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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 $\text{Ab}$.

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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.

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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

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