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. Shift operator is a decategorified analogue of triangulated category.
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) Σ {\displaystyle \Sigma } from D to D. It is common to write X [ n ] = Σ n X {\displaystyle X[n]=\Sigma ^{n}X} for integers n. A triangle (X, Y, Z, u, v, w) consists of three objects X, Y, and Z, together with morphisms u : X → Y {\displaystyle u\colon X\to Y} , v : Y → Z {\displaystyle v\colon Y\to Z} and w : Z → X [ 1 ] {\displaystyle w\colon Z\to X[1]} . Triangles are generally written in the unravelled form:
X →
u Y →
v Z →
w X [ 1 ] , {\displaystyle X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1],}
or
X →
u Y →
v Z →
w {\displaystyle X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}}
for short. A triangulated category is an additive category D with a translation functor and a class of triangles, called exact triangles (or distinguished triangles), satisfying the following properties (TR 1), (TR 2), (TR 3) and (TR 4). (These axioms are not entirely independent, since (TR 3) can be derived from the others.)
TR 1 For every object X, the following triangle is exact:
X → id X → 0 → X [ 1 ] {\displaystyle X{\overset {\text{id}}{\to }}X\to 0\to X[1]}
For every morphism u : X → Y {\displaystyle u\colon X\to Y} , there is an object Z (called a cone or cofiber of the morphism u) fitting into an exact triangle
X →
u Y → Z → X [ 1 ] {\displaystyle X{\xrightarrow {{} \atop u}}Y\to Z\to X[1]}
The name "cone" comes from the cone of a map of chain complexes, which in turn was inspired by the mapping cone in topology. It follows from the other axioms that an exact triangle (and in particular the object Z) is determined up to isomorphism by the morphism X → Y {\displaystyle X\to Y} , although not always up to a unique isomorphism. Every triangle isomorphic to an exact triangle is exact. This means that if
X →
u Y →
v Z →
w X [ 1 ] {\displaystyle X{\xrightarrow {{} \atop u}}Y{\xrightarrow {{} \atop v}}Z{\xrightarrow {{} \atop w}}X[1]}
… excerpt ends here. Continue reading the full article.
