Triangulated category

Definition

Let $\mathcal{T}$ be an additive category and let $\Sigma \colon \mathcal{T} \to \mathcal{T}$ be an additive functor that is an automorphism, the shift, also written $X \mapsto X[1]$. A triangle is a diagram $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X$. A morphism of triangles is a triple of morphisms $(f, g, h)$ between the first, second and third objects. It must make the three squares commute, the last one with $\Sigma f$. A triangulated category is such a pair $(\mathcal{T}, \Sigma)$ with a class of triangles, the distinguished triangles, such that:

1. (TR1) for each object $X$, the triangle $X \xrightarrow{\id} X \to 0 \to \Sigma X$ is distinguished. Each triangle isomorphic to a distinguished triangle is distinguished. Each morphism $u \colon X \to Y$ is the first morphism of a distinguished triangle $X \xrightarrow{u} Y \to Z \to \Sigma X$;
2. (TR2) a triangle $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X$ is distinguished if and only if $Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X \xrightarrow{-\Sigma u} \Sigma Y$ is distinguished;
3. (TR3) for distinguished triangles $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} \Sigma X$ and $X' \xrightarrow{u'} Y' \xrightarrow{v'} Z' \xrightarrow{w'} \Sigma X'$ and morphisms $f \colon X \to X'$, $g \colon Y \to Y'$ with $g u = u' f$, there is $h \colon Z \to Z'$ such that $(f, g, h)$ is a morphism of triangles;
4. (TR4) the octahedral axiom holds.