# Generator (category theory)

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In category theory in mathematics 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∈I* and morphism , such that the compositions . If the family consists of a single object *G*, we say it is a *generator*.

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.

## References

- {{#invoke:citation/CS1|citation

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