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Suppose $\mathcal{T}$ is a triangulated category and $X$ is an object of it, if there exists a morphisms \begin{equation} p_i: X \rightarrow X, i=1,2 \end{equation} such that \begin{equation} p_1^2=p_1,~p_2^2=p_2,~p_1 p_2=0,~p_1+p_2=1 \end{equation} Do there exist objects $X_1$ and $X_2$ sucht that $X \simeq X_1 \oplus X_2$ and $p_i$ are the corresponding composition of projection morphisms and injection morphisms?

If $\mathcal{T}$ is abelian, then this question is trivial. But if it is only a triangulated category, then $\text{Im}\,p_i$ or $\text{Ker}\,p_i$ might not exist! So I do not know how to prove it or give a conterexample!

Edit: what happens if we make additional assumption that $\text{Ker}\,p_1$ and $\text{Im}\,p_1$ exists?

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  • $\begingroup$ There exists some triangulated categories which are not pseudo-abelian. I assume that if you start with a non pseudo-abelian additive category, then its homotopy category (category of complexes and morphims of complexes up to homotopy) is not idempotent complete in general. I just know some examples of triangulated categories constructed using pseudo-abelianization (though I never bother to prove that it is a necessary step). However if $\mathcal{T}$ has infinite direct sum, then it is idempotent complete. Also, if $\ker p_1$ (or $Im p_1$) exists, then $X$ indeed decomposes as $X_1\oplus X_2$. $\endgroup$ – Roland Jun 1 '17 at 15:25

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