I am interested in abstracting the Scott topology from Domains to Categories. One can find a definition of a continuous functor which is just such an abstraction:
A functor F:C→D is continuous if it preserves all small (weighted) limits that exist in C, i.e. if for every small category diagram A:E→C in C there is an isomorphism F(limA)≃lim(F∘A).F(\lim A) \simeq \lim (F\circ A).
I found this on the n-Lab. What I am interested in is a notion of compact element which is defined for Domains as this:
if k is less than the sup of any directed subset D, then there is an element x in D such that k is less than x.
Any reference would be good. My intuitions are telling me that if a category is compact, in this sense, it should have a kind of finite presentation. This would be an abstraction from a dcpo of groups where the compact elements are finitely generated. I realize this might not make sense so any help would be great.