What should the morphisms in the Category of Directed Sets be? Directed sets 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?
 A: 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.
