Theorems · Definition · category theory

CategoryTheory.ObjectProperty.HasInducedTStructure.casesOn

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} 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] →
              {P : CategoryTheory.ObjectProperty C} →
                {t : CategoryTheory.Triangulated.TStructure C} →
                  [inst_6 : P.IsTriangulated] →
                    {motive : P.HasInducedTStructure t → Sort u} →
                      (t_1 : P.HasInducedTStructure t) →
                        ((exists_triangle_zero_one :
                              ∀ (A : C),
                                P A →
                                  ∃ X Y,
                                    ∃ (_ : t.IsLE X 0) (_ : t.IsGE Y 1),
                                      ∃ f g h,
                                        ∃ (_ :
                                          CategoryTheory.Pretriangulated.Triangle.mk f g h ∈
                                            CategoryTheory.Pretriangulated.distinguishedTriangles),
                                          P.isoClosure X ∧ P.isoClosure Y) →
                            motive ⋯) →
                          motive t_1

Defined in

Mathlib.CategoryTheory.Triangulated.TStructure.Induced

Cited by

1 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasZeroObjectCategoryTheory.HasShiftCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.ObjectProperty.IsTriangulated