| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.DiagonalResolution |
|
|
| namespace OAI |
|
|
| open CategoryTheory Limits SimplicialObject Simplicial Opposite AlgebraicTopology |
| open HomologicalComplex HomologicalComplex₂ |
| namespace BiResolution |
|
|
| open DiagonalResolution |
|
|
| noncomputable def row (p : SimplexCategory) : SimplicialObject (D ⥤ A) where |
| obj q := free (p,q.unop) |
| map f := pre (Prod.mkHom (𝟙 p) f.unop) |
| map_id q := by simpa only [unop_id,←prod_id] using pre_id (p,q.unop) |
| map_comp f g := by |
| simpa only [prod_comp, Prod.mkHom, unop_comp, Category.id_comp] using |
| pre_comp (Prod.mkHom (𝟙 p) g.unop) (Prod.mkHom (𝟙 p) f.unop) |
|
|
| noncomputable def rowMap {p p' : SimplexCategory} (f : p ⟶ p') : row p' ⟶ row p where |
| app q := pre (Prod.mkHom f (𝟙 q.unop)) |
| naturality q q' g := by |
| dsimp [row] |
| rw [←pre_comp,←pre_comp] |
| congr 1 |
|
|
| noncomputable def chains : SimplicialObject (SimplicialObject (D ⥤ A)) where |
| obj p := row p.unop |
| map f := rowMap f.unop |
| map_id p := by |
| apply NatTrans.ext |
| funext q |
| dsimp [rowMap] |
| simpa only [unop_id,←prod_id,row] using pre_id (p.unop,q.unop) |
| map_comp f g := by |
| apply NatTrans.ext |
| funext q |
| dsimp [rowMap, row] |
| simpa only [prod_comp,Prod.mkHom,unop_comp,Category.id_comp] using |
| pre_comp (Prod.mkHom g.unop (𝟙 q.unop)) (Prod.mkHom f.unop (𝟙 q.unop)) |
|
|
| noncomputable def complex : HomologicalComplex₂ (D ⥤ A) (ComplexShape.down ℕ) (ComplexShape.down ℕ) := |
| AlternatingFaceMapComplex.obj (chains ⋙ alternatingFaceMapComplex (D ⥤ A)) |
|
|
| noncomputable def total : ChainComplex (D ⥤ A) ℕ := complex.total (ComplexShape.down ℕ) |
|
|
| instance projective (n : ℕ) : Projective (total.X n) := by |
| change Projective (∐ (fun ij : {ij : ℕ × ℕ // ij.1+ij.2=n} => free (⦋ij.1.1⦌,⦋ij.1.2⦌))) |
| have (ij : {ij : ℕ × ℕ // ij.1+ij.2=n}) : Projective (free (⦋ij.1.1⦌,⦋ij.1.2⦌)) := |
| (evaluationAdjunctionRight A _).map_projective Z inferInstance |
| infer_instance |
|
|
| noncomputable def rowAt (d : D) (p : SimplexCategory) : SimplicialObject A := |
| row p ⋙ (evaluation D A).obj d |
|
|
| noncomputable def chainsAt (d : D) : SimplicialObject (SimplicialObject A) := |
| chains ⋙ (SimplicialObject.whiskering _ _).obj ((evaluation D A).obj d) |
|
|
| noncomputable def point (d : D) : SimplicialObject A where |
| obj q := ∐ (fun _ : q.unop ⟶ d.2 => Z) |
| map f := Sigma.desc (fun g => Sigma.ι (fun _ : _ ⟶ d.2 => Z) (f.unop ≫ g)) |
| map_id q := by apply Sigma.hom_ext; intro g; simp |
| map_comp f g := by apply Sigma.hom_ext; intro h; simp [Category.assoc] |
|
|
| @[simp] lemma chainsAt_obj_map (d : D) (p : SimplexCategoryᵒᵖ) |
| {q q' : SimplexCategoryᵒᵖ} (f : q ⟶ q') : |
| ((chainsAt d).obj p).map f = (pre (Prod.mkHom (𝟙 p.unop) f.unop)).app d := rfl |
| @[simp] lemma chainsAt_map_app (d : D) {p p' : SimplexCategoryᵒᵖ} (f : p ⟶ p') |
| (q : SimplexCategoryᵒᵖ) : |
| ((chainsAt d).map f).app q = (pre (Prod.mkHom f.unop (𝟙 q.unop))).app d := rfl |
| noncomputable def augmentation (d : D) : chainsAt d ⟶ (SimplicialObject.const _).obj (point d) where |
| app p := |
| { app q := Sigma.desc (fun f : (p.unop,q.unop) ⟶ d => Sigma.ι (fun _ : q.unop ⟶ d.2 => Z) f.2) |
| naturality q q' f := by |
| apply Sigma.hom_ext |
| intro g |
| simp [point,pre] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc, |
| Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] } |
| naturality p p' f := by |
| apply NatTrans.ext |
| funext q |
| apply Sigma.hom_ext |
| intro g |
| simp [pre] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] |
| rfl |
|
|
| noncomputable def augmentedAt (d : D) : SimplicialObject.Augmented (SimplicialObject A) where |
| left := chainsAt d |
| right := point d |
| hom := augmentation d |
| noncomputable def section_ (d : D) : point d ⟶ (chainsAt d).obj (op ⦋0⦌) where |
| app q := Sigma.desc (fun f : q.unop ⟶ d.2 => |
| Sigma.ι (fun _ : (⦋0⦌,q.unop) ⟶ d => Z) (zeroSimplex d.1,f)) |
| naturality q q' f := by |
| apply Sigma.hom_ext |
| intro g |
| simp [point,pre] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] |
| rfl |
| |
| open SSet.Augmented.StandardSimplex in |
| noncomputable def shift_ (d : D) (n : ℕ) : |
| (chainsAt d).obj (op ⦋n⦌) ⟶ (chainsAt d).obj (op ⦋n+1⦌) where |
| app q := Sigma.desc (fun f : (⦋n⦌,q.unop) ⟶ d => |
| Sigma.ι (fun _ : (⦋n+1⦌,q.unop) ⟶ d => Z) (shift f.1,f.2)) |
| naturality q q' f := by |
| apply Sigma.hom_ext |
| intro g |
| simp [pre] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc, |
| Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] |
| noncomputable def extra (d : D) : (augmentedAt d).ExtraDegeneracy where |
| s' := section_ d |
| s := shift_ d |
| s'_comp_ε := by |
| apply NatTrans.ext |
| funext q |
| apply Sigma.hom_ext |
| intro f |
| dsimp only [augmentedAt,section_,shift_,augmentation,SimplicialObject.δ,SimplicialObject.σ,NatTrans.comp_app,NatTrans.id_app] |
| simp |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc, Category.comp_id] |
| s₀_comp_δ₁ := by |
| apply NatTrans.ext |
| funext q |
| apply Sigma.hom_ext |
| intro f |
| dsimp only [augmentedAt,section_,shift_,augmentation,SimplicialObject.δ,SimplicialObject.σ,NatTrans.comp_app,NatTrans.id_app] |
| simp only [Quiver.Hom.unop_op, chainsAt_map_app, pre, prod_comp, Prod.mkHom, Category.id_comp] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc, |
| Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] |
| simp only [shift_δ₁] |
| rfl |
| s_comp_δ₀ n := by |
| apply NatTrans.ext |
| funext q |
| apply Sigma.hom_ext |
| intro f |
| dsimp only [augmentedAt,section_,shift_,augmentation,SimplicialObject.δ,SimplicialObject.σ,NatTrans.comp_app,NatTrans.id_app] |
| simp only [Quiver.Hom.unop_op, chainsAt_map_app, pre, prod_comp, Prod.mkHom, Category.id_comp] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] |
| simp only [shift_δ₀, Prod.eta] |
| exact (Category.comp_id _).symm |
| s_comp_δ n i := by |
| apply NatTrans.ext |
| funext q |
| apply Sigma.hom_ext |
| intro f |
| dsimp only [augmentedAt,section_,shift_,augmentation,SimplicialObject.δ,SimplicialObject.σ,NatTrans.comp_app,NatTrans.id_app] |
| simp only [Quiver.Hom.unop_op, chainsAt_map_app, pre, prod_comp, Prod.mkHom, Category.id_comp] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc, |
| Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] |
| simp only [shift_δ_succ] |
| rfl |
| s_comp_σ n i := by |
| apply NatTrans.ext |
| funext q |
| apply Sigma.hom_ext |
| intro f |
| dsimp only [augmentedAt,section_,shift_,augmentation,SimplicialObject.δ,SimplicialObject.σ,NatTrans.comp_app,NatTrans.id_app] |
| simp only [Quiver.Hom.unop_op, chainsAt_map_app, pre, prod_comp, Prod.mkHom, Category.id_comp] |
| erw [Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc, |
| Sigma.ι_comp_desc_assoc, Sigma.ι_comp_desc] |
| simp only [shift_σ_succ] |
| rfl |
|
|
| end BiResolution |
|
|
| end OAI |
|
|
