| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.EZNatural |
| import OAI.GroupTheory.SimpleAmenable.Homology.TotalComparison |
|
|
| namespace OAI |
|
|
| section |
| open CategoryTheory Limits SimplicialObject Simplicial Opposite AlgebraicTopology |
| open HomologicalComplex HomologicalComplex₂ |
| namespace EilenbergZilber |
|
|
| open DiagonalResolution ModelAssembly |
| variable {X Y : Dᵒᵖ ⥤ Type} (φ : X ⟶ Y) |
| noncomputable def bisimplicialMap : bisimplicial X ⟶ bisimplicial Y := |
| Functor.curry.map (Functor.whiskerLeft |
| (CategoryTheory.prodOpEquiv SimplexCategory (D:=SimplexCategory)).inverse φ) |
| lemma level_homology_isIso |
| (hφ : ∀ p q,IsIso (SSet.homologyMap ((bisimplicialMap φ).app p) Z q)) (n : ℕ) : |
| IsIso (homologyMap (diagonalMap φ) n) := by |
| have : IsIso (homologyMap (total.map (twoMap φ) c) n) := by |
| apply TotalComparison.total_homology_isIso |
| intro p q |
| exact hφ (op ⦋p⦌) q |
| have he : homologyMap (diagonalMap φ) n = |
| (homologyIsoN X n).hom ≫ homologyMap (total.map (twoMap φ) c) n ≫ (homologyIsoN Y n).inv := by |
| rw [←Category.assoc,←homology_natural,Category.assoc,Iso.hom_inv_id,Category.comp_id] |
| rw [he] |
| infer_instance |
| end EilenbergZilber |
|
|
| end |
|
|
| section |
|
|
| open CategoryTheory MonoidalCategory Simplicial Opposite |
| namespace NerveHomotopy |
|
|
| universe u v |
| variable {C D : Type u} [Category.{v} C] [Category.{v} D] |
| variable {F G : C ⥤ D} |
|
|
| def cylinder (α : F ⟶ G) : C × Fin 2 ⥤ D := |
| CategoryTheory.Functor.uncurry.obj (ComposableArrows.mk₁ α).flip |
|
|
| noncomputable def homotopyMap (α : F ⟶ G) : nerve C ⊗ Δ[1] ⟶ nerve D where |
| app n := ↾fun x => x.1.prod' ((SSet.stdSimplex.objEquiv x.2).toOrderHom.toFunctor) ⋙ cylinder α |
| naturality _ _ f := by ext x; rfl |
| @[simp] lemma homotopyMap_zero (α : F ⟶ G) : SSet.ι₀ ≫ homotopyMap α=nerveMap F := by |
| ext n x |
| apply CategoryTheory.Functor.ext |
| · intro i j f |
| erw [eqToHom_refl, Category.id_comp, Category.comp_id] |
| change F.map (x.map f) ≫ 𝟙 _=F.map (x.map f) |
| simp |
| · intro i |
| rfl |
| @[simp] lemma homotopyMap_one (α : F ⟶ G) : SSet.ι₁ ≫ homotopyMap α=nerveMap G := by |
| ext n x |
| apply CategoryTheory.Functor.ext |
| · intro i j f |
| erw [eqToHom_refl, Category.id_comp, Category.comp_id] |
| change G.map (x.map f) ≫ 𝟙 _=G.map (x.map f) |
| simp |
| · intro i |
| rfl |
|
|
| noncomputable def ofNatTrans (α : F ⟶ G) : SSet.Homotopy (nerveMap F) (nerveMap G) where |
| h := homotopyMap α |
| h₀ := homotopyMap_zero α |
| h₁ := homotopyMap_one α |
| rel := by ext n x; exact x.1.property.elim |
| |
| universe w |
| variable {A : Type w} [Category A] [Preadditive A] |
| [Limits.HasCoproducts.{max u v} A] [CategoryWithHomology A] |
|
|
| noncomputable def homologyIso (e : C ≌ D) (R : A) (n : ℕ) : |
| (nerve C).homology R n ≅ (nerve D).homology R n where |
| hom := SSet.homologyMap (nerveMap e.functor) R n |
| inv := SSet.homologyMap (nerveMap e.inverse) R n |
| hom_inv_id := by |
| rw [←SSet.homologyMap_comp] |
| have h := (ofNatTrans e.unitIso.hom).congr_homologyMap R n |
| change SSet.homologyMap (nerveMap (e.functor ⋙ e.inverse)) R n = _ |
| rw [←h] |
| exact SSet.homologyMap_id _ _ _ |
| inv_hom_id := by |
| rw [←SSet.homologyMap_comp] |
| have h := (ofNatTrans e.counitIso.hom).congr_homologyMap R n |
| change SSet.homologyMap (nerveMap (e.inverse ⋙ e.functor)) R n = _ |
| rw [h] |
| exact SSet.homologyMap_id _ _ _ |
|
|
| noncomputable instance homologyMap_isIso (F : C ⥤ D) [F.IsEquivalence] |
| (R : A) (n : ℕ) : IsIso (SSet.homologyMap (nerveMap F) R n) := |
| (homologyIso F.asEquivalence R n).isIso_hom |
|
|
| end NerveHomotopy |
|
|
| end |
|
|
| open CategoryTheory SimplicialObject Simplicial Opposite AlgebraicTopology |
| namespace SimplicialDiagonal |
| open DiagonalResolution |
|
|
| abbrev I := SimplexCategoryᵒᵖ |
| def uncurry (X : I ⥤ SSet) : DiagonalResolution.Dᵒᵖ ⥤ Type := |
| (prodOpEquiv SimplexCategory (D:=SimplexCategory)).functor ⋙ Functor.uncurry.obj X |
| def diagonal : (I ⥤ SSet) ⥤ SSet where |
| obj X := EilenbergZilber.diag ⋙ uncurry X |
| map f := Functor.whiskerLeft EilenbergZilber.diag |
| (Functor.whiskerLeft (prodOpEquiv SimplexCategory (D:=SimplexCategory)).functor (Functor.uncurry.map f)) |
| map_id _ := by ext; rfl |
| map_comp _ _ := by ext; rfl |
| def rowIso (X : I ⥤ SSet) (p : I) : |
| (EilenbergZilber.bisimplicial (uncurry X)).obj p ≅ X.obj p := |
| NatIso.ofComponents (fun _ => Iso.refl _) (by |
| intro q q' f |
| simp [EilenbergZilber.bisimplicial,uncurry,Functor.uncurry]) |
| private theorem homologyMap_comp_iso {W X Y Z : SSet} (e : W ≅ X) (f : X ⟶ Y) |
| (e' : Y ≅ Z) (q : ℕ) (hf : IsIso (SSet.homologyMap f DiagonalResolution.Z q)) : |
| IsIso (SSet.homologyMap (e.hom ≫ f ≫ e'.hom) DiagonalResolution.Z q) := by |
| let := hf |
| rw [SSet.homologyMap_comp, SSet.homologyMap_comp] |
| infer_instance |
|
|
| private theorem row_homologyMap_isIso {X Y : I ⥤ SSet} (f : X ⟶ Y) |
| (hf : ∀ p q, IsIso (SSet.homologyMap (f.app p) DiagonalResolution.Z q)) |
| (p : SimplexCategoryᵒᵖ) (q : ℕ) : |
| IsIso (SSet.homologyMap ((EilenbergZilber.bisimplicialMap |
| (Functor.whiskerLeft (prodOpEquiv SimplexCategory (D:=SimplexCategory)).functor |
| (Functor.uncurry.map f))).app p) DiagonalResolution.Z q) := by |
| have he : (EilenbergZilber.bisimplicialMap |
| (Functor.whiskerLeft (prodOpEquiv SimplexCategory (D:=SimplexCategory)).functor |
| (Functor.uncurry.map f))).app p = |
| (rowIso X p).hom ≫ f.app p ≫ (rowIso Y p).inv := by |
| ext k x |
| rfl |
| rw [he] |
| exact homologyMap_comp_iso (rowIso X p) (f.app p) (rowIso Y p).symm q (hf p q) |
|
|
| lemma homologyMap_isIso {X Y : I ⥤ SSet} (f : X ⟶ Y) |
| (hf : ∀ p q, IsIso (SSet.homologyMap (f.app p) DiagonalResolution.Z q)) (n : ℕ) : |
| IsIso (SSet.homologyMap (diagonal.map f) DiagonalResolution.Z n) := by |
| exact EilenbergZilber.level_homology_isIso _ (row_homologyMap_isIso f hf) n |
|
|
| def nerveDiagonal : (I ⥤ Cat.{0,0}) ⥤ SSet := |
| (Functor.whiskeringRight I Cat SSet).obj nerveFunctor ⋙ diagonal |
| lemma nerveDiagonal_isIso {X Y : I ⥤ Cat.{0,0}} (f : X ⟶ Y) |
| (hf : ∀ p, (f.app p).toFunctor.IsEquivalence) (n : ℕ) : |
| IsIso (SSet.homologyMap (nerveDiagonal.map f) DiagonalResolution.Z n) := by |
| apply homologyMap_isIso |
| intro p q |
| have := hf p |
| exact NerveHomotopy.homologyMap_isIso (f.app p).toFunctor DiagonalResolution.Z q |
| end SimplicialDiagonal |
|
|
| end OAI |
|
|
