Theorems · Definition · category theory

CochainComplex.HomComplex.CohomologyClass.equivOfIsKProjective

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      {K L : CochainComplex C ℤ} →
        {n : ℤ} →
          [inst_2 :
              CategoryTheory.Localization.HasSmallLocalizedShiftedHom
                (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) ℤ K L] →
            [K.IsKProjective] →
              CochainComplex.HomComplex.CohomologyClass K L n ≃
                CategoryTheory.Localization.SmallShiftedHom (HomologicalComplex.quasiIso C (ComplexShape.up ℤ)) K L n

When K is a K-projective cochain complex, cohomology classes in CohomologyClass K L n identify to elements in a type SmallShiftedHom relatively to quasi-isomorphisms.

Defined in

Mathlib.Algebra.Homology.DerivedCategory.KProjective

Cited by

2 results in Mathlib

Foundations

Depth 109 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.Localization.HasSmallLocalizedShiftedHomCochainComplex.IsKProjective