Generator (category theory)

In category theory in mathematics a family of generators (or family of separators) of a category ${\mathcal {C}}$ is a collection $\{G_{i}\in Ob({\mathcal C})|i\in I\}$ of objects, indexed by some set I, such that for any two morphisms $f,g:X\rightarrow Y$ in ${\mathcal {C}}$, if $f\neq g$ then there is some i∈I and morphism $h:G_{i}\rightarrow X$, such that the compositions $f\circ h\neq g\circ h$. If the family consists of a single object G, we say it is a generator (or separator).

Generators are central to the definition of Grothendieck categories.

The dual concept is called a cogenerator or coseparator.

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. In fact, any nonempty set is a generator.
• In the category of sets, any set with at least two objects is a cogenerator.

References

• 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