Theorems · Inductive type · category theory

CategoryTheory.Functor.IsTriangulated

{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] →
        [inst_2 : CategoryTheory.HasShift C ℤ] →
          [inst_3 : CategoryTheory.HasShift D ℤ] →
            (F : CategoryTheory.Functor C D) →
              [F.CommShift ℤ] →
                [inst_5 : CategoryTheory.Limits.HasZeroObject C] →
                  [inst_6 : CategoryTheory.Limits.HasZeroObject D] →
                    [inst_7 : CategoryTheory.Preadditive C] →
                      [inst_8 : CategoryTheory.Preadditive D] →
                        [inst_9 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] →
                          [inst_10 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] →
                            [CategoryTheory.Pretriangulated C] → [CategoryTheory.Pretriangulated D] → Prop

A functor which commutes with the shift by ℤ is triangulated if it sends distinguished triangles to distinguished triangles.

Defined in

Mathlib.CategoryTheory.Triangulated.Functor

Cited by

30 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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