Triangulated category

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 distinguished triangle is a sequence X → Y → Z → X → Y {displaystyle X o Y o Z o X o Y} which is 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 X [ n ] = T n X {displaystyle X=T^{n}X} and likewise for morphisms from X to Y. A triangle (X, Y, Z, u, v, w) consists of 3 objects X, Y, and Z, together with morphisms u : X → Y, v : Y → Z and w : Z → X. Triangles are generally written in the unravelled form: