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.
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).
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(idA)(x) ≃ x,
- ∀x ∈ F(A), F(g∘f)(x) ≃ F(g)(F(f)(x)),
where x ≃ y means x ≤ y and y ≤ x.