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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Splitting in triangulated categories

Using the axioms for a triangulated category, is it possible to prove the following: $A\stackrel{0}{\to}B\to A\oplus B\to$ is a distinguished triangle. From the first axiom, the map ...
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If a colimit of distinguished triangles exists, is it also a distinguished triangle?

Consider the following situation in some triangulated category: We are given a collection of distinguished triangles $A_n \to B_n \to C_n \to A_n[1]$ indexed by the natural numbers, together with maps ...
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Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?

Let $\mathcal{A}$, $\mathcal{B}$ be two triangulated categories and $F: \mathcal{A}\to \mathcal{B}$ be a triangulated functor between them. Then $F$ induces an homomorphism between their Grothendieck ...