If a colimit of distinguished triangles exists, is it also a distinguished triangle? Consider the following situation in some triangulated category: We are given a collection of distinguished triangles $A_n \to B_n \to C_n \to A_n[1]$ indexed by the natural numbers, together with maps (in the obvious sense) from the n'th triangle to the (n+1)'st. If $A \to B \to C$ is the colimit of this system of triangles, is it also a distinguished triangle?
It would be really interesting to have a proof or a counterexample, or possible a proof depending on some additional hypotheses on the triangulated category in question.
 A: I strongly suspect that the answer is 'no'. You see: when you use the triangulated version of the definition of (countable) homotopy limits, you question becomes a partial case of the following one: can a morphism of distinguished triangles be completed to a $3\times3$-commutative diagram whose rows and columns are distinguished triangles? The answer to the latter question is 'no' (though any commutative square could be completed to such a diagram); looking at it one can probably construct a counterexample for your original question.
A: this is true in the topological setting. cofibrations can be thought of as colimits, they are actually colimits of a diagram (i have been told, but i cant recall the example) and colimits commute with colimits. In the setting i am thinking of cofibrations are the distinguished triangles. So i doubt you will find a counterexample in general since the result is definitely true for at least one triangulated category. Unfortunately, i do not see how this could be extended to other triangulated categories.
It does not seem like a result that would be true in general given my above reasons for believing the result. I will ask about the diagram it is the colimit/homotopy colimit of.
Is there a particular triangulated category you are interested in?
A: (Deligne has an assertion that a certain triangle in the ind-category of a triangulated category can be written as a limit of distinguished triangles there; see SGA 4, XVII, Lemme 1.2.2.1, http://www.msri.org/publications/books/sga/sga/4-3/4-3t_275.html . Does not seem to answer the question, though.)
