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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?

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When using directed sets to take limits of points in topology or taking direct limits, it is well known that the limit is the same if you only consider a cofinal portion of the directed set. Therefore in the category of directed sets, each directed set should be isomorphic to any cofinal subset. I suspect that the category of directed sets is related to the Tukey ordering on directed sets. I will probably answer this question soon once I figure everything out. – Joseph Van Name Jul 10 '13 at 17:31
Also, the category of directed sets should be a full subcategory of the category of filtrant categories (whatever that is) since the notion of a filtrant category is the categorization of the notion of a directed set. By the way, this is an interesting question. – Joseph Van Name Jul 10 '13 at 17:47
Thanks Joeseph! It's been sitting in the back of my mind for a couple days. – Tyler Bryson Jul 10 '13 at 17:56
Look here for a collection of papers of Douglas et al. The morphisms are there continuous set maps between direct intervals. – Marc Palm Jul 10 '13 at 20:09
Doesn't the answer to the question depend ultimately on what you want to do with this category? I can imagine different uses of the class of directed sets, which have different natural morphisms. – Joel David Hamkins Jul 10 '13 at 23:35
up vote 5 down vote accepted

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.

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Thanks for this Joseph. I'm reading about the Tukey ordering now. – Tyler Bryson Jul 11 '13 at 3:21

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