 
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Categories of Complexes; Distinguished Triangles
February 8, 2007
Definition 1. An abelian category is a category satisfying the following four axioms:
(1) For any two objects A and B in A, the set of morphisms Hom(A, B) is endowed with the structure
of an abelian group, and composition of morphisms is biadditive.
(2) There is a "zero object" 0 with the property that Hom(0, 0) = 0. (This property implies that it is
unique up to unique isomorphism, and that Hom(0, A) = Hom(A, 0) = 0 for all other objects A.)
(3) For any two objects A and B, there is a "biproduct" AB. It is equipped with morphisms as shown
below:
A
i1
GG A B
p1
oo
p2
GG B
i2
oo
