Theorems · Theorem · category theory

CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_hom_naturality_assoc

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u_1} [inst_1 : CategoryTheory.Category.{v_1, u_1} D]
  [inst_2 : CategoryTheory.Abelian C] [inst_3 : CategoryTheory.HasInjectiveResolutions C]
  [inst_4 : CategoryTheory.Abelian D] {X Y : C} (f : X ⟶ Y) (I : CategoryTheory.InjectiveResolution X)
  (J : CategoryTheory.InjectiveResolution Y) (φ : I.cocomplex ⟶ J.cocomplex),
  CategoryTheory.CategoryStruct.comp (I.ι.f 0) (φ.f 0) = CategoryTheory.CategoryStruct.comp f (J.ι.f 0) →
    ∀ (F : CategoryTheory.Functor C D) [inst_5 : F.Additive] {Z : HomotopyCategory D (ComplexShape.up ℕ)}
      (h :
        (HomotopyCategory.quotient D (ComplexShape.up ℕ)).obj
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj J.cocomplex) ⟶
          Z),
      CategoryTheory.CategoryStruct.comp (F.rightDerivedToHomotopyCategory.map f)
          (CategoryTheory.CategoryStruct.comp (J.isoRightDerivedToHomotopyCategoryObj F).hom h) =
        CategoryTheory.CategoryStruct.comp (I.isoRightDerivedToHomotopyCategoryObj F).hom
          (CategoryTheory.CategoryStruct.comp
            ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
              ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map φ))
            h)

Defined in

Mathlib.CategoryTheory.Abelian.RightDerived

Cited by

1 results in Mathlib

Foundations

Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasInjectiveResolutionsCategoryTheory.AbelianCategoryTheory.Functor.Additive