Applications of Homological Algebra Introduction to Perverse Sheaves Spring 2007 P. Achar Summary: Applications of Homological Algebra Introduction to Perverse Sheaves Spring 2007 P. Achar Triangulated Categories and t-Structures March 27, 2007 Definition 1. A triangulated category is an additive category C equipped with (a) a shift functor [1] : C C and (b) a class of triangles X Y Z X[1], called distinguished triangles, satisfying the following axioms: (-2) The shift functor is an equivalence categories. In particular, it has an inverse, denoted [-1] : C C. (-1) Every triangle isomorphic to a distinguished triangle is a distinguished triangle. (0) Every morphism f : X Y can be completed to a distinguished triangle X f - Y Z X[1]. (1) (Identity) (2) (Rotation) (3) (Square Completion) (4) (Octahedral Property) (The last four axioms are the properties of distinguished triangles in the derived category that were estab- lished in an earlier set of notes.) Example 2. The derived category of an abelian category, with its usual shift functor and its usual notion of distinguished triangles, is a triangulated category. The same is true of the homotopy category of complexes Collections: Mathematics