Definition:Category of Ordered Sets

Definition
The category of ordered sets, denoted $\mathbf{OrdSet}$, is the metacategory with:


 * objects: All ordered sets;
 * morphisms: All increasing mappings.

This forms a metacategory, as shown on Category of Ordered Sets is Category.

Note
The reason to call $\mathbf{OrdSet}$ a metacategory is foundational; allowing it to be a category would bring us to axiomatic troubles.

Also known as
Similar to some sources referring to an ordered set as a poset, $\mathbf{OrdSet}$ is also referred to as the category of posets, and consequently denoted $\mathbf{Pos}$.