Let $\mathcal{C}$ be a category, and $f,g:A\to B$ a pair of distinct morphisms. A morphism $h:X\to A$ is said to distinguish or separate $f$ and $g$ if $f\circ h\ne g\circ h$ For example, if $f\ne g:A\to B$ then $1_A$ on $A$ distinguishes $f$ and $g$

A set $S=\lbrace X_i\mid i\in I\rbrace$ of objects (indexed by a set $I$ is called a generating set of $\mathcal{C}$ if any pair of distinct morphisms $f,g:A\to B$ can be distinguished by a morphism with domain in $S$ and codomain $A$ In other words, there is $h:X_i\to A$ for some $i\in I$ such that $f\circ h\ne g\circ h$ If $\lbrace X\rbrace$ is a generating family of $\mathcal{C}$ then $X$ is called a generator of $\mathcal{C}$ Any set of morphisms containing a generator is a generating set.

Examples

1. In Set, the category of sets, any singleton is a generator. Suppose $f,g:A\to B$ are distinct functions, so that $f(x)\ne g(x)$ for some $x\in A$ Let $\lbrace y\rbrace$ be any singleton. Then $h:\lbrace y\rbrace \to A$ defined by $h(y)=x$ is the function distinguishing $f$ and $g$ for $f\circ h(y)=f(x)\ne g(x)=g\circ h(y)$
2. In Rng, the category of rings, the ring $\mathbb{Z}$ is a generator. If $f,g:R\to S$ are distinct ring homomorphisms, say, $f(r)\ne g(r)$ for some $r\in R$ Then the ring homomorphism $h:\mathbb{Z}\to R$ given by $h(1)=r$ distinguishes $f$ and $g$

Remark. A projective object that is also a generator is called a progenerator.

Bibliography

1

F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)