| import OAI.Algebra.Finitistic.Bar.CoproductHomotopy |
| import OAI.Algebra.Finitistic.Bar.CoproductHomBijections |
| import Mathlib.Algebra.Homology.DerivedCategory.KProjective |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| noncomputable section |
| namespace DerivedBar |
| universe u v w |
| variable {C : Type u} [Category.{v} C] [Abelian C] |
| [HasColimitsOfShape (Discrete ℕ) C] [HasDerivedCategory.{w} C] |
| (F : Discrete ℕ ⥤ CochainComplex C ℤ) (s : Cocone F) (hs : IsColimit s) |
| [s.pt.IsKProjective] [∀ r, (F.obj r).IsKProjective] |
|
|
| /-- An actual coproduct of K-projective cochain complexes with K-projective |
| coproduct object remains a coproduct in the genuine derived category. -/ |
| def kProjectiveCoproductIsColimit : IsColimit (DerivedCategory.Q.mapCocone s) := by |
| let H := HomotopyCategory.quotient C (.up ℤ) |
| let hsH := homotopyCoproductIsColimit F s hs |
| let hsD := mapCoproductOfHomBijections DerivedCategory.Qh (F ⋙ H) (H.mapCocone s) hsH |
| (fun T => CochainComplex.IsKProjective.Qh_map_bijective s.pt T) |
| (fun r T => CochainComplex.IsKProjective.Qh_map_bijective (F.obj r) T) |
| letI : PreservesColimit F (H ⋙ DerivedCategory.Qh) := |
| preservesColimit_of_preserves_colimit_cocone hs hsD |
| letI : PreservesColimit F (DerivedCategory.Q : CochainComplex C ℤ ⥤ _) := |
| preservesColimit_of_natIso F (DerivedCategory.quotientCompQhIso C) |
| exact isColimitOfPreserves DerivedCategory.Q hs |
| end DerivedBar |
|
|
| end |
|
|
| end OAI |
|
|
