## You are here

Homemonic

## Primary tabs

# monic

A morphism $f\colon A\to B$ in a category is called a monic morphism, or *monomorphism*, if it can be cancelled from the left — for any object $C$ and any morphisms $g_{1},g_{2}\colon\ C\to A$ we have $f\circ g_{1}=f\circ g_{2}$ if and only if $g_{1}=g_{2}$.

A morphism $f:A\to B$ in a category is called a *split monomorphism*
if there exists a morphism $g\colon B\to A$ such that $g\circ f=\operatorname{id}_{A}$. Note that every split monomorphism is a monomorphism;
if $f$ is a split monomorphism and $f\circ h=f\circ k$, then one has
$g\circ(f\circ h)=g\circ(f\circ k)$. By associativity, $(g\circ f)\circ h=(g\circ f)\circ k$; by definition of split monomorphism,
$\operatorname{id}_{a}\circ h=\operatorname{id}_{a}\circ k$; by definition of
identity, $h=k$, so $f$ is a monomorphism. Split monomorphisms are also
known as *sections* and *coretractions*.

The notion of epimorphism is dual to that of monomorphism. An epimorphism of a category is a monomorphism of the dual category and vice versa.

A monomorphism in the category of sets is simply a one-to-one function. Moreover, in the category of sets all monomorphisms are split monomorphisms.

## Mathematics Subject Classification

18A20*no label found*18-00

*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