ext; exact Prod.ext (f.apply_symm_apply _) rfl
noncomputable def arrow {U V : FiniteSetGroupoid} (f : U ⟶ V) : object P l U ⟶ object P l V where
arrow := relabelArrow (pointMap P l f) f (fun _ => rfl)
positional _ := rfl
labelled _ := rfl
noncomputable def functor : FiniteSetGroupoid ⥤ Labelled a n where
obj := object P l
map := arrow P l
map_id _ := by apply Hom.ext; apply Arrow.ext; apply Equiv.ext; intro x; rfl
map_comp _ _ := by apply Hom.ext; apply Arrow.ext; apply Equiv.ext; intro x; rfl
@[simp] lemma arrow_apply {U V : FiniteSetGroupoid} (f : U ⟶ V)
(x : (object P l U).polygon.Point) :
((arrow P l f).arrow.toEquiv x).val = (f x.val.1,x.val.2) := rfl
noncomputable def tensorPoint (U V : FiniteSetGroupoid) :
(sum (object P l U) (object P l V)).polygon.Point ≃ (object P l (U⊗V)).polygon.Point where
toFun x := ⟨x.val,by
obtain ⟨y,rfl⟩ := (sumPointEquiv (object P l U).polygon (object P l V).polygon).surjective x
cases y with
| inl y => simpa only [sumPointEquiv_inl,object] using y.property
| inr y => simpa only [sumPointEquiv_inr,object] using y.property⟩
invFun x := ⟨x.val,by
change x.val.2 ∈ (Fin.addCases (motive := fun _ => polygonAlgebra a)
(fun _ => P) (fun _ => P) x.val.1).val
refine Fin.addCases (motive := fun i => x.val.2 ∈ (Fin.addCases
(motive := fun _ => polygonAlgebra a) (fun _ => P) (fun _ => P) i).val)
(fun i => ?_) (fun i => ?_) x.val.1
· simpa only [Fin.addCases_left,object] using x.property
· simpa only [Fin.addCases_right,object] using x.property⟩
left_inv _ := rfl
right_inv _ := rfl
noncomputable def tensorArrow (U V : FiniteSetGroupoid) :
(object P l U) ⊗ (object P l V) ⟶ object P l (U⊗V) where
arrow := relabelArrow (tensorPoint P l U V) id (fun _ => rfl)
positional _ := rfl
labelled x := by
change l.val = Fin.addCases (fun _ => l.val) (fun _ => l.val) x.val.1
refine Fin.addCases (motive := fun i => l.val = Fin.addCases
(motive := fun _ => Fin n → CutRing × CutRing) (fun _ => l.val) (fun _ => l.val) i)
(fun i => ?_) (fun i => ?_) x.val.1
· simp only [Fin.addCases_left]
· simp only [Fin.addCases_right]
@[simp] lemma tensorArrow_apply (U V : FiniteSetGroupoid)
(x : (sum (object P l U) (object P l V)).polygon.Point) :
((tensorArrow P l U V).arrow.toEquiv x).val=x.val := rfl
noncomputable def unitArrow : 𝟙_ (Labelled a n) ⟶ object P l (𝟙_ FiniteSetGroupoid) where
arrow := relabelArrow {
toFun x := Fin.elim0 x.val.1
invFun x := Fin.elim0 x.val.1
left_inv x := Fin.elim0 x.val.1
right_inv x := Fin.elim0 x.val.1 } Fin.elim0 (fun x => Fin.elim0 x.val.1)
positional x := Fin.elim0 x.val.1
labelled x := Fin.elim0 x.val.1
end SimpleAmenable.PolygonObject.Labelled.Cell
end
end OAI
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