Theorems · Theorem · category theory

CategoryTheory.Triangulated.TStructure.triangle_iso_exists

∀ {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) {T T' : CategoryTheory.Pretriangulated.Triangle C},
  T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →
    T' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →
      ∀ (e : T.obj₂ ≅ T'.obj₂) (a b : ℤ),
        t.IsLE T.obj₁ a →
          t.IsGE T.obj₃ b →
            t.IsLE T'.obj₁ a →
              t.IsGE T'.obj₃ b →
                autoParam (a < b) CategoryTheory.Triangulated.TStructure.triangle_iso_exists._auto_1 →
                  ∃ e', e'.hom.hom₂ = e.hom

If a < b, then an isomorphism T.obj₂ ≅ T'.obj₂ extends to an isomorphism T ≅ T' of distinguished triangles when for a t-structure, both T.obj₁ and T'.obj₁ are ≤ a and both T.obj₃ and T'.obj₃ are ≥ b.

Defined in

Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Cited by

5 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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