Definition:Preorder Category

Definition

Definition 1

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 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.

Definition 2

Let $\mathbf C$ be a metacategory.

Then $\mathbf C$ is a preorder category if and only if

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

Equivalence of Definitions

That the above definitions are equivalent is shown on Equivalence of Definitions of Preorder Category.

Also see

• Definition:Order Category

• Preorder Category is Category
• Category Induces Preorder
• Preorder Induced by Preorder Category

• Results about preorder categories can be found here.