Theorems · Theorem · category theory

CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_inv_naturality_assoc

∀ {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 Y : C} (f : X ⟶ Y) (P : CategoryTheory.ProjectiveResolution X)
  (Q : CategoryTheory.ProjectiveResolution Y) (φ : P.complex ⟶ Q.complex),
  CategoryTheory.CategoryStruct.comp (φ.f 0) (Q.π.f 0) = CategoryTheory.CategoryStruct.comp (P.π.f 0) f →
    ∀ (F : CategoryTheory.Functor C D) [inst_5 : F.Additive] (n : ℕ) {Z : D} (h : (F.leftDerived n).obj Y ⟶ Z),
      CategoryTheory.CategoryStruct.comp (P.isoLeftDerivedObj F n).inv
          (CategoryTheory.CategoryStruct.comp ((F.leftDerived n).map f) h) =
        CategoryTheory.CategoryStruct.comp
          ((HomologicalComplex.homologyFunctor D (ComplexShape.down ℕ) n).map
            ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ))
          (CategoryTheory.CategoryStruct.comp (Q.isoLeftDerivedObj F n).inv h)

Defined in

Mathlib.CategoryTheory.Abelian.LeftDerived

Cited by

0 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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