Smallest Element is Initial Object
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Theorem
Let $\mathbf P$ be an order category.
Suppose the objects $\mathbf P_0$ of $\mathbf P$, considered as an ordered set, have a smallest element $p$.
Then $p$ is an initial object of $\mathbf P$.
Proof
Since $p$ is the smallest element of $\mathbf P_0$, we have:
- $\forall q \in \mathbf P_0: p \le q$
that is, for every object $q$ of $\mathbf P$ there is a unique morphism $p \to q$.
That is, $p$ is initial.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $5$