Definition:Order Category/Definition 1

Definition
Let $\struct {S, \preceq}$ be an ordered set.

One can interpret $\struct {S, \preceq}$ as being a category, with:

More formally, we let the morphisms be the elements of the relation ${\preceq} \subseteq S \times S$.

Thus, $a \to b$ in fact denotes the ordered pair $\tuple {a, b}$.

The category that so arises is called an order category.

Also see

 * Equivalence of Definitions of Order Category