Discrete Category is Order Category

Theorem

Let $\map {\mathbf{Dis} } S$ be a discrete category.

Then $\map {\mathbf{Dis} } S$ is also an order category.

Proof

We have, for any morphism $a \to b$ in $\map {\mathbf{Dis} } S$ that $a = b$.

Thus we see that $\map {\mathbf{Dis} } S$ will be an order category if and only if:

$\forall a, b \in S: a \preceq b \iff a = b$

holds for some ordering $\preceq$ on $S$.

The trivial ordering on $S$ accomplishes this.

Hence the result.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.12$