Let $C$ a triangulated category and $(Y_a)$ a direct system of objects of $C$. Is it true that $colimY_a\cong \oplus Y_a$?
My supervisor told me that he thinks this is not true, but didn't tell me why. My thoughts are the following: We know that $colimY_a= \oplus Y_a/S$ so we get the short exact sequence $0\longrightarrow S\longrightarrow \oplus Y_a\longrightarrow colimY_a\longrightarrow 0$. This sequence gives rise to the exact triangle $S\longrightarrow \oplus Y_a\longrightarrow colimY_a\longrightarrow \Omega^{-1}S$. From this triangle we get the short exact sequence $0\longrightarrow \oplus Y_a\longrightarrow colimY_a\longrightarrow \Omega^{-1}S\longrightarrow 0$. It is easy to check that this is truly an exact sequence, since the above triangle is exact. Since this is an exact sequence, the morphism $\oplus Y_a\longrightarrow colimY_a$ is a monomorphism, but it is also an epimorphism by its definition. Thus, it is an isomorphism.
Is there any obvious mistake in all this?
Suppose that $\mathsf{A}$ is the category of chain complexes of modules over your favorite ring and $\mathsf{C}$ is the associated homotopy category. A short exact sequence in $\mathsf{A}$ will yield an exact triangle in $\mathsf{C}$, and to some extent this works in the other direction, but it's complicated. If $$ 0 \to X \to Y \to Z \to 0 $$ is short exact of modules (and hence chain complexes) in $\mathsf{A}$, then there are contractible chain complexes $W_i$ such that there is a short exact sequence $$ 0 \to \Omega Z \oplus W_1 \to X \oplus W_2 \to Y \oplus W_3 \to 0. $$ So although there is a monomorphism $X \to Y$ in $\mathsf{A}$, there need be no epimorphism $X \to Y$, unless you can somehow arrange for $W_2$ to be zero.
For the same reason (you can add and remove contractible chain complexes whenever you want), there are no sensible notions of monomorphism or epimorphism in the triangulated category $\mathsf{C}$.