Set the framework to be a triangulated category with all set indexed coproducts. In "Relative Homological Algebra and Purity in Triangulated Categories", J. of Algebra 227, (2000), pp. 268 361, (available at the author home page: http://www.math.uoi.gr/~abeligia/) Apostolos Beligiannis said that the homotopy colimit of a diagram of towers whose terms are triangles is also a triangle, provided that the triangulated category has a model; he gives also a reference, namely the Hovey's book: M. Hovey, "Model Categories", Mathematical Surveys and Monographs 63, AMS, Providence, RI, 1999. My problem is that the reference is not precise, and I do not know too much about model categories. Thus my questions are the following: Why is it true that in a triagulated categories with a model, the homotopy colimit of a tower of triangles is a triangle? Where can I find this result in Hovey's book? Finally what about the dual, namely under what extent is it true that the the homotopy limit of a tower of triangles is a triangle?
