Discrete Category is Order Category
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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): $\S 1.4.12$