Theorems · Theorem · category theory

CategoryTheory.InjectiveResolution.rightDerivedToHomotopyCategory_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),
  (CategoryTheory.NatTrans.rightDerivedToHomotopyCategory α).app X =
    CategoryTheory.CategoryStruct.comp (P.isoRightDerivedToHomotopyCategoryObj F).hom
      (CategoryTheory.CategoryStruct.comp
        ((HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
          ((CategoryTheory.NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex))
        (P.isoRightDerivedToHomotopyCategoryObj G).inv)

Defined in

Mathlib.CategoryTheory.Abelian.RightDerived

Cited by

1 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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