Definition:Preorder Category

Definition
Let $\left({S, \precsim}\right)$ be a preordered set.

One can interpret $\left({S, \precsim}\right)$ as being a category, with:


 * objects: The elements of $S$;
 * morphisms: 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 $\precsim$, interpreted as a relation, i.e. a subset of $S \times S$.

Thus, $a \to b$ in fact denotes the ordered pair $\left({a, b}\right)$.

The category that so arises is called a preorder category.

Poset Category
Naturally enough, when $\left({S, \precsim}\right)$ is in fact a poset, one calls the associated category a poset category.

Also defined as
Some sources define an arbitrary metacategory $\mathbf C$ to be a preorder category iff:


 * For all objects $C, C'$ of $\mathbf C$, there is at most one morphism $f: C \to C'$

$\mathbf C$ is then said to be a poset category iff also:


 * Whenever $f: C \to C'$ is an isomorphism, $C = C'$

Also see

 * Preorder Category is Category
 * Category Induces Preorder
 * Preorder Induced by Preorder Category