GENERATOR (CATEGORY THEORY)

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 $f,\; g:\; X$
ightarrow Y in $mathcal\; C$, there is a morphism $h\; :\; G$
ightarrow X, such that the compositions $f\; circ\; h$
eq g circ h.

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Examples

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 $f(x)$
eq g(x). Hence the map $mathbf\; Z$
ightarrow X, $n\; mapsto\; n\; cdot\; x$ suffices.

★ Similarly, the one-point set is a generator for the category of sets.