Triangulated categories without models

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. Introduction. Triangulated categories are fundamental tools in both algebra and topology. In algebra they often arise as the stable category of a Frobenius category ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear as a full triangulated subcategory of the homotopy category of a Quillen stable model category [Hov99, 7.1]. The triangulated categories which belong, up to exact equivalence, to one of these two families will be termed algebraic and topological, respectively. We borrow this terminology from [Kel06, 3.6] and [Sch06]. Algebraic triangulated categories are generally also topological, but there are many well-known examples of topological triangulated categories which are not algebraic. In the present paper we exhibit examples of triangulated categories which are neither algebraic nor topological. As far as we know, these are the first examples of this kind. Even worse (or better, depending on the perspective), our examples do not even admit non-trivial exact functors to or from algeThe first author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract. Mathematics Subject Classification (1991): 18E30, 55P42