Some new axioms for triangulated categories

Abstract

and this mapping cone should, by rights, be a triangle. It is, for instance, true that any homological functor applied to (*) gives a long exact sequence. Unfortunately, the world of triangulated categories is a bad one, and (*) need not be a triangle. It is however true that, givenf and g, there exists an h for which (*) is a triangle (see Theorem 1.8); not all morphisms of triangles are equal. Some are better than others. It turns out that Theorem 1.8 is equivalent to the octahedral axiom. In the first two sections of this article, we study two possible notions of “good” morphisms between triangles, and we quickly decide that neither is satisfactory. The morphisms do not compose well; the composite of good morphisms need not be good. Worse still, the non-category of good morphisms is non-additive; the sum of two good morphisms need not be good. The problem goes right back to the definition of a triangulated category.