In category theory in mathematics a family of generators (or family of separators) of a category C is a collection { G i ∈ O b ( C ) | i ∈ I } of objects, indexed by some set I, such that for any two morphisms f , g : X → Y in C , if f ≠ g then there is some i∈I and morphism h : G i → X , such that the compositions f ∘ h ≠ g ∘ h . 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.