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 $f\circ h\neq g\circ h$.

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.