Theorems · Theorem · category theory

HomotopyCategory.distinguished_iff_iso_trianglehOfDegreewiseSplit

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  [inst_2 : CategoryTheory.Limits.HasZeroObject C] [inst_3 : CategoryTheory.Limits.HasBinaryBiproducts C]
  (T : CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))),
  T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles ↔
    ∃ S σ, Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ)

Defined in: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit
Cited by: 0 results in Mathlib
Foundations: Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive CategoryTheory.Limits.HasZeroObject CategoryTheory.Limits.HasBinaryBiproducts

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Setstatement · cited by 53,352
CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homproof · cited by 32,603
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CategoryTheory.Functor.compproof · cited by 6,529
CategoryTheory.Isostatement and proof · cited by 3,963
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CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.shiftFunctorproof · cited by 1,553
CategoryTheory.Limits.HasZeroObjectstatement and proof · cited by 1,298
ComplexShape.upstatement and proof · cited by 1,123
CochainComplexstatement and proof · cited by 1,016

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