Theorems · Theorem · category theory

CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_inv_naturality

∀ {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.HasProjectiveResolutions C]
  [inst_4 : CategoryTheory.Abelian D] {X Y : C} (f : X ⟶ Y) (P : CategoryTheory.ProjectiveResolution X)
  (Q : CategoryTheory.ProjectiveResolution Y) (φ : P.complex ⟶ Q.complex),
  CategoryTheory.CategoryStruct.comp (φ.f 0) (Q.π.f 0) = CategoryTheory.CategoryStruct.comp (P.π.f 0) f →
    ∀ (F : CategoryTheory.Functor C D) [inst_5 : F.Additive],
      CategoryTheory.CategoryStruct.comp (P.isoLeftDerivedToHomotopyCategoryObj F).inv
          (F.leftDerivedToHomotopyCategory.map f) =
        CategoryTheory.CategoryStruct.comp
          (((F.mapHomologicalComplex (ComplexShape.down ℕ)).comp
                (HomotopyCategory.quotient D (ComplexShape.down ℕ))).map
            φ)
          (Q.isoLeftDerivedToHomotopyCategoryObj F).inv

Defined in

Mathlib.CategoryTheory.Abelian.LeftDerived

Cited by

1 results in Mathlib

Foundations

Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasProjectiveResolutionsCategoryTheory.AbelianCategoryTheory.Functor.Additive