Morphism in Preorder Category is Epic
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Theorem
Let $\mathbf P$ be a preorder category.
Let $f \in \mathbf P_1$ be a morphism.
Then $f$ is epic.
Proof
Suppose that $g, h \in \mathbf P_1$ are morphisms such that:
- $g \circ f = h \circ f$
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 epic.
$\blacksquare$
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 $2.2$