# Generator (category theory)

In category theory in mathematics a family of generators (or family of separators) of a category ${\displaystyle {\mathcal {C}}}$ is a collection ${\displaystyle \{G_{i}\in Ob({\mathcal {C}})|i\in I\}}$ of objects, indexed by some set I, such that for any two morphisms ${\displaystyle f,g:X\rightarrow Y}$ in ${\displaystyle {\mathcal {C}}}$, if ${\displaystyle f\neq g}$ then there is some i∈I and morphism ${\displaystyle h:G_{i}\rightarrow X}$, such that the compositions ${\displaystyle f\circ h\neq g\circ h}$. 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.

## Examples

• In the category of abelian groups, the group of integers ${\displaystyle \mathbf {Z} }$ is a generator: If f and g are different, then there is an element ${\displaystyle x\in X}$, such that ${\displaystyle f(x)\neq g(x)}$. Hence the map ${\displaystyle \mathbf {Z} \rightarrow X,}$ ${\displaystyle n\mapsto n\cdot x}$ 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.