Triangulated category

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. 
Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space "X" to complexes of sheaves, viewed as objects of the derived category of sheaves on "X". More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry conjecture predicts that the derived category of a Calabi–Yau manifold is equivalent to the Fukaya category of its "mirror" symplectic manifold.
History.
Triangulated categories were introduced independently by Dieter Puppe (1962) and Jean-Louis Verdier (1963), although Puppe's axioms were less complete (lacking the octahedral axiom (TR 4)). Puppe was motivated by the stable homotopy category. Verdier's key example was the derived category of an abelian category, which he also defined, developing ideas of Alexander Grothendieck. The early applications of derived categories included coherent duality and Verdier duality, which extends Poincaré duality to singular spaces.
Definition.
A shift or translation functor on a category "D" is an additive automorphism (or for some authors, an auto-equivalence) formula_1 from "D" to "D". It is common to write formula_2 for integers "n".
A triangle ("X", "Y", "Z", "u", "v", "w") consists of three objects "X", "Y", and "Z", together with morphisms formula_3, formula_4 and formula_5. Triangles are generally written in the unravelled form: