THC Science

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[edit]

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 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.