Functor between Order Categories
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Theorem
Let $\struct {S, \preceq}$ and $\struct {T, \preceq'}$ be ordered sets.
Let $\mathbf S$ and $\mathbf T$ be their associated order categories, respectively.
Let $F: \mathbf S \to \mathbf T$ be a functor.
Then its object functor $F: S \to T$ is a monotone mapping.
Proof
Suppose that for some $a, b \in S$, we have:
- $a \preceq b$
Then there is a morphism $a \to b$ in $\mathbf S$.
As $F$ is a functor, it follows that there is a morphism:
- $F a \to F b$
in $\mathbf T$ as well, that is:
- $F a \preceq' F b$
Hence the result.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.8$