[Directed sets](http://ncatlab.org/nlab/show/direction) are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ and $q \preceq r$. Equivalently, they may be defined as thin categories in which every finite diagram admits a cocone.


In either case, there is some intuition as to what a morphism of directed sets ought to be: in the first case, perhaps monotone functions; in the second, a functor such that there exists a cocone over any finite diagram that maps to a cocone over the image.


However, these aren't the only two descriptions of directed sets nor are the two suggested definitions equivalent. Finally, I haven't been able to find a source which describes a (the?) category of directed sets. Is there a consensus on the 'right' definition of a morphism of directed sets? Further, are there any good resources on the properties of the category of directed sets?



What should the morphisms in the Category of Directed Sets be?
Here is a partial answer.


Given a directed set $D$, we say that a subset $A\subseteq D$ is cofinal if for each $d\in D$ there is some $a\in A$ with $d\leq a$. We say that $A\subseteq D$ is bounded if $A\subseteq\downarrow d=\{x\in D|x\leq d\}$ for some $d\in D$ and we say that $A$ is unbounded if it is not bounded. Let $D,E$ be two directed sets. Then we say that a function $f:D\rightarrow E$ (not necessarily order preserving) is a cofinal map (also called a convergent map) if the image of every cofinal subset of $D$ is a cofinal subset of $E$. A function $g:E\rightarrow D$ (not necessarily order preserving) is said to be unbounded (also called a Tukey map) if the image of every unbounded subset of $E$ is an unbounded subset of $D$. If $D,E$ are posets, then there is an unbounded map $f:D\rightarrow E$ if and only if there is a cofinal map $g:E\rightarrow D$, and in either case we say that $D$ is Tukey reducible to $E$ and we write $D\leq\_{T}E$. The Tukey ordering $\leq\_{T}$ in a sense measures how big your directed set is. Clearly the class of directed sets can be made a category where the morphisms are either the cofinal maps or the unbounded maps, and these categories are enough for us to define the Tukey ordering. I conjecture that one can take a quotient category of one of these categories to get a more natural category of directed sets(i.e. where a directed set is isomorphic to each of its cofinal subsets), but I don't see how exactly to go about this. Perhaps someone else will give a better and more complete answer to this question.


