Generator (category theory)

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

{{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=Categories for the Working Mathematician | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998}}, p. 123, section V.7

External links

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Category:Category theory