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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?

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  • $\begingroup$ 'where the functor...' - which functor? And what is the question here. MO is not particularly suited to 'have any thoughts'-type questions. $\endgroup$
    – David Roberts
    Sep 23, 2011 at 2:57
  • $\begingroup$ I know some category theory, but I don't know all the jargon you're using ("dcpo" in particular). Can you provide a bit more details? (You will probably benefit from such an exercise as well: I find it very valuable to practice writing down definitions.) Also, with @David Roberts, I'm not a fan of "does anyone have any thoughts" questions, although it looks like you do have some precise mathematical questions to ask, and those should be better highlighted. $\endgroup$ Sep 23, 2011 at 3:31
  • $\begingroup$ Then again, The Google tells me that "dcpo" means "directed complete partial order". I don't understand "where the functor is taken as the ordering relation" of course. $\endgroup$ Sep 23, 2011 at 3:32
  • $\begingroup$ Probably you just mean that that the order relation is supposed to be realized by some prescribed functors? $\endgroup$ Sep 23, 2011 at 3:33
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    $\begingroup$ My thoughts are "this question is not very clear" and "a dcpo cannot be a topos". Is this a random question, or did it actually occur in real work? $\endgroup$ Sep 23, 2011 at 4:50

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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.

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