instIsHomologicalAddCommGrpCatObjOppositeFunctorPreadditiveCoyoneda

English

The concluding form of the mk' construction, giving an IsTriangulated structure from octahedral data.

Русский

Завершение конструкции mk' даёт структуру IsTriangulated из данных октaэдра.

LaTeX

$$$\\text{IsTriangulated.mk'} \\Rightarrow \\text{IsTriangulated} C$$$

Lean4

instance (A : Cᵒᵖ) : (preadditiveCoyoneda.obj A).IsHomological where
  exact T
    hT := by
    rw [ShortComplex.ab_exact_iff]
    intro (x₂ : A.unop ⟶ T.obj₂) (hx₂ : x₂ ≫ T.mor₂ = 0)
    obtain ⟨x₁, hx₁⟩ := T.coyoneda_exact₂ hT x₂ hx₂
    exact ⟨x₁, hx₁.symm⟩

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