Theorems · Definition · category theory

CategoryTheory.ProjectiveResolution.isoLeftDerivedObj

{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] →
                      (n : ℕ) →
                        (F.leftDerived n).obj X ≅
                          (HomologicalComplex.homologyFunctor D (ComplexShape.down ℕ) n).obj
                            ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)

We can compute a left derived functor using a chosen projective resolution.

Defined in

Mathlib.CategoryTheory.Abelian.LeftDerived

Cited by

8 results in Mathlib

Foundations

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

Assumes

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