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
 : C C and (b) a class of triangles X Y Z X, called distinguished triangles, satisfying the
(-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
- Y Z X.
(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