Theorems · Definition · category theory

CategoryTheory.ProjectiveResolution.iso

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      [inst_2 : CategoryTheory.HasProjectiveResolutions C] →
        {X : C} →
          (P : CategoryTheory.ProjectiveResolution X) →
            (CategoryTheory.projectiveResolutions C).obj X ≅
              (HomotopyCategory.quotient C (ComplexShape.down ℕ)).obj P.complex

If P : ProjectiveResolution X, then the chosen (projectiveResolutions C).obj X is isomorphic (in the homotopy category) to P.complex.

Defined in

Mathlib.CategoryTheory.Abelian.Projective.Resolution

Cited by

7 results in Mathlib

Foundations

Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasProjectiveResolutions