Theorems · Theorem · category theory

DerivedCategory.left_fac_of_isStrictlyLE_of_isStrictlyGE

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
  [inst_2 : HasDerivedCategory C] {X Y : CochainComplex C ℤ} (a b : ℤ) [X.IsStrictlyLE b] [Y.IsStrictlyGE a]
  [Y.IsStrictlyLE b] (f : DerivedCategory.Q.obj X ⟶ DerivedCategory.Q.obj Y),
  ∃ Y',
    ∃ (_ : Y'.IsStrictlyGE a) (_ : Y'.IsStrictlyLE b),
      ∃ 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 with X strictly ≤ b, and Y strictly ≥ a and ≤ b can be written as f = Q.map g ≫ inv (Q.map s) with g : X ⟶ Y' and s : Y ⟶ Y' a quasi-isomorphism with Y' strictly ≥ a and ≤ b.

Defined in

Mathlib.Algebra.Homology.DerivedCategory.Fractions

Cited by

0 results in Mathlib

Foundations

Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianHasDerivedCategoryCochainComplex.IsStrictlyLECochainComplex.IsStrictlyGECochainComplex.IsStrictlyLE