# Proof of epimormphism?

1. Sep 19, 2013

### jhendren

How would one show a morphism is an epimorphism iff it is surjective (ONTO)?

2. Sep 19, 2013

### Office_Shredder

Staff Emeritus
What is your definition of epimorphism if it's not that it is a surjective map?

3. Sep 19, 2013

### micromass

Staff Emeritus
An epimorphism in category theory is a morphism $f:X\rightarrow Y$ such that if $g,h:Y\rightarrow Z$ are morphisms such that $g\circ f = h\circ f$, then $g=h$.

It is certainly not always true that an epimorphism is surjective. It depends on what category you work in. So, to the OP, you are talking about an epimorphism between which structures? Sets? Groups?

4. Nov 17, 2013