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A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.
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Proof that the homotopy category of a stable $\infty$-category is triangulated
I did not go through the details, but I think the following happens essentially:
In classical category theory, we have for a functor $\mathcal{C}^0 \to \mathcal{D}$ an essentially unique left Kan ext …