Theorems · Definition · category theory

CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj

{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 : C} →
                (P : CategoryTheory.ProjectiveResolution X) →
                  (F : CategoryTheory.Functor C D) →
                    [inst_5 : F.Additive] →
                      F.leftDerivedToHomotopyCategory.obj X ≅
                        ((F.mapHomologicalComplex (ComplexShape.down ℕ)).comp
                              (HomotopyCategory.quotient D (ComplexShape.down ℕ))).obj
                          P.complex

If P : ProjectiveResolution Z and F : C ⥤ D is an additive functor, this is an isomorphism between F.leftDerivedToHomotopyCategory.obj X and the complex obtained by applying F to P.complex.

Defined in

Mathlib.CategoryTheory.Abelian.LeftDerived

Cited by

8 results in Mathlib

Foundations

Depth 109 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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