Theorems · Theorem · category theory

CategoryTheory.Pretriangulated.exists_iso_binaryBiproduct_of_distTriang

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroObject C]
  [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C]
  [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [hC : CategoryTheory.Pretriangulated C],
  ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles,
    T.mor₃ = 0 →
      ∃ e,
        CategoryTheory.CategoryStruct.comp T.mor₁ e.hom = CategoryTheory.Limits.biprod.inl ∧
          T.mor₂ = CategoryTheory.CategoryStruct.comp e.hom CategoryTheory.Limits.biprod.snd

Defined in

Mathlib.CategoryTheory.Triangulated.Pretriangulated

Cited by

1 results in Mathlib

Foundations

Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroObjectCategoryTheory.HasShiftCategoryTheory.PreadditiveCategoryTheory.Functor.AdditiveCategoryTheory.Pretriangulated