Theorems · Theorem · category theory

DerivedCategory.left_fac

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
  [inst_2 : HasDerivedCategory C] {X Y : CochainComplex C ℤ} (f : DerivedCategory.Q.obj X ⟶ DerivedCategory.Q.obj Y),
  ∃ Y' g s,
    ∃ (x : CategoryTheory.IsIso (DerivedCategory.Q.map s)),
      f = CategoryTheory.CategoryStruct.comp (DerivedCategory.Q.map g) (CategoryTheory.inv (DerivedCategory.Q.map s))

Any morphism f : Q.obj X ⟶ Q.obj Y in the derived category can be written as f = Q.map g ≫ inv (Q.map s) with g : X ⟶ Y' and s : Y ⟶ Y' a quasi-isomorphism.

Defined in

Mathlib.Algebra.Homology.DerivedCategory.Fractions

Cited by

1 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianHasDerivedCategory