Theorems · Theorem · category theory

CategoryTheory.ProjectiveResolution.fromLeftDerivedZero_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.HasProjectiveResolutions C]
  [inst_4 : CategoryTheory.Abelian D] {X : C} (P : CategoryTheory.ProjectiveResolution X)
  (F : CategoryTheory.Functor C D) [inst_5 : F.Additive],
  F.fromLeftDerivedZero.app X =
    CategoryTheory.CategoryStruct.comp (P.isoLeftDerivedObj F 0).hom
      (CategoryTheory.CategoryStruct.comp
        (ChainComplex.isoHomologyι₀ ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)).hom
        (P.fromLeftDerivedZero' F))

Defined in

Mathlib.CategoryTheory.Abelian.LeftDerived

Cited by

0 results in Mathlib

Foundations

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

Assumes

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