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. While Ord is a category with different properties, the category of preordered groups, denoted OrdGrp, presents a more complex picture, nonetheless both imply preordered connections.^[1]

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.

See also

• FinOrd
• Simplex category

References

1. Clementino, Maria Manuel; Martins-Ferreira, Nelson; Montoli, Andrea (1 October 2019). "On the categorical behaviour of preordered groups". Journal of Pure and Applied Algebra. pp. 4226–4245. doi:10.1016/j.jpaa.2019.01.006.