# Category Theory/Categories of ordered sets

Definition (category of preordered sets):

The category ${\displaystyle {\textbf {Ord}}}$ of preordered sets is the category whose objects ${\displaystyle \operatorname {Ob} ({\textbf {Ord}})}$ are given by the preordered sets and whose morphisms are given by the order homomorphisms.

Definition (category of posets):

The category ${\displaystyle {\textbf {Pos}}}$ of posets is the category whose objects are the posets and whose morphisms are the order homomorphisms.

Directly from the definitions, we have:

Proposition (posets form full subcategory of preordered sets):

${\displaystyle {\textbf {Pos}}}$ is a full subcategory of ${\displaystyle {\textbf {Ord}}}$.

Proposition (products in the category of preordered sets):

Let ${\displaystyle (S_{\alpha })_{\alpha \in A}}$ be a family of preordered sets. Then a product of this family in the category ${\displaystyle {\textbf {Ord}}}$ is given by the set ${\displaystyle \prod _{\alpha \in A}S_{\alpha }}$ together with the product order, where the projections are given by the functions

${\displaystyle p_{\beta }:\prod _{\alpha \in A}S_{\alpha }\to S_{\beta },p_{\beta }((s_{\alpha })_{\alpha \in A})=s_{\beta }}$.

Proof: The functions ${\displaystyle p_{\beta }}$ are order homomorphisms, because if ${\displaystyle \beta \in A}$ and ${\displaystyle (s_{\alpha })_{\alpha \in A}\leq (t_{\alpha })_{\alpha \in A}}$, then ${\displaystyle p_{\beta }((s_{\alpha })_{\alpha \in A})=s_{\beta }\leq t_{\beta }=p_{\beta }((t_{\alpha })_{\alpha \in A})}$ by definition. Thus, we have a cone. Moreover, if the order homomorphisms ${\displaystyle q_{\alpha }:Q\to S_{\alpha }}$ define another cone, then as for the set product, the function given by

${\displaystyle f:Q\to \prod _{\alpha \in A},f(x)=(q_{\alpha }(x))_{\alpha \in A}}$

is the unique function from ${\displaystyle Q}$ to ${\displaystyle \prod _{\alpha \in A}S_{\alpha }}$ such that ${\displaystyle q_{\alpha }=p_{\alpha }\circ f}$ for all ${\displaystyle \alpha \in A}$, and it is an order homomorphism, because if ${\displaystyle x\leq y}$, then for all ${\displaystyle \alpha \in A}$ we have ${\displaystyle q_{\alpha }(x)\leq q_{\alpha }(y)}$. ${\displaystyle \Box }$

Proposition (coproducts in the category of preordered sets):

Let ${\displaystyle (S_{\alpha })_{\alpha \in A}}$ be a family of preordered sets. Then a coproduct of this family in the category ${\displaystyle {\textbf {Ord}}}$ is given by the parallel composition of the ${\displaystyle S_{\alpha }}$, where the inclusions are given by the functions

${\displaystyle i_{\beta }:S_{\beta }\to \bigsqcup _{\alpha \in A}S_{\alpha },i_{\beta }(s_{\beta })=(s_{\beta },\beta )}$.

Proof: The ${\displaystyle i_{\beta }}$ are order homomorphisms by definition of the parallel order, so that we do have a cocone. Suppose now that the maps ${\displaystyle j_{\alpha }:S_{\alpha }\to D}$ define another cocone in the category ${\displaystyle {\textbf {Ord}}}$. Then the unique function ${\displaystyle f:\bigsqcup _{\alpha \in A}S_{\alpha }\to D}$ such that ${\displaystyle f\circ i_{\alpha }=j_{\alpha }}$ for all ${\displaystyle \alpha \in A}$ is given by

${\displaystyle f:\bigsqcup _{\alpha \in A}S_{\alpha }\to D,f((s,\alpha ))=j_{\alpha }(s)}$

as in set theory, and it is an order homomorphism because if ${\displaystyle (s,\beta )\leq (t,\beta )}$, then ${\displaystyle s\leq _{\beta }t}$ by def. of the parallel order and consequently ${\displaystyle f((s,\beta ))=j_{\beta }(s)\leq j_{\beta }(t)=f((t,\beta ))}$. ${\displaystyle \Box }$

Proposition (products and coproducts in the category of posets):

In the category ${\displaystyle \mathbf {Pos} }$, products and coproducts are given by the respective products and coproducts in the category ${\displaystyle \mathbf {Ord} }$.

Proof: This follows since limits and colimits are preserved when restricting to a full subcategory. ${\displaystyle \Box }$