Triangulated category

A triangulated category is a pretriangulated category whose distinguished triangles also satisfy the octahedral axiom. Its structure packages an additive shift and a class of triangles behaving like mapping-cone sequences.

Derived categories and homotopy categories of suitable stable settings are central examples. Some authors include the octahedral axiom in the word “pretriangulated”; the terminology here follows mathlib, where Pretriangulated supplies the earlier axioms and IsTriangulated supplies octahedrality.