Surjection iff Epimorphism in Category of Sets

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Theorem

Let $\mathbf{Set}$ be the category of sets.

Let $f: X \to Y$ be a morphism in $\mathbf{Set}$, i.e. a mapping.


Then $f$ is a surjection if and only if it is an epimorphism.


Proof

Necessary Condition

Suppose that $f$ is surjective.

Suppose further that we have mappings $g, h: Y \to Z$ such that $g \ne h$.

Then necessarily there exists some $y \in Y$ such that $\map g y \ne \map h y$ by Equality of Mappings.


As $f$ is surjective, it follows that there is an $x \in X$ such that $\map f x = y$.

Hence, we conclude that:

$\map g {\map f x} \ne \map h {\map f x}$

which, again by Equality of Mappings, means that $g \circ f \ne h \circ f$.

Therefore, $f$ is epic, by the Rule of Transposition.

$\Box$


Sufficient Condition

Suppose that $f: X \twoheadrightarrow Y$ is an epimorphism.

By definition of surjection, it will suffice to show that:

$\forall y \in Y: \exists x \in X: \map f x = y$


Let us reason by contradiction.

So suppose $f$ were not surjective.


Then there would be an $y_0 \in Y$ such that:

$\forall x \in X: \map f x \ne y_0$

Consider the mappings defined by:

$g: Y \to Y \cup \set Y, \map g y := y$
$h: Y \to Y \cup \set Y, \map h y := \begin{cases} y & \text{if } y \ne y_0 \\

Y & \text{if } y = y_0 \end{cases}$

The assumption on $f$ yields that $g \circ f = h \circ f$.

Since $h \ne g$, it follows that $f$ cannot be epic.


This contradiction shows that $f$ is necessarily surjective.

$\blacksquare$


Sources

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