  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 \w{\mapa(A,Y)} (up to $A$-equivalence). We describe a recognition
principle, modelled on the classical ones for loop spaces, but using the more
general notion of an \emph{\Ama[.]} It has an associated transfinite procedure
for recovering \w{\CWA Y} from \w[,]{\mapa(A,Y)} inspired by Dror-Farjoun's
construction of \ww{\CWA{}}-approximations.
