Theorems · Theorem · category theory

CategoryTheory.InjectiveResolution.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.InjectiveResolution Y)
  [inst_3 : HasDerivedCategory C] {n : ℕ} (f : X ⟶ R.cocomplex.X n) (m : ℕ) (hm : n + 1 = m)
  (hf : CategoryTheory.CategoryStruct.comp f (R.cocomplex.d n m) = 0),
  (R.extMk f m hm hf).hom =
    (CategoryTheory.ShiftedHom.mk₀ 0 ⋯ ((DerivedCategory.singleFunctorIsoCompQ C 0).hom.app X)).comp
      ((CategoryTheory.ShiftedHom.map
            (CochainComplex.HomComplex.Cocycle.equivHomShift.symm
              (CochainComplex.HomComplex.Cocycle.fromSingleMk
                (CategoryTheory.CategoryStruct.comp f (R.cochainComplexXIso (↑n) n ⋯).inv) ⋯ ↑m ⋯ ⋯))
            DerivedCategory.Q).comp
        (CategoryTheory.ShiftedHom.mk₀ 0 ⋯
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (DerivedCategory.Q.map R.ι'))
            ((DerivedCategory.singleFunctorIsoCompQ C 0).inv.app Y)))
        ⋯)
      ⋯

Defined in

Mathlib.CategoryTheory.Abelian.Injective.Ext

Cited by

1 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasExtHasDerivedCategory