Generator (category theory)

In mathematics, specifically category theory, a family of generators (or family of separators) of a category is a collection of objects, indexed by some set I, such that for any two morphisms in if then there is some i in I and some morphism such that 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.


  • In the category of abelian groups, the group of integers is a generator: If f and g are different, then there is an element , such that . Hence the map suffices.
  • Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator.
  • In the category of sets, any set with at least two objects is a cogenerator.
  • In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand. Consequently, a generator module is faithful, i.e. has zero annihilator.


  • Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2, p. 123, section V.7

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