Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let A and B are two objects in a category $\mathcal{C}$ and $u \in Hom(A,B)$. We say $u $ is a monomorphism if and only if for any $X \in Obj(\mathcal{C})$ and $v \in Hom(X,A)$, the map $f:Hom(X,A)\rightarrow Hom(X,B)$ given by $f(v)=uv $ is a monomorphism.

My question:

Is there a same kind of if and only if statement for category epimorphism ?

share|cite|improve this question
up vote 8 down vote accepted

Yes (though every definition can be viewed as an "if and only if statement"). The definition of an epimorphism in a category $\mathcal{C}$ is an arrow $f:X\to Y$ such that whenever $Z$ is an object and $u,v:Y\to Z$ are arrows, $u\circ f=v\circ f$ implies $u=v$. In other words, $f:X\to Y$ is an epimorphism if and only if for all objects $Z$ the map $f^*:\mathrm{Hom}_{\mathcal{C}}(Y,Z)\to\mathrm{Hom}_{\mathcal{C}}(X,Z)$ given by $f^*(u)=u\circ f$, is injective.

share|cite|improve this answer
thanks Keenan Kidwell – ASB Jun 24 '14 at 7:38

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.