Generator (category theory)

In category theory in mathematics a generator (or separator) of a category ${\mathcal {C}}$ is an object G of the category, such that for any two different morphisms $f,g:X\rightarrow Y$ in ${\mathcal {C}}$ , there is a morphism $h:G\rightarrow X$ , such that the compositions $f\circ h\neq g\circ h$ .

Generators are central to the definition of Grothendieck categories.

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.