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.

Reference

Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2 , p. 123, section V.7

External links

Generator in nLab