Generator (category theory)

In mathematics, specifically category theory, a family of generators (or family of separators) of a category ${\mathcal {C}}$ is a collection $\{G_{i}\in Ob({\mathcal {C}})\mid i\in I\}$ of objects, indexed by some set I, such that for any two morphisms $f,g:X\to Y$ in ${\mathcal {C}},$ if $f\neq g$ then there is some i in I and some morphism $h:G_{i}\to X$ such that $f\circ h\neq g\circ h.$ 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.
In the category of modules over a ring R, a generator in a finite direct sum with itself contains an isomorphic copy of R as a direct summand. Consequently, a generator module is faithful, i.e. has zero annihilator.

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

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