Hi,


Suppose you are given a dcpo of categories, where the functor is taken as the ordering relation. This collection is a category too and it has limits, or perhaps cartesian products. This comes from the shape of what the supremum looks like. This dcpo might also be a topos. Does anyone have any thoughts on this?
 Also, what are the compact elements? That is to say, what universal properties do the compact elements have?



a dcpo of categories has limits and compact categories
To answer your more pointed form of the question: if $P$ is a poset viewed as a category then the limit of a functor $F:I \to P$ is the same thing as $\inf \lbrace F(i) : i\in I \rbrace$ and the colimit is the same thing as $\sup \lbrace F(i) : i\in I\rbrace$. If I'm not mistaken a directed complete partial order is just a partially ordered set in which every directed subset has a supremum. So a dcpo doesn't have all limits or colimits in general.


Note that your example of taking a collection of categories with a preorder defined by the existence of functors between them doesn't amount to much without restricting the class of functors allowed (because of the existence of constant functors). On the other hand, if you use a restricted class of functors such that you do obtain a directed complete partial order then the first paragraph applies. In other words, knowing that it forms a dcpo doesn't provide you with limits although it does provide you with *some* colimits.


Perhaps I have misunderstood what you're getting at, but it is also possible that these elementary observations will be helpful.


