Take the 2-minute tour ×
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.

Let $C$ be a category. Recall that a morphism $f : a \to b$ in $C$ is said to be a monomorphism if, for any morphisms $g_1, g_2 : c \to a$, it is true that $f g_1 = f g_2$ implies $g_1 = g_2$. Equivalently, $f$ is a monomorphism if and only if it is injective on generalized points in the sense that the induced map $\text{Hom}(c, a) \to \text{Hom}(c, b)$ given by composition with $f$ is an injection for all $c$.

Is there a corresponding term for morphisms which are surjective on generalized points? Note that any such morphism is a retraction, hence an epimorphism. Indeed, the induced map $\text{Hom}(b, a) \to \text{Hom}(b, b)$ is surjective, so there exists $g \in \text{Hom}(b, a)$ such that $fg = \text{id}_b$. But the converse fails since there exist epimorphisms which are not retractions, such as the quotient $\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ in $\text{Ab}$.

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

Oh, ha! These morphisms are precisely the retractions. If there exists $g \in \text{Hom}(b, a)$ such that $fg = \text{id}_b$, then for every $h \in \text{Hom}(c, b)$ it follows that $gh \in \text{Hom}(c, a)$ maps to $h$ under the map induced by $f$. That's curious.

Retractions are also known as split epimorphisms, so I suppose that's my answer.

share|improve this answer
1  
Yuan: And $f$ is an epimorphism iff $f$ is injective on generalized copoints, and $f$ is a section iff $f$ is surjective on generalized copoints. (Actually I made up the term “generalized copoints”, maybe there already is a name.) –  beroal Jul 4 '11 at 17:02
add comment

Your Answer

 
discard

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.