Generator (category theory)

In mathematics, specifically category theory, a family of generators (or family of separators) of a category ${𝒞}$ is a collection ${𝒢}\subseteq Ob({𝒞})$ of objects in ${𝒞}$, such that for any two distinct morphisms $f,g:X\to Y$ in ${𝒞}$, that is with $f\ne g$, there is some $G$ in ${𝒢}$ and some morphism $h:G\to X$ such that $f\circ h\ne g\circ h.$ If the collection 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 $f(x)\ne g(x)$. Hence the map $\mathbf{𝐙}\to X,$ $n\mapsto n\cdot x$ 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 elements 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

• separator in nLab