One more variation on the ``internal lfp'' theme. Continuing Paul Taylor's
The idea is that an object X is finitely presentable if homming from it preserves filtered colimits. (It is also interesting to investigate directed unions, and filtered colimits of surjections.)
But which homming functor do we mean?
, one might also ask - which filteredness do we mean? In other words, in the condition `all finite diagrams can be coned', there might be several inequivalent finiteness notions to consider. For example, several such notions appeared in early works by Kock, Lecouturier and Mikkelsen, by Johnstone and Linton; Japie Vermeulen was investigating embeddability into a K-finite object; and, as far as I know, Richard Squire has discovered a whole infinite sequence of (intuitionistically, but not classically) inequivalent finiteness notions. I would appreciate references to any other investigations of finite presentability in this context. Mamuka Jibladze