Theorems · Theorem · category theory

CategoryTheory.isTriangulated_of_essSurj_mapComposableArrows_two

∀ {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] [inst_4 : CategoryTheory.Limits.HasZeroObject C]
  [inst_5 : CategoryTheory.Limits.HasZeroObject D] [inst_6 : CategoryTheory.Preadditive C]
  [inst_7 : CategoryTheory.Preadditive D] [inst_8 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
  [inst_9 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [inst_10 : CategoryTheory.Pretriangulated C]
  [inst_11 : CategoryTheory.Pretriangulated D] (F : CategoryTheory.Functor C D) [inst_12 : F.CommShift ℤ]
  [F.IsTriangulated] [(F.mapComposableArrows 2).EssSurj] [CategoryTheory.IsTriangulated C],
  CategoryTheory.IsTriangulated D

If F : C ⥤ D is a triangulated functor from a triangulated category, then D is also triangulated if tuples of composable arrows in D can be lifted to C.

Defined in

Mathlib.CategoryTheory.Triangulated.Functor

Cited by

1 results in Mathlib

Foundations

Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Limits.HasZeroObjectCategoryTheory.Limits.HasZeroObjectCategoryTheory.PreadditiveCategoryTheory.PreadditiveCategoryTheory.Functor.AdditiveCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.PretriangulatedCategoryTheory.Functor.CommShiftCategoryTheory.Functor.IsTriangulatedCategoryTheory.Functor.EssSurjCategoryTheory.IsTriangulated