Heller triangulated categories

Abstract

Let $\mathcal{E}$ be a Frobenius category. Let $\underset {=} {\mathcal{E}}$ denote its stable category. The shift functor on $\underline {E}$ induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in $\underset {=} {\mathcal{E}}$. Shifting a complex by 3 positions yields an outer shift functor on this category. Passing to quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomorphic, via an isomorphism satisfying yet a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a Verdier triangulation on $\underset {=} {\mathcal{E}}$, except for the octahedral axiom. We generalise the notion of acyclic complexes such that the accordingly enlarged version of Heller’s construction includes octahedra.