Let $\mathcal{C}$ be a category in which for every pair of objects $A,B$ , 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: $$A\le B \qquad \mbox{iff} \qquad \hom(A,B)\ne \varnothing.$$ 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.

Given a partially ordered category $\mathcal{C}$ , the partial order defined on the objects of $\mathcal{C}$ may be extended to a partial order on the morphisms of $\mathcal{C}$ . Formally, let $f,g$ be morphisms in $\mathcal{C}$ . Define $f\le g$ iff $\operatorname{dom}(f)\le \operatorname{dom}(g)$ and $\operatorname{codom}(f)\le \operatorname{codom}(g)$ . This binary relation can be immediately verified as a partial order (on the class of all morphisms), and an extension of the partial order on objects of $\mathcal{C}$ , since every object is identified the corresponding identity morphism. Furthermore, the partial order respects composition, if $f\le g$ , then

• $f\circ h \le g\circ h$ , provided that $\operatorname{dom}(f)=\operatorname{dom}(g)=\operatorname{codom}(h)$ ,
• $h\circ f \le h\circ g$ , provided that $\operatorname{codom}(f)=\operatorname{codom}(g)=\operatorname{dom}(h)$ .

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 $A,B$ of objects
• a directed category, which is a partially ordered category in which for every pair $A,B$ of objects, there is an object $C$ such that $\hom(A,C)$ and $\hom(B,C)$ are both non-empty.
• The definition given above of a partially ordered category can be generalized. Instead of having a partial order on the objects of the category, the partial order is now on the morphisms. Let $\mathcal{C}$ be a category such that a partial order $\le$ is given on the class of morphisms of $\mathcal{C}$ . This partial order induces a partial order on the objects: $A\le B$ iff $1_A\le 1_B$ . Call $\mathcal{C}$ a partially ordered category if
  1. if $f\le g$ , then $\operatorname{dom}(f)\le \operatorname{dom}(g)$ and $\operatorname{codom}(f)\le \operatorname{codom}(g)$ ,
  2. if $f_1\le g_1$ and $f_2\le g_2$ such that $f_1\circ f_2$ and $g_1\circ g_2$ are defined, then $f_1\circ f_2 \le g_1\circ g_2$ ,
  3. for any $A$ and $f$ such that $A\le \operatorname{dom}(f)$ , there exists a unique $g$ such that $g\le f$ with $\operatorname{dom}(g)=A$ ,
  4. for any $A$ and $f$ such that $A\le \operatorname{codom}(f)$ , there exists a unique $g$ such that $g\le f$ with $\operatorname{codom}(g)=A$ .
  It's not hard to see that the above definition does generalize the one given at the beginning of the entry. For example, to see 3., let $f:B\to C$ . So $A\le B$ , which means there is a unique $h:A\to B$ . Then $g= f\circ h:A\to C$ is the desired morphism: $g$ is unique, $\operatorname{dom}(g)=A$ , and $g\le f$ .

Bibliography

1

A. J. Berrick, M. E. Keating, Categories and Modules, with K-theory in View, Cambridge (2000).

2

C. Hollins, Extending The Ehresmann-Schein-Nambooripad Theorem, (2009)