Distinguished triangle

Summary

A distinguished triangle is a sequence $X \to Y \to Z \to X[1]$ of morphisms in a triangulated category that belongs to the chosen class of exact triangles. It is the analogue of a short exact sequence: $Z$ plays the role of a cokernel of $X \to Y$, and $X[1]$ records the connecting map. In the derived category, every short exact sequence of complexes gives a distinguished triangle. In the homotopy category of chain complexes the standard example is built from the mapping cone.