File size: 8,531 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Simplicial.TripleReindex
import OAI.GroupTheory.SimpleAmenable.Configurations.EmptyStage
import OAI.GroupTheory.SimpleAmenable.Simplicial.FiberSum
namespace OAI
open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory
namespace BarFiberProduct
universe homSource homTarget
variable {A C D:Type} [Category.{homSource} A] [Category.{homTarget} C]
[Category.{homTarget} D]
[MonoidalCategory A] [MonoidalCategory C] [MonoidalCategory D]
[BraidedCategory A] [BraidedCategory C] [BraidedCategory D]
noncomputable instance prodFunctorBraided (F:A⥤C) (G:A⥤D) [F.Braided] [G.Braided] :
(F.prod' G).Braided where
braided X Y := by
apply CategoryTheory.Prod.hom_ext
· simp only [CategoryTheory.prod_comp_fst,Functor.prod'_μ_fst]
change Functor.LaxMonoidal.μ F X Y ≫ F.map (β_ X Y).hom =
(β_ (F.obj X) (F.obj Y)).hom ≫ Functor.LaxMonoidal.μ F Y X
exact Functor.Braided.braided (F:=F) X Y
· simp only [CategoryTheory.prod_comp_snd,Functor.prod'_μ_snd]
change Functor.LaxMonoidal.μ G X Y ≫ G.map (β_ X Y).hom =
(β_ (G.obj X) (G.obj Y)).hom ≫ Functor.LaxMonoidal.μ G Y X
exact Functor.Braided.braided (F:=G) X Y
end BarFiberProduct
namespace IntervalBar.Diagram
variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] [SymmetricCategory C]
lemma bar₃Map_id_homology (j:ℕ) :
SSet.homologyMap (bar₃Map (𝟭 C)) DiagonalResolution.Z j=𝟙 _ := by
have := bar₃Map_homology_isIso (𝟭 C) j
apply (cancel_epi (SSet.homologyMap (bar₃Map (𝟭 C)) DiagonalResolution.Z j)).mp
rw [Category.comp_id,← SSet.homologyMap_comp,←bar₃Map_comp]
exact bar₃Map_eq_of_monoidalNatTrans (Functor.leftUnitor (𝟭 C)).hom j
end IntervalBar.Diagram
namespace SimpleAmenable.PolygonObject.FiniteSetGroupoid
open BarFiberProduct IntervalBar.Diagram BarFinitePower
open LabelledStage.Stage
noncomputable def merge : (Bool → FiniteSetGroupoid) ⥤ FiniteSetGroupoid :=
(project Bool true).prod' (project Bool false) ⋙ tensor FiniteSetGroupoid
noncomputable instance : merge.Braided := by unfold merge; infer_instance
noncomputable def selected (b k:Bool) : FiniteSetGroupoid ⥤ FiniteSetGroupoid :=
match b,k with
| false,false => 𝟭 _
| true,true => 𝟭 _
| _,_ => terminal.{0} ⋙ emptyFunctor.{0}
noncomputable instance selectedMonoidal (b k:Bool) : (selected b k).Monoidal := by
cases b <;> cases k <;> unfold selected <;> infer_instance
noncomputable instance selectedBraided (b k:Bool) : (selected b k).Braided := by
cases b <;> cases k <;> unfold selected <;> infer_instance
noncomputable def insertFiber (b:Bool) : FiniteSetGroupoid ⥤ (Bool→FiniteSetGroupoid) :=
Functor.pi' (selected b)
noncomputable instance (b:Bool) : (insertFiber b).Braided := by unfold insertFiber; infer_instance
noncomputable def insertCoordinate (b k:Bool) : insertFiber b ⋙ project Bool k ≅ selected b k := Iso.refl _
noncomputable instance (b k:Bool) : NatTrans.IsMonoidal (insertCoordinate b k).hom where
unit := by
simp [insertCoordinate,insertFiber,project,toEquiv]
cases b <;> cases k <;> rfl
tensor U V := by
change Functor.LaxMonoidal.μ (selected b k) U V ≫ 𝟙 _ =
(𝟙 _ ⊗ₘ 𝟙 _) ≫ Functor.LaxMonoidal.μ (selected b k) U V
rw [id_tensorHom_id,Category.comp_id,Category.id_comp]
lemma selected_homology (b k:Bool) (j:ℕ) (hj:0<j) :
SSet.homologyMap (bar₃Map (selected b k)) DiagonalResolution.Z j =
if b=k then 𝟙 _ else 0 := by
cases b <;> cases k
· exact bar₃Map_id_homology j
· change SSet.homologyMap (bar₃Map (terminal.{0} ⋙ emptyFunctor.{0})) DiagonalResolution.Z j=0
rw [bar₃Map_comp,SSet.homologyMap_comp]
have hz := bar₃_acyclic (C:=Terminal.{0}) (fun x y => ⟨eqToHom (Subsingleton.elim x y)⟩) j hj
rw [hz.eq_of_src (SSet.homologyMap (bar₃Map emptyFunctor.{0}) DiagonalResolution.Z j) 0,comp_zero]
· change SSet.homologyMap (bar₃Map (terminal.{0} ⋙ emptyFunctor.{0})) DiagonalResolution.Z j=0
rw [bar₃Map_comp,SSet.homologyMap_comp]
have hz := bar₃_acyclic (C:=Terminal.{0}) (fun x y => ⟨eqToHom (Subsingleton.elim x y)⟩) j hj
rw [hz.eq_of_src (SSet.homologyMap (bar₃Map emptyFunctor.{0}) DiagonalResolution.Z j) 0,comp_zero]
· exact bar₃Map_id_homology j
noncomputable def mergeInsertIso (b:Bool) : insertFiber b ⋙ merge ≅ 𝟭 FiniteSetGroupoid := by
cases b
· exact NatIso.ofComponents (fun U => asIso (C := FiniteSetGroupoid)
(X := sum empty U) (Y := U) (left U)) (by
intro U V f
change sumHom (𝟙 empty) f ≫ left V = left U ≫ f
exact (leftUnitor_naturality f))
· exact NatIso.ofComponents (fun U => asIso (C := FiniteSetGroupoid)
(X := sum U empty) (Y := U) (right U)) (by
intro U V f
change sumHom f (𝟙 empty) ≫ right V = right U ≫ f
exact (rightUnitor_naturality f))
lemma merge_μ (U V:Bool→FiniteSetGroupoid) :
Functor.LaxMonoidal.μ merge U V=tensorμ (U true) (U false) (V true) (V false) := by
unfold merge
rw [Functor.LaxMonoidal.comp_μ]
change tensorμ (U true) (U false) (V true) (V false) ≫
sumHom (Functor.LaxMonoidal.μ ((project Bool true).prod' (project Bool false)) U V).1
(Functor.LaxMonoidal.μ ((project Bool true).prod' (project Bool false)) U V).2 = _
rw [Functor.prod'_μ_fst,Functor.prod'_μ_snd]
change tensorμ (U true) (U false) (V true) (V false) ≫ sumHom (𝟙 _) (𝟙 _) = _
rw [sumHom_id]
apply Equiv.ext; intro x; rfl
lemma mergeInsert_false_μ (U V:FiniteSetGroupoid) :
Functor.LaxMonoidal.μ (insertFiber false ⋙ merge) U V = tensorμ empty U empty V := by
rw [Functor.LaxMonoidal.comp_μ,merge_μ]
change tensorμ _ _ _ _ ≫ sumHom (𝟙 _) (𝟙 _) = _
rw [sumHom_id]
apply Equiv.ext; intro x; rfl
lemma mergeInsert_true_μ (U V:FiniteSetGroupoid) :
Functor.LaxMonoidal.μ (insertFiber true ⋙ merge) U V = tensorμ U empty V empty := by
rw [Functor.LaxMonoidal.comp_μ,merge_μ]
change tensorμ _ _ _ _ ≫ sumHom (𝟙 _) (𝟙 _) = _
rw [sumHom_id]
apply Equiv.ext; intro x; rfl
noncomputable instance mergeInsertMonoidal (b:Bool) : NatTrans.IsMonoidal (mergeInsertIso b).hom where
unit := by cases b <;> apply Equiv.ext <;> intro x <;> exact Fin.elim0 x
tensor U V := by
cases b
· rw [mergeInsert_false_μ]
change tensorμ empty U empty V ≫ left (sum U V) = sumHom (left U) (left V) ≫ 𝟙 _
rw [Category.comp_id,tensorμ_eq]
dsimp only [CategoryStruct.comp,instGroupoid,toEquiv]
apply Equiv.ext; intro x
obtain ⟨z,rfl⟩ := (sumEquiv (sum empty U) (sum empty V)).surjective x
cases z with
| inl z =>
obtain ⟨t,rfl⟩ := (sumEquiv empty U).surjective z
cases t with
| inl t => exact Fin.elim0 t
| inr t =>
simp [toEquiv,sumHom,assoc,swap,left,inv_eq,Equiv.trans_apply,MonoidalCategory.tensorObj]
dsimp only [sum,empty]
rw [Equiv.symm_apply_apply]
rfl
| inr z =>
obtain ⟨t,rfl⟩ := (sumEquiv empty V).surjective z
cases t with
| inl t => exact Fin.elim0 t
| inr t =>
simp [toEquiv,sumHom,assoc,swap,left,inv_eq,Equiv.trans_apply,MonoidalCategory.tensorObj]
dsimp only [sum,empty]
rw [Equiv.symm_apply_apply]
rfl
· rw [mergeInsert_true_μ]
change tensorμ U empty V empty ≫ right (sum U V) = sumHom (right U) (right V) ≫ 𝟙 _
rw [Category.comp_id,tensorμ_eq]
dsimp only [CategoryStruct.comp,instGroupoid,toEquiv]
apply Equiv.ext; intro x
obtain ⟨z,rfl⟩ := (sumEquiv (sum U empty) (sum V empty)).surjective x
cases z with
| inl z =>
obtain ⟨t,rfl⟩ := (sumEquiv U empty).surjective z
cases t with
| inr t => exact Fin.elim0 t
| inl t =>
simp [toEquiv,sumHom,assoc,swap,right,inv_eq,Equiv.trans_apply,MonoidalCategory.tensorObj]
dsimp only [sum,empty]
rw [Equiv.symm_apply_apply]
rfl
| inr z =>
obtain ⟨t,rfl⟩ := (sumEquiv V empty).surjective z
cases t with
| inr t => exact Fin.elim0 t
| inl t =>
simp [toEquiv,sumHom,assoc,swap,right,inv_eq,Equiv.trans_apply,MonoidalCategory.tensorObj]
dsimp only [sum,empty]
rw [Equiv.symm_apply_apply]
rfl
end SimpleAmenable.PolygonObject.FiniteSetGroupoid
end OAI
|