Theorems · Theorem · category theory

CategoryTheory.ProjectiveResolution.extMk_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)
  [inst_3 : HasDerivedCategory C] {n : ℕ} (f : R.complex.X n ⟶ Y) (m : ℕ) (hm : n + 1 = m)
  (hf : CategoryTheory.CategoryStruct.comp (R.complex.d m n) f = 0),
  (R.extMk f m hm hf).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
              (CochainComplex.HomComplex.Cocycle.toSingleMk
                (CategoryTheory.CategoryStruct.comp (R.cochainComplexXIso (-↑n) n ⋯).hom f) ⋯ (-↑m) ⋯ ⋯))
            DerivedCategory.Q).comp
        (CategoryTheory.ShiftedHom.mk₀ 0 ⋯ ((DerivedCategory.singleFunctorIsoCompQ C 0).inv.app Y)) ⋯)
      ⋯

Defined in

Mathlib.CategoryTheory.Abelian.Projective.Ext

Cited by

1 results in Mathlib

Foundations

Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasExtHasDerivedCategory