Distinguished triangle

Remarks

In a distinguished triangle, consecutive maps compose to zero: $v u = 0$, $w v = 0$ and $(\Sigma u) w = 0$. For each object $W$, applying $\Hom(W, -)$ gives a long exact sequence $\cdots \to \Hom(W, X) \to \Hom(W, Y) \to \Hom(W, Z) \to \Hom(W, \Sigma X) \to \cdots$. By TR1 and TR3, the third object $Z$ is determined by $u$ up to a non-unique isomorphism; it is called a cone of $u$. Example: $X \xrightarrow{\id} X \to 0 \to \Sigma X$ is distinguished for every $X$, by the axiom TR1. Non-example: if $X \ne 0$, then $X \to 0 \to 0 \to \Sigma X$ is not distinguished. Indeed, the long exact sequence for $W = \Sigma X$ would give $\Hom(\Sigma X, 0) \to \Hom(\Sigma X, \Sigma X) \to \Hom(\Sigma X, \Sigma 0)$ exact, which forces $\Hom(\Sigma X, \Sigma X) = 0$, but $\id_{\Sigma X} \ne 0$. In the homotopy category of complexes, the distinguished triangles are those isomorphic to $X \xrightarrow{f} Y \to \operatorname{Cone}(f) \to X[1]$ for a chain map $f$.