# Definition:Order Category

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## 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:

- 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.