# Category of preordered sets

In mathematics, the category **Ord** has preordered sets as objects and order-preserving functions as morphisms. This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving.

The monomorphisms in **Ord** are the injective order-preserving functions.

The empty set (considered as a preordered set) is the initial object of **Ord**, and the terminal objects are precisely the singleton preordered sets. There are thus no zero objects in **Ord**.

The categorical product in **Ord** is given by the product order on the cartesian product.

We have a forgetful functor **Ord** → **Set** that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore **Ord** is a concrete category. This functor has a left adjoint (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to that set equipped with the total relation).

## 2-category structure

The set of morphisms (order-preserving functions) between two preorders actually has more structure than that of a set. It can be made into a preordered set itself by the pointwise relation:

- (
*f*≤*g*) ⇔ (∀*x**f*(*x*) ≤*g*(*x*))

This preordered set can in turn be considered as a category, which makes **Ord** a 2-category (the additional axioms of a 2-category trivially hold because any equation of parallel morphisms is true in a posetal category).

With this 2-category structure, a pseudofunctor F from a category *C* to **Ord** is given by the same data as a 2-functor, but has the relaxed properties:

- ∀
*x*∈ F(*A*), F(*id*_{A})(*x*) ≃*x*,

- ∀
*x*∈ F(*A*), F(*g*∘*f*)(*x*) ≃ F(*g*)(F(*f*)(*x*)),

where *x* ≃ *y* means *x* ≤ *y* and *y* ≤ *x*.