Morphism in Preorder Category is Monic

Theorem

Let $\mathbf P$ be a preorder category.

Let $f \in \mathbf P_1$ be a morphism.

Then $f$ is monic.

Proof

Suppose that $g, h \in \mathbf P_1$ are morphisms such that:

$f \circ g = f \circ h$

In particular then, $g$ and $h$ have equal domain and codomain.

Since $\mathbf P$ is a preorder category, there is at most one morphism between any two objects.

Thus necessarily $g = h$, and hence $f$ is monic.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.1$: Example $2.4$
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.8$: Exercise $2.2$