Find a category $K$ which is cocomplete and in which every object is a directed colimit of finitely presentable objects, although $K$ is not locally presentable.

My attempt was the category Ord, the category of ordinal numbers. But it does not work, in fact not any ordinal is colimit of finitely presentable ones. Does anyone have any better example?

  • $\begingroup$ I'm confused by your claim about Ord. Are you thinking of some other structure than the full subcategory of Pos, which has the arrow as a dense generator? $\endgroup$ – Kevin Carlson May 15 '17 at 17:07

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