\begin{definition}[Definition:Preorder Category/Definition 1]
Let $\left({S, \precsim}\right)$ be a preordered set.
One can interpret $\left({S, \precsim}\right)$ as being a category, with:
{{DefineCategory
|ob = The elements of $S$
|mor = Precisely one morphism $a \to b$ for every $a, b \in S$ with $a \precsim b$
}}
More formally, we let the morphisms be the elements of the relation ${\precsim} \subseteq S \times S$.
Thus, $a \to b$ in fact denotes the ordered pair $\left({a, b}\right)$.
The category that so arises is called a '''preorder category'''.
\end{definition}