## You are here

Homegenerator of a category

## Primary tabs

# generator of a category

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 $f\circ h\neq g\circ h$. For example, if $f\neq g:A\to B$, then $1_{A}$ on $A$ distinguishes $f$ and $g$.

A set $S=\{X_{i}\mid i\in I\}$ 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 $f\circ h\neq g\circ h$. If $\{X\}$ 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 $f(x)\neq g(x)$ for some $x\in A$. Let $\{y\}$ be any singleton. Then $h:\{y\}\to A$ defined by $h(y)=x$ is the function distinguishing $f$ and $g$: for $f\circ h(y)=f(x)\neq g(x)=g\circ h(y)$.

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, $f(r)\neq g(r)$ 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*.

# References

- 1
F. Borceux
*Basic Category Theory, Handbook of Categorical Algebra I*, Cambridge University Press, Cambridge (1994)

## Mathematics Subject Classification

18A99*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections