| 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 |
|
|
