Proving Epimorphism: Understanding Surjective Morphisms

[jhendren]

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

 

[Office_Shredder]

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

 

[micromass]

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

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?

 

[PSarkar]

Must be on sets. Here is a Wikipedia page on it: https://en.wikipedia.org/wiki/Surjective_function#Surjections_as_epimorphisms