Theorems · Theorem · category theory

CategoryTheory.InjectiveResolution.rightDerived_app_eq

∀ {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] {F G : CategoryTheory.Functor C D} [inst_5 : F.Additive] [inst_6 : G.Additive]
  (α : F ⟶ G) {X : C} (P : CategoryTheory.InjectiveResolution X) (n : ℕ),
  (CategoryTheory.NatTrans.rightDerived α n).app X =
    CategoryTheory.CategoryStruct.comp (P.isoRightDerivedObj F n).hom
      (CategoryTheory.CategoryStruct.comp
        ((HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) n).map
          ((CategoryTheory.NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex))
        (P.isoRightDerivedObj G n).inv)

A component of the natural transformation between right-derived functors can be computed using a chosen injective resolution.

Defined in

Mathlib.CategoryTheory.Abelian.RightDerived

Cited by

0 results in Mathlib

Foundations

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

Assumes

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