# The Category of Posets in Category Theory

In the category of **posets (partially ordered sets)**, there can be at most one direct morphism $a \rightarrow b$ between any pair of objects $a, b$ within set P. Morphisms in this context are defined as order relations.

Typically, a poset is a set \(P\) with an order relation \(\leq\) that is reflexive, antisymmetric, and transitive.

$$ Poset (P, \leq) $$

This implies that within the set:

- each element is equivalent to itself \( a = a \) (reflexivity),
- if \(a \leq b\) and \(b \leq a\), then \(a = b\) (antisymmetry),
- if \(a \leq b\) and \(b \leq c\), then \(a \leq c\) (transitivity).

These properties qualify posets as a category (\(P, \leq\)) with specific structures:

**Objects**: The elements of the set \(P\).**Morphisms**: Functions between two objects \(a, b \in P\) that exist only if \(a \leq b\). There is a unique morphism for each pair \(a \leq b\), representing the order relation between \(a\) and \(b\). All order relations adhere to associativity.**Note**: Like other categories, all morphisms between two objects are part of the hom-set(a, b). Generally, all morphisms in set \(P\) are part of the set \(Mor(P)\).**Morphism Composition**: If there are morphisms from \(a\) to \(b\) and from \(b\) to \(c\), reflecting \(a \leq b\) and \(b \leq c\), a direct and unique morphism from \(a\) to \(c\) must exist due to the transitivity of the order.**Identity Morphism**: Each element \(a \in P\) inherently possesses an identity morphism due to the reflexive nature of the order relation.

Thus, a poset meets all the criteria of a category.

In essence, category theory applied to posets provides a structured and abstract framework for analyzing order relations within partially ordered sets.

## An Illustrative Example

Each natural number \(n \in \mathbb{N}\) can be considered a set that includes all natural numbers less than \(n\), from \(0\) to \(n-1\).

$$ \mathbb{N} = \{ 0,1,2,3,4,5,... \} $$

The natural number \(0\) is defined as the empty set \(\emptyset\) since it lacks predecessors.

$$ 0 = \{ \} $$

Conversely, the natural number \(1\) is a singleton set, having only \(0\) as its predecessor.

$$ 1 = \{ 0 \} $$

Following this pattern, the natural number \(2\) is defined by its predecessors \(0\) and \(1\).

$$ 2 = \{ 0,1 \} $$

This structure implies that **each natural number inherently forms a category** where the objects are the numbers less than itself, and the morphisms reflect the natural or inclusion relations among these numbers.

For example, consider the natural number \(3\) as a category.

**The category of number 3** consists of all natural numbers preceding it:

$$ 3 = \{0, 1, 2\} $$

Here, the **objects** are the numbers \(0\), \(1\), and \(2\).

The **morphisms** are established through the natural order (\(\leq\)), where a morphism exists from one number to another if the former is less than or equal to the latter.

The morphisms in this category include:

$$ id_0: 0 \rightarrow 0 $$

$$ id_1: 1 \rightarrow 1 $$

$$ id_2: 2 \rightarrow 2 $$

$$ f: 0 \rightarrow 1 $$

$$ g: 0 \rightarrow 2 $$

$$ h: 1 \rightarrow 2 $$

The first three morphisms are identity morphisms, as each element relates to itself in the order \( \leq \). The other morphisms represent ascending order relations among different objects: \( 0 \leq 1 \), \( 0 \leq 2 \), and \( 1 \leq 2 \).

The respective hom-sets are as follows:

**hom(0, 0)**: Contains only the identity of \(0\), \(\text{id}_0\).**hom(1, 1)**: Contains only the identity of \(1\), \(\text{id}_1\).**hom(2, 2)**: Contains only the identity of \(2\), \(\text{id}_2\).**hom(0, 1)**: Contains the morphism \(f\), representing \(0 \leq 1\).**hom(0, 2)**: Contains the morphism \(g\), representing \(0 \leq 2\).**hom(1, 2)**: Contains the morphism \(h\), representing \(1 \leq 2\).

**Note**: The hom-sets hom(1, 0), hom(2, 0), and hom(2, 1) are empty because no order relations descend from a larger to a smaller number.

The morphism \(f\) composed with \(h\) results in \(g\), demonstrating the transitivity of the natural order.

$$ h \circ f = g $$

Graphically, this category can be depicted as follows:

This diagram uses arrows to indicate morphisms, showing an ascending order relation.

The **set of all morphisms** in this category includes six morphisms:

$$ Mor(3) = \{ id_0, id_1, id_2, f, g, h \} $$

In summary, this example illustrates how a set of natural numbers less than a given number can be organized into a category, with objects and morphisms reflecting the natural order relations. Here, the numbers themselves serve as objects, and their relational orderings as morphisms.