English
A triangle in the homotopy category is distinguished iff it is isomorphic to a triangle of the form trianglehOfDegreewiseSplit S σ for some degreewise split short complex S and splitting σ.
Русский
Треугольник в гомотопической категории является distinguнed, если и только если он изоморфен треугольнику вида trianglehOfDegreewiseSplit S σ для некоторого разложения по степеням короткой раздельной последовательности S и разбиения σ.
LaTeX
$$$T \in distTriang _ \; \iff \; \exists S,\sigma,\ Nonempty (T \cong \mathrm{trianglehOfDegreewiseSplit}(S,\sigma))$$$
Lean4
theorem distinguished_iff_iso_trianglehOfDegreewiseSplit (T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))) :
(T ∈ distTriang _) ↔
∃ (S : ShortComplex (CochainComplex C ℤ)) (σ : ∀ n, (S.map (HomologicalComplex.eval C _ n)).Splitting),
Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ) :=
by
constructor
· intro hT
obtain ⟨K, L, φ, ⟨e⟩⟩ := inv_rot_of_distTriang _ hT
exact
⟨_, _,
⟨(triangleRotation _).counitIso.symm.app _ ≪≫
(rotate _).mapIso e ≪≫ CochainComplex.mappingCone.trianglehRotateIsoTrianglehOfDegreewiseSplit φ⟩⟩
· rintro ⟨S, σ, ⟨e⟩⟩
rw [rotate_distinguished_triangle, rotate_distinguished_triangle]
refine
isomorphic_distinguished _ ?_ _
((rotate _ ⋙ rotate _).mapIso e ≪≫ CochainComplex.trianglehOfDegreewiseSplitRotateRotateIso S σ)
exact ⟨_, _, _, ⟨Iso.refl _⟩⟩
