 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Triangulated Categories and tStructures
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
