Theorems · Theorem · category theory

CategoryTheory.Functor.isTriangulated_iff_comp_right

∀ {C : Type u_1} {D : Type u_2} {E : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Category.{v_3, u_3} E]
  [inst_3 : CategoryTheory.HasShift C ℤ] [inst_4 : CategoryTheory.HasShift D ℤ] [inst_5 : CategoryTheory.HasShift E ℤ]
  [inst_6 : CategoryTheory.Limits.HasZeroObject C] [inst_7 : CategoryTheory.Limits.HasZeroObject D]
  [inst_8 : CategoryTheory.Limits.HasZeroObject E] [inst_9 : CategoryTheory.Preadditive C]
  [inst_10 : CategoryTheory.Preadditive D] [inst_11 : CategoryTheory.Preadditive E]
  [inst_12 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
  [inst_13 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive]
  [inst_14 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor E n).Additive] [inst_15 : CategoryTheory.Pretriangulated C]
  [inst_16 : CategoryTheory.Pretriangulated D] [inst_17 : CategoryTheory.Pretriangulated E]
  {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D E} {H : CategoryTheory.Functor C E} (e : F.comp G ≅ H)
  [inst_18 : F.CommShift ℤ] [inst_19 : G.CommShift ℤ] [inst_20 : H.CommShift ℤ]
  [CategoryTheory.NatTrans.CommShift e.hom ℤ] [G.IsTriangulated] [G.Full] [G.Faithful],
  F.IsTriangulated ↔ H.IsTriangulated

Defined in

Mathlib.CategoryTheory.Triangulated.Functor

Cited by

0 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Limits.HasZeroObjectCategoryTheory.Limits.HasZeroObjectCategoryTheory.Limits.HasZeroObjectCategoryTheory.PreadditiveCategoryTheory.PreadditiveCategoryTheory.PreadditiveCategoryTheory.Functor.AdditiveCategoryTheory.Functor.AdditiveCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.PretriangulatedCategoryTheory.PretriangulatedCategoryTheory.Functor.CommShiftCategoryTheory.Functor.CommShiftCategoryTheory.Functor.CommShiftCategoryTheory.NatTrans.CommShiftCategoryTheory.Functor.IsTriangulatedCategoryTheory.Functor.FullCategoryTheory.Functor.Faithful