# Category of preordered sets

The category Ord has preordered sets as objects and monotonic functions as morphisms. This is a category because the composition of two monotonic functions is monotonic and the identity map is monotonic.

The monomorphisms in Ord are the injective monotonic functions.

The empty set (considered as a preordered set) is the initial object of Ord; any singleton preordered set is a terminal object. There are thus no zero objects in Ord.

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

We have a forgetful functor OrdSet which assigns to each preordered set the underlying set, and to each monotonic 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 (monotonic 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 (idA) (x) ≃ x
∀ x ∈ F(A), F (g ∘ f) (x) ≃ F(g) (F(f) x)

where x ≃ y means x ≤ y ∧ y ≤ x.