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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | 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
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