cone of a map of chain complexes, which in turn was inspired by the mapping cone in topology. It follows from the other axioms that an exact triangle (and in particular the object Z) is determined up to isomorphism by the morphism , although not always up to a unique isomorphism. Every triangle isomorphic to an exact triangle is exact. This means that if is an exact triangle, and , , and are isomorphisms, then is also an exact triangle. TR 2 If is an exact triangle, then so are the two rotated triangles and In view of the last triangle, the object Z[−1] is called a fiber of the morphism . The second rotated triangle has a more complex form when and are not isomorphisms but only mutually inverse equivalences of categories, since is a morphism from to , and to obtain a morphism to one must compose with the natural transformation . This leads to complex questions about possible axioms one has to impose on the natural transformations making and into a pair of inverse equivalences. Due to this issue, the assumption that and are mutually inverse isomorphisms is the usual choice in the definition of a triangulated category. TR 3 Given two exact triangles and a map between the first morphisms in each triangle, there exists a morphism between the third objects in each of the two triangles that makes everything commute. That is, in the following diagram (where the two rows are exact triangles and f and g are morphisms such that gu = u′f), there exists a map h (not necessarily unique) making all the squares commute: TR 4: The octahedral axiom Let and be morphisms, and consider the composed morphism . Form exact triangles for each of these three morphisms according to TR 1. The octahedral axiom states (roughly) that the three mapping cones can be made into the vertices of an exact triangle so that "everything commutes". More formally, given exact triangles , there exists an exact triangle such that This axiom is called the "octahedral axiom" because drawing all the objects and morphisms gives the skeleton of an octahedron, four of whose faces are exact triangles. The presentation here is Verdier's own, and appears, complete with octahedral diagram, in . In the following diagram, u and v are the given morphisms, and the primed letters are the cones of various maps (chosen