Octahedral axiom

Summary

The octahedral axiom is the fourth axiom, TR4, in the definition of a triangulated category. It relates the cones of two composable morphisms $f$ and $g$ to the cone of their composite $g f$ by a further distinguished triangle. It is the triangulated analogue of the third isomorphism theorem $(C/A)/(B/A) \cong C/B$. Its name comes from drawing the four triangles involved as the faces of an octahedron.