Theorems · Theorem · category theory

CategoryTheory.Functor.mem_homologicalKernel_trW_iff

∀ {C : Type u_1} {A : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.HasShift C ℤ]
  [inst_2 : CategoryTheory.Category.{v_3, u_3} A] (F : CategoryTheory.Functor C A)
  [inst_3 : CategoryTheory.Limits.HasZeroObject C] [inst_4 : CategoryTheory.Preadditive C]
  [inst_5 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_6 : CategoryTheory.Pretriangulated C]
  [inst_7 : CategoryTheory.Abelian A] [F.IsHomological] [inst_9 : F.ShiftSequence ℤ] {X Y : C} (f : X ⟶ Y),
  F.homologicalKernel.trW f ↔ ∀ (n : ℤ), CategoryTheory.IsIso ((F.shift n).map f)

Defined in

Mathlib.CategoryTheory.Triangulated.HomologicalFunctor

Cited by

1 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.HasShiftCategoryTheory.CategoryCategoryTheory.Limits.HasZeroObjectCategoryTheory.PreadditiveCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.AbelianCategoryTheory.Functor.IsHomologicalCategoryTheory.Functor.ShiftSequence