Definition:Order Category
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Definition
Definition 1
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.
Definition 2
Let $\mathbf C$ be a metacategory.
Then $\mathbf C$ is an order category if and only if:
- 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.