Generator (category theory)

In category theory in mathematics a family of generators (or family of separators) of a category ${\mathcal {C}}$ is a collection $\{G_{i}\in Ob({\mathcal {C}})|i\in I\}$ of objects, indexed by some set I, such that for any two morphisms $f,g:X\rightarrow Y$ in ${\mathcal {C}}$ , if $f\neq g$ then there is some i∈I and morphism $h:G_{i}\rightarrow X$ , such that the compositions $f\circ h\neq g\circ h$ . 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 $\mathbf {Z}$ is a generator: If f and g are different, then there is an element $x\in X$ , such that $f(x)\neq g(x)$ . Hence the map $\mathbf {Z} \rightarrow X,$ $n\mapsto n\cdot x$ 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.