Strongly fibered objects and spaces

Given a model category C enriched over groupoids, we study and characterize the class of strongly fibered objects F(K) in terms of homotopy limits, with respect to a suitable subcategory K of C. We also show that the strong shape category for the pair (C,K) turns out to be isomorphic to the usual shape category for the pair (C,F(K)). This applies, in particular, to topological shape and strong shape.