Theorems · Theorem · category theory

CategoryTheory.Functor.map_distinguished_iff

∀ {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) [inst_4 : 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] [inst_11 : CategoryTheory.Pretriangulated C]
  [inst_12 : CategoryTheory.Pretriangulated D] [F.IsTriangulated] [F.Full] [F.Faithful]
  (T : CategoryTheory.Pretriangulated.Triangle C),
  F.mapTriangle.obj T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles ↔
    T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles

Defined in

Mathlib.CategoryTheory.Triangulated.Functor

Cited by

3 results in Mathlib

Foundations

Depth 53 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.PretriangulatedCategoryTheory.Functor.IsTriangulatedCategoryTheory.Functor.FullCategoryTheory.Functor.Faithful