# Definition:Order Category

## Definition

### Definition 1

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

One can interpret $\left({S, \preceq}\right)$ 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 $\left({a, b}\right)$.

The category that so arises is called an order category.

### Definition 2

Let $\mathbf C$ be a metacategory.

Then $\mathbf C$ is an order category iff:

For all objects $C, C'$ of $\mathbf C$, there is at most one morphism $f: C \to C'$
Whenever $f: C \to C'$ is an isomorphism, $C = C'$

Thus, an order category is a skeletal preorder category.

## Also known as

In sources which address ordered sets as posets, such a category is usually named poset category.

## Also see

• Results about order categories can be found here.