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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | 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
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