Theorems · Theorem · category theory

CategoryTheory.ProjectiveResolution.extEquivCohomologyClass_symm_mk_hom

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
  [inst_2 : CategoryTheory.HasExt C] {X Y : C} (R : CategoryTheory.ProjectiveResolution X) {n : ℕ}
  [inst_3 : HasDerivedCategory C]
  (x : CochainComplex.HomComplex.Cocycle R.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) ↑n),
  (R.extEquivCohomologyClass.symm (CochainComplex.HomComplex.CohomologyClass.mk x)).hom =
    (CategoryTheory.ShiftedHom.mk₀ 0 ⋯
          (CategoryTheory.CategoryStruct.comp ((DerivedCategory.singleFunctorIsoCompQ C 0).hom.app X)
            (CategoryTheory.inv (DerivedCategory.Q.map R.π')))).comp
      ((CategoryTheory.ShiftedHom.map (CochainComplex.HomComplex.Cocycle.equivHomShift.symm x) DerivedCategory.Q).comp
        (CategoryTheory.ShiftedHom.mk₀ 0 ⋯ ((DerivedCategory.singleFunctorIsoCompQ C 0).inv.app Y)) ⋯)
      ⋯

Defined in

Mathlib.CategoryTheory.Abelian.Projective.Ext

Cited by

3 results in Mathlib

Foundations

Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasExtHasDerivedCategory