Theorems · Theorem · category theory

CategoryTheory.ObjectProperty.mem_of_hasInductedTStructure

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  [inst_2 : CategoryTheory.Limits.HasZeroObject C] [inst_3 : CategoryTheory.HasShift C ℤ]
  [inst_4 : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [inst_5 : CategoryTheory.Pretriangulated C]
  (P : CategoryTheory.ObjectProperty C) [inst_6 : P.IsTriangulated] (t : CategoryTheory.Triangulated.TStructure C)
  [P.IsClosedUnderIsomorphisms] [P.HasInducedTStructure t],
  ∀ T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles,
    ∀ (n₀ n₁ : ℤ), n₀ + 1 = n₁ → t.IsLE T.obj₁ n₀ → P T.obj₂ → t.IsGE T.obj₃ n₁ → P T.obj₁ ∧ P T.obj₃

Defined in

Mathlib.CategoryTheory.Triangulated.TStructure.Induced

Cited by

0 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasZeroObjectCategoryTheory.HasShiftCategoryTheory.Functor.AdditiveCategoryTheory.PretriangulatedCategoryTheory.ObjectProperty.IsTriangulatedCategoryTheory.ObjectProperty.IsClosedUnderIsomorphismsCategoryTheory.ObjectProperty.HasInducedTStructure