Distinguished triangle

In mathematics, a triangulated category is a category together with the additional structure of a "translation functor" and a class of "distinguished triangles". Prominent examples are the derived category of an abelian category and the stable homotopy category of spectra (more generally, the homotopy category of a stable ∞-category), both of which carry the structure of a triangulated category in a natural fashion. The distinguished triangles generate the long exact sequences of homology; they play a role akin to that of short exact sequences in abelian categories.

A t-category is a triangulated category with a t-structure.

The notion of a derived category was introduced by Jean-Louis Verdier (1963) in his Ph.D. thesis, based on the ideas of Grothendieck. He also defined the notion of a triangulated category, based upon the observation that a derived category had some special "triangles", by writing down axioms for the basic properties of these triangles. A very similar set of axioms was written down at about the same time by Dold and Puppe (1961).

A translation functor on a category D is an automorphism (or for some authors, an auto-equivalence) T from D to D. One usually uses the notation ${\displaystyle X[n]=T^{n}X}$ and likewise for morphisms from X to Y.

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