Theorems · Theorem · category theory

CategoryTheory.Adjunction.isTriangulated_leftAdjoint

∀ {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.Limits.HasZeroObject C]
  [inst_3 : CategoryTheory.Limits.HasZeroObject D] [inst_4 : CategoryTheory.Preadditive C]
  [inst_5 : CategoryTheory.Preadditive D] [inst_6 : CategoryTheory.HasShift C ℤ] [inst_7 : CategoryTheory.HasShift D ℤ]
  [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} {G : CategoryTheory.Functor D C}
  (adj : F ⊣ G) [inst_12 : F.CommShift ℤ] [inst_13 : G.CommShift ℤ] [adj.CommShift ℤ] [G.IsTriangulated],
  F.IsTriangulated

The left adjoint of a triangulated functor is triangulated.

Defined in

Mathlib.CategoryTheory.Triangulated.Adjunction

Cited by

1 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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