Every preorder $P$ has an associated structure of a category. Before describing what this category is, we first associate $P$ with a simpler structure, that of a precategory.

Let's call this $\operatorname{PreCat}(P)$ The objects of this precategory are elements of $P$ and for every $a,b\in P$ $\hom(a,b)$ is either a singleton if $a\le b$ or the empty set otherwise. The category associated with $P$ is the category generated by enlarging $\operatorname{PreCat}(P)$ For now, call this category $\operatorname{Cat}(P)$ Then we see that the objects of $\operatorname{Cat}(P)$ are again elements of $P$ and for every $a,b\in P$ $\hom(a,b)$ is the set of all finite chains $f$ from $a$ to $b$

With this association, we see the following constructs also have the structure of a category:

• a poset: here, a morphism in $\hom(a,b)$ is a finite chain from $a$ to $b$ where successive nodes are related such that the subsequent node covers the prior node
• a partition of a (non-empty) set (a set with an equivalence relation): $\hom(a,b)$ is non-empty iff $a$ and $b$ belong to the same partition
• a lattice: every pair of objects have a product and a coproduct
• a well-ordered set, in particular an ordinal: if $\hom(a,b)$ is non-empty, it is a singleton. For example, ${n}$ is the category consisting of objects $0,1,\ldots,n-1$ and if $a\le b$ a morphism in $\hom(a,b)$ is the chain $a\to a+1 \to \cdots \to b-1 \to b$