Theorems · Theorem · category theory

CategoryTheory.Triangulated.TStructure.isIso_truncGE_map_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
  [inst_2 : CategoryTheory.Limits.HasZeroObject C] [inst_3 : CategoryTheory.HasShift C ℤ]
  [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_5 : CategoryTheory.Pretriangulated C]
  (t : CategoryTheory.Triangulated.TStructure C) {Y Z : C} (g : Y ⟶ Z) (n₀ n₁ : ℤ),
  n₀ + 1 = n₁ →
    (CategoryTheory.IsIso ((t.truncGE n₁).map g) ↔
      ∃ X f h,
        ∃ (_ :
          CategoryTheory.Pretriangulated.Triangle.mk f (CategoryTheory.CategoryStruct.comp g ((t.truncGEπ n₁).app Z))
              h ∈
            CategoryTheory.Pretriangulated.distinguishedTriangles),
          t.IsLE X n₀)

Defined in

Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Cited by

2 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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