Definition:Order Category/Definition 1

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

One can interpret $\left({S, \preceq}\right)$ as being a category, with:

More formally, we let the morphisms be the elements of $\preceq$, interpreted as a relation, i.e. a subset of $S \times S$.

Thus, $a \to b$ in fact denotes the ordered pair $\left({a, b}\right)$.

The category that so arises is called a poset category.

Also see

 * Preorder Category