Theorems · Theorem · category theory

CategoryTheory.Triangulated.AbelianSubcategory.exists_distinguished_triangle_of_epi

∀ {C : Type u_1} {A : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Limits.HasZeroObject C] [inst_2 : CategoryTheory.Preadditive C]
  [inst_3 : CategoryTheory.HasShift C ℤ] [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
  [inst_5 : CategoryTheory.Pretriangulated C] [inst_6 : CategoryTheory.Category.{v_2, u_2} A]
  {ι : CategoryTheory.Functor A C},
  (∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ (CategoryTheory.shiftFunctor C n).obj (ι.obj Y)), n < 0 → f = 0) →
    ∀ [inst_7 : CategoryTheory.Preadditive A] [ι.Full] [ι.Faithful] [ι.Additive],
      CategoryTheory.Triangulated.AbelianSubcategory.admissibleMorphism ι = ⊤ →
        ∀ {X₂ X₃ : A} (π : X₂ ⟶ X₃) [CategoryTheory.Epi π],
          ∃ X₁ i δ,
            CategoryTheory.Pretriangulated.Triangle.mk (ι.map i) (ι.map π) δ ∈
              CategoryTheory.Pretriangulated.distinguishedTriangles

Defined in

Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory

Cited by

0 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroObjectCategoryTheory.PreadditiveCategoryTheory.HasShiftCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulCategoryTheory.Functor.AdditiveCategoryTheory.Epi