PlanetMath: partially ordered category

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partially ordered category

(Definition)

Let $\mathcal{C}$ be a category in which for every pair of objects , the collection $\hom(A,B)$ is a set. $\mathcal{C}$ is called a partially ordered category if $\hom(A,B)\cup \hom(B,A)$ has at most one element. The reason why it is called partially ordered is because we can put a partial order on the objects of $\mathcal{C}$ , as follows:

$\displaystyle A\le B$ iff $\displaystyle \qquad \hom(A,B)\ne \varnothing.$

It is easily verified that $\le$ is a partial order on $\operatorname{Ob}(\mathcal{C})$ : clearly, $\hom(A,A)$ is a singleton as it contains the identity morphism, so that $\le$ is reflexive; if $\hom(A,B)\ne \varnothing$ and $A\ne B$ , then $\hom(B,A)=\varnothing$ by definition, and so $\le$ is antisymmetric; finally, if neither $\hom(A,B)$ nor $\hom(B,C)$ are empty, then $\hom(A,C)$ can not be empty, as it contains the composition of the elements in $\hom(A,B)$ and $\hom(B,C)$ , and thus $\le$ is transitive.

It is easy to see that any poset can be realized as categroy, and in fact, a partially ordered category.

Remarks. From a partially ordered category, one may also define

a linearly ordered category, which is partially category in which $\hom(A,B)\cup \hom(B,A)$ has at least one element for every pair of objects
a directed category, which is a partially ordered category in which for every pair of objects, there is an object such that $\hom(A,C)$ and $\hom(B,C)$ are both non-empty.