Theorems · Definition · category theory

CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj

{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 : C} →
                (I : CategoryTheory.InjectiveResolution X) →
                  (F : CategoryTheory.Functor C D) →
                    [inst_5 : F.Additive] →
                      F.rightDerivedToHomotopyCategory.obj X ≅
                        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).comp
                              (HomotopyCategory.quotient D (ComplexShape.up ℕ))).obj
                          I.cocomplex

If I : InjectiveResolution Z and F : C ⥤ D is an additive functor, this is an isomorphism between F.rightDerivedToHomotopyCategory.obj X and the complex obtained by applying F to I.cocomplex.

Defined in

Mathlib.CategoryTheory.Abelian.RightDerived

Cited by

8 results in Mathlib

Foundations

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

Assumes

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