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My question is basically stated in the title. Does somebody know an explicit description of the filtered colimits in the category of complete semilattices?

I am happy to provide background explaining why I am interested in the question, but I do not think it is necessary to understand what I would like to know.

Can somebody help me? I am very sure this has been worked out somewhere (however, an online-search has not led me to an answer).

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    $\begingroup$ Maybe I'm misunderstanding the definition of you category, but is this the same as algebras over the power set monad on Set? $\endgroup$
    – Dylan Wilson
    Commented Aug 6, 2013 at 15:17
  • $\begingroup$ (because the category of algebras over any monad on Set is cocomplete, and there's a concrete description of colimits) $\endgroup$
    – Dylan Wilson
    Commented Aug 6, 2013 at 15:18
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    $\begingroup$ Sebastian, what are the maps in your category? Obviously they should be order-preserving, but do you want them to preserve joins and/or meets? If it's just join-preservation, then as Dylan suggests, you're looking at the category of algebras for the powerset monad on Set. This still doesn't give you an easy answer, though. This monad isn't finitary, i.e. doesn't preserve filtered colimits, which means that filtered colimits in your category aren't calculated as in Set. They do exist, but they're not quite so easy to describe. $\endgroup$
    – Tom Leinster
    Commented Aug 8, 2013 at 12:42
  • $\begingroup$ Yes, Tom, exactly. Dylan is also right. I was aware of exactly that. That is, I know (in principle) how to construct the filtered colimits. But as you say, Tom, I also found the task to be "not quite so easy". I thought that maybe somebody knows the answer by heart, saving me some time. Since I do of course not expect anybody to do the calculation for me, I will probably now try to do it myself. $\endgroup$
    – Niemi
    Commented Aug 8, 2013 at 16:25

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