distinguished_cocone_triangle₁ — Mathlib

English

In the triangle setting, any morphism f from X to T.obj₂ that composes with T.mor₂ to zero factors through T.obj₁: f = g ≫ T.mor₁ for some g.

Русский

В треугольнике любой морфизм f: X → T.obj₂ с условием f ≫ T.mor₂ = 0 факторизуется через T.obj₁: f = g ≫ T.mor₁.

LaTeX

$$coyoneda_exact₂$$

Lean4

/-- Any morphism `Y ⟶ Z` is part of a distinguished triangle `X ⟶ Y ⟶ Z ⟶ X⟦1⟧` -/
theorem distinguished_cocone_triangle₁ {Y Z : C} (g : Y ⟶ Z) :
    ∃ (X : C) (f : X ⟶ Y) (h : Z ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C :=
  by
  obtain ⟨X', f', g', mem⟩ := distinguished_cocone_triangle g
  exact ⟨_, _, _, inv_rot_of_distTriang _ mem⟩

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