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The category of modules over the endomorphism algebra of a rigid object in aHom-finite triangulated category C has been given two different descriptions:On the one hand, as shown by Osamu Iyama and Yuji Yoshino, it is equivalent toan ideal quotient of a subcategory of C. On the other hand, Aslak Buan andRobert Marsh proved that this module category is also equivalent to somelocalisation of C. In this paper, we give a conceptual interpretation, inspiredfrom homotopical algebra, of this double description. Our main aim, yet to beacheived, is to generalise Buan-Marsh's result to the case of Hom-infinitecluster categories. We note that, contrary to the more common case where amodel category is a module category whose homotopy category is triangulated, weconsider here some triangulated categories whose homotopy categories are modulecategories.

From triangulated categories to module categories via homotopical algebra