Why it is a distinguished triangle?

Question

The following paper gives an alternative formulation of the definition of epaisee subcategory: http://ac.els-cdn.com/0022404989900819/1-s2.0-0022404989900819-main.pdf?_tid=76c221e6-7282-11e6-afcb-00000aab0f27&acdnat=1472981714_a8d233643dcc094e8f98bf3ff4b81986

Can anyone tell me in the proof of proposition 1.3, why the sequence $X_2 [-1] \rightarrow X_1 \rightarrow X \rightarrow $ is a distinguished triangle? Thank you.

Answer 1

It's a general and well-known fact about triangulated categories that for any objects $X_1$ and $X_2$, $$X_2[-1]\to X_1\to X_1\oplus X_2\to X_2$$ is a distinguished triangle, where the first map is the zero map and the other maps are the obvious ones.

You can find a complete proof in Neeman's "Triangulated Categories", Corollary 1.2.7, but the proof is quite straightforward.

Sketch: complete the zero map $X_2[-1]\to X_1$ to a distinguished triangle $$X_2[-1]\to X_1\to Y\to X_2$$ and consider the long exact sequences obtained by applying the cohomological functors $\text{Hom}(X_2,-)$ and $\text{Hom}(-,X_1)$ to this triangle.