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:

Objects:		The elements of $S$
Morphisms:		Precisely one morphism $a \to b$ for every $a, b \in S$ with $a \preceq b$

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

• Results about order categories can be found here.

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.8$