Characterizing shape theories by Kan extensions

Every pair (C,K) of categories, where K is a proreflective subcategory of C, generates a shape theory. As a main result in this paper we give a characterization of such pairs, showing that these are exactly those having the property that every functor F :K→A has a Kan extension Ran F :C→A, which is preserved by all functors commuting with inverse limits.