Introduction: family of generators and generator of a category[edit | edit source]
Let be a category. A family of its objects is said to be a
family of generators of if for every pair of distinct morphisms
there is a morphism for some index such that .
One notes that in an additive category, is a family of generators if and only if for each nonzero morphism in there is a morphism such that .
An object in is called a generator for
if with being a family of generators for .
Equivalently, (viz. Mitchell) is a generator for if and only if the
set-valued functor is an imbedding functor.
Proper generator of a Grothendieck category[edit | edit source]