# 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**.

## Examples

- 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.

## References

- 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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