2 9 A ug 2 00 5 A definition of triangulated categories following Heller Matthias

Let E be a Frobenius category, let E denote its homotopy category. The shift functor on E induces a first shift functor on the category of acyclic complexes with entries in E by pointwise application. Shifting a complex by 3 positions yields a second shift functor on this category. Passing to the quotient modulo split acyclic complexes, Heller remarked that these two shift functors become isomorphic, via an isomorphism satisfying still a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a triangulation on E , except for the octahedral axiom. We generalize the notion of acyclic complexes such that the accordingly enlarged version of Heller’s construction includes octahedra.