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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
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