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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