Heller triangulated categories

Abstract

Let E be a Frobenius category, let_E_ denote its stable 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.