completeDistinguishedTriangleMorphism

English

A construction producing an isomorphism of triangles from a given arrow iso between mor₁ components.

Русский

Построение изоморфизма треугольников из изоморфизма стрел между mor1-компонентами.

LaTeX

$$$(T_1 \\cong T_2) \\text{ exists given morphisms } e_1,e_2$$$

Lean4

/-- A chosen extension of a commutative square into a morphism of distinguished triangles. -/
@[simps hom₁ hom₂]
def completeDistinguishedTriangleMorphism (T₁ T₂ : Triangle C) (hT₁ : T₁ ∈ distTriang C) (hT₂ : T₂ ∈ distTriang C)
    (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) : T₁ ⟶ T₂ :=
  have h := complete_distinguished_triangle_morphism _ _ hT₁ hT₂ a b comm
  { hom₁ := a
    hom₂ := b
    hom₃ := h.choose
    comm₁ := comm
    comm₂ := h.choose_spec.1
    comm₃ := h.choose_spec.2 }

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