Generator (category theory)

In mathematics, specifically category theory, a family of generators (or family of separators) of a category formula_1 is a collection formula_2 of objects, indexed by some set "I", such that for any two morphisms formula_3 in formula_4 if formula_5 then there is some "i" in "I" and some morphism formula_6 such that formula_7 If the family consists of a single object "G", we say it is a generator (or separator). 
Generators are central to the definition of Grothendieck categories.
The dual concept is called a cogenerator or coseparator.