The category

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

 Ord has preordered sets

Preorder

In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...

 as objects and monotonic function

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

s as morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s. This is a category because the composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

 of two monotonic functions is monotonic and the identity map is monotonic.

The monomorphism

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

s in Ord are the injective monotonic functions.

The empty set

Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

 (considered as a preordered set) is the initial object

Initial object

In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...

 of Ord; any singleton preordered set is a terminal object. There are thus no zero objects in Ord.

The product

Product (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

 in Ord is given by the product order

Product order

In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product A × B. Giventwo pairs and in A × B, one sets ≤...

 on the cartesian product

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

.

We have a forgetful functor

Forgetful functor

In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...

 Ord → Set

Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

 which assigns to each preordered set the underlying set, and to each monotonic function the underlying function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. This functor is faithful, and therefore Ord is a concrete category

Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...

. This functor has a left adjoint

Adjoint functors

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...

 (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

2-category

In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...

 (the additional axioms of a 2-category trivially hold because any equation of parallel morphisms is true in a posetal category

Posetal category

In mathematics, a posetal category is a category whose homsets each contain at most one morphism. As such a posetal category amounts to a preordered set...

).

With this 2-category structure, a pseudofunctor F from a category C to Ord is given by the same data as a 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 ∧ y ≤ x.