Theorems · Theorem · category theory

CategoryTheory.Triangulated.TStructure.triangle_map_ext

∀ {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} {f₁ f₂ : T ⟶ T'},
  T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →
    T' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →
      ∀ (a b : ℤ),
        t.IsLE T.obj₁ a →
          t.IsGE T'.obj₃ b →
            autoParam (f₁.hom₂ = f₂.hom₂) CategoryTheory.Triangulated.TStructure.triangle_map_ext._auto_1 →
              autoParam (a ≤ b) CategoryTheory.Triangulated.TStructure.triangle_map_ext._auto_3 → f₁ = f₂

Two morphisms T ⟶ T' between distinguished triangles must coincide when they coincide on the middle object, and there are integers a ≤ b such that for a t-structure, we have T.obj₁ ≤ a and T'.obj₃ ≥ b.

Defined in

Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Cited by

1 results in Mathlib

Foundations

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

Assumes

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