Canonical triangle for diagonal map in a triangulated category

Question

Let $\mathcal{T}$ be a triangulated category and $X \in \mathcal{T}$ an object. Note as $\Delta:X \to X \oplus X$ the diagonal map $\Delta = id_X \oplus id_X$.

Is there a canonical completion to a distinguished triangle $$ X \xrightarrow{\Delta} X \oplus X \to A $$ for an arbitrary triangulated category $\mathcal{T}$?

I'm not very versed in triangulated categories and although my intuition tells me the completion should be $A = X$, with the map given by $\pi_1 - \pi_2$, where $\pi_i$ is the projection on the $i$-th coordinate, I've been unable to reach a conclusion.

Thanks in advance.

Answer 1

Yes, that’s right. In any additive category the diagonal is a split monomorphism which is a kernel for $\pi_1-\pi_2$. Split short exact sequences in triangulated categories are distinguished triangles, with trivial connecting maps, and in fact give the only examples of monomorphisms and epimorphisms of any kind in triangulated categories.