| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Homology.ExactShapeHomology |
|
|
| namespace OAI |
|
|
| section |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory Simplicial Opposite |
| namespace ComponentStable |
| open FreeChains ComponentTranslation |
|
|
| variable {C:Type} [Groupoid.{0} C] [MonoidalCategory C] [SymmetricCategory C] |
| noncomputable abbrev H (q:ℕ) := SSet.homologyFunctor Z q |
| noncomputable abbrev object (q:ℕ) := colimit (homologyDiagram (C:=C) q) |
| noncomputable def inclusion (p:Skeleton C) (q:ℕ) : |
| (nerve (Fiber p)).homology Z q ⟶ (nerve C).homology Z q := |
| SSet.homologyMap (nerveMap (property p).ι) Z q |
| noncomputable def unionCocone (q:ℕ) : |
| Cocone (ComponentCoproduct.diagram C ⋙ H q) where |
| pt := object (C:=C) q |
| ι := Discrete.natTrans (fun p=>colimit.ι (homologyDiagram (C:=C) q) ((ActionCategory.objEquiv _ _) p.as)) |
| noncomputable def toStable (q:ℕ) : (nerve C).homology Z q ⟶ object (C:=C) q := |
| (ComponentCoproduct.homologyIsColimit C q).desc (unionCocone q) |
| omit [SymmetricCategory C] in |
| @[reassoc (attr:=simp)] lemma inclusion_toStable (p:Skeleton C) (q:ℕ) : |
| inclusion p q ≫ toStable q=colimit.ι (homologyDiagram q) ((ActionCategory.objEquiv _ _) p) := |
| (ComponentCoproduct.homologyIsColimit C q).fac (unionCocone q) ⟨p⟩ |
| instance epi_toStable (q:ℕ) : Epi (toStable (C:=C) q) where |
| left_cancellation f g h := by |
| apply colimit.hom_ext |
| intro stage |
| obtain ⟨component,rfl⟩ := (ActionCategory.objEquiv (Skeleton C) (Skeleton C)).surjective stage |
| have hp:=congrArg (fun k=>inclusion component q ≫ k) h |
| simp only [←Category.assoc,inclusion_toStable] at hp |
| change colimit.ι (homologyDiagram (C:=C) q) ((ActionCategory.objEquiv _ _) component) ≫ f = |
| colimit.ι (homologyDiagram (C:=C) q) ((ActionCategory.objEquiv _ _) component) ≫ g |
| exact hp |
| omit [SymmetricCategory C] in |
| lemma translate_toStable (r p s:Skeleton C) (h:r*p=s) (q:ℕ) : |
| SSet.homologyMap (nerveMap (translate r p s h)) Z q ≫ inclusion s q ≫ toStable q = |
| inclusion p q ≫ toStable q := by |
| rw [inclusion_toStable,inclusion_toStable] |
| exact colimit.w (homologyDiagram q) (show (ActionCategory.objEquiv _ _) p ⟶ |
| (ActionCategory.objEquiv _ _) s from ⟨r,h⟩) |
| variable {D E:Type} [Category.{0} D] [Category.{0} E] |
| omit [MonoidalCategory C] [SymmetricCategory C] in |
| lemma map_comp (F:C⥤D) (G:D⥤E) (q:ℕ) : |
| SSet.homologyMap (nerveMap (F⋙G)) Z q= |
| SSet.homologyMap (nerveMap F) Z q ≫ SSet.homologyMap (nerveMap G) Z q := |
| by |
| change SSet.homologyMap (nerveMap F ≫ nerveMap G) Z q=_ |
| rw [SSet.homologyMap_comp] |
| omit [MonoidalCategory C] [SymmetricCategory C] in |
| lemma map_const (U:D) (q:ℕ) (hq:q≠0) : SSet.homologyMap (nerveMap ((Functor.const C).obj U)) Z q=0 := by |
| have h : nerveMap ((Functor.const C).obj U)=SSet.const (nerveEquiv.symm U) := by |
| ext n x; rfl |
| rw [h] |
| exact ConnectedProduct.const_homology_zero _ q hq |
| end ComponentStable |
|
|
| end |
|
|
| end OAI |
|
|
