Generator (category theory)

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In category theory in mathematics a generator 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 $f \circ h \neq g \circ h$ .

[edit] 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.