Theorems · Theorem · category theory

CategoryTheory.Functor.complete_distinguished_essImageDistTriang_morphism

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] (L : CategoryTheory.Functor C D)
  [inst_2 : CategoryTheory.HasShift C ℤ] [inst_3 : CategoryTheory.Preadditive C]
  [inst_4 : CategoryTheory.Limits.HasZeroObject C] [inst_5 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
  [inst_6 : CategoryTheory.Pretriangulated C] [inst_7 : CategoryTheory.HasShift D ℤ] [inst_8 : L.CommShift ℤ],
  (∀ (T₁' T₂' : CategoryTheory.Pretriangulated.Triangle C),
      T₁' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →
        T₂' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →
          ∀ (a : L.obj T₁'.obj₁ ⟶ L.obj T₂'.obj₁) (b : L.obj T₁'.obj₂ ⟶ L.obj T₂'.obj₂),
            CategoryTheory.CategoryStruct.comp (L.map T₁'.mor₁) b =
                CategoryTheory.CategoryStruct.comp a (L.map T₂'.mor₁) →
              ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b) →
    ∀ (T₁ T₂ : CategoryTheory.Pretriangulated.Triangle D),
      T₁ ∈ L.essImageDistTriang →
        T₂ ∈ L.essImageDistTriang →
          ∀ (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂),
            CategoryTheory.CategoryStruct.comp T₁.mor₁ b = CategoryTheory.CategoryStruct.comp a T₂.mor₁ →
              ∃ c,
                CategoryTheory.CategoryStruct.comp T₁.mor₂ c = CategoryTheory.CategoryStruct.comp b T₂.mor₂ ∧
                  CategoryTheory.CategoryStruct.comp T₁.mor₃ ((CategoryTheory.shiftFunctor D 1).map a) =
                    CategoryTheory.CategoryStruct.comp c T₂.mor₃

Defined in

Mathlib.CategoryTheory.Localization.Triangulated

Cited by

1 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasShiftCategoryTheory.PreadditiveCategoryTheory.Limits.HasZeroObjectCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.HasShiftCategoryTheory.Functor.CommShift