Definition:Preorder Category
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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
Equivalence of Definitions
That the above definitions are equivalent is shown on Equivalence of Definitions of Preorder Category.
Also see
- Results about preorder categories can be found here.