so that every exact triangle has an X, a Y, and a Z letter). Various arrows have been marked with [1] to indicate that they are of "degree 1"; e.g. the map from Z′ to X is in fact from Z′ to X[1]. The octahedral axiom then asserts the existence of maps f and g forming an exact triangle, and so that f and g form commutative triangles in the other faces that contain them: Two different pictures appear in ( also present the first one). The first presents the upper and lower pyramids of the above octahedron and asserts that given a lower pyramid, one can fill in an upper pyramid so that the two paths from Y to Y′, and from Y′ to Y, are equal (this condition is omitted, perhaps erroneously, from Hartshorne's presentation). The triangles marked + are commutative and those marked "d" are exact: The second diagram is a more innovative presentation. Exact triangles are presented linearly, and the diagram emphasizes the fact that the four triangles in the "octahedron" are connected by a series of maps of triangles, where three triangles (namely, those completing the morphisms from X to Y, from Y to Z, and from X to Z) are given and the existence of the fourth is claimed. One passes between the first two by "pivoting" about X, to the third by pivoting about Z, and to the fourth by pivoting about X′. All enclosures in this diagram are commutative (both trigons and the square) but the other commutative square, expressing the equality of the two paths from Y′ to Y, is not evident. All the arrows pointing "off the edge" are degree 1: This last diagram also illustrates a useful intuitive interpretation of the octahedral axiom. In triangulated categories, triangles play the role of exact sequences, and so it is suggestive to think of these objects as "quotients", and . In those terms, the existence of the last triangle expresses on the one hand (looking at the triangle ), and (looking at the triangle ). Putting these together, the octahedral axiom asserts the "third isomorphism theorem": If the triangulated category is the derived category D(A) of an abelian category A, and X, Y, Z are objects of A viewed as complexes concentrated in degree 0, and the maps and are monomorphisms in A, then the cones of these morphisms in D(A) are