actually isomorphic to the quotients above in A. Finally, formulates the octahedral axiom using a two-dimensional commutative diagram with 4 rows and 4 columns. also give generalizations of the octahedral axiom. Properties Here are some simple consequences of the axioms for a triangulated category D. Given an exact triangle in D, the composition of any two successive morphisms is zero. That is, vu = 0, wv = 0, u[1]w = 0, and so on. Given a morphism , TR 1 guarantees the existence of a cone Z completing an exact triangle. Any two cones of u are isomorphic, but the isomorphism is not always uniquely determined. Every monomorphism in D is the inclusion of a direct summand, , and every epimorphism is a projection . A related point is that one should not talk about "injectivity" or "surjectivity" for morphisms in a triangulated category. Every morphism that is not an isomorphism has a nonzero "cokernel" Z (meaning that there is an exact triangle ) and also a nonzero "kernel", namely Z[−1]. Non-functoriality of the cone construction One of the technical complications with triangulated categories is the fact the cone construction is not functorial. For example, given a ring and the partial map of distinguished trianglesin , there are two maps which complete this diagram. This could be the identity map, or the zero mapboth of which are commutative. The fact there exist two maps is a shadow of the fact that a triangulated category is a tool which encodes homotopy limits and colimit. One solution for this problem was proposed by Grothendieck where not only the derived category is considered, but the derived category of diagrams on this category. Such an object is called a Derivator. Examples Are there better axioms? Some experts suspectpg 190 (see, for example, ) that triangulated categories are not really the "correct" concept. The essential reason is that the cone of a morphism is unique only up to a non-unique isomorphism. In particular, the cone of a morphism does not in general depend functorially on the morphism (note the non-uniqueness in axiom (TR 3), for example). This non-uniqueness is a potential source of errors. The axioms work adequately in practice, however, and there is a great deal of literature devoted to their study. Derivators One alternative proposal is the theory of derivators proposed in Pursuing stacks by Grothendieck in the 80spg 191, and later developed