Recognizing mapping spaces

Given a fixed object A in a suitable pointed simplicial model category C, we study the problem of recovering the target Y from the pointed mapping space   (up to A-equivalence). We describe a recognition principle, modeled on the classical ones for loop spaces, but using the more general notion of an A-mapping algebra. It has an associated transfinite procedure for recovering   from   , inspired by Dror-Farjoun’s construction of   -approximations.