Using results of Escardó-Flagg  and Hofmann , we characterize continuous maps which are injective with respect to special classes of embeddings. These characterizations are based on a fibrewise way-below relation which generalizes the one used by Richter in his characterization of fibrewise exponentiability .
(Joint work with Francesca Cagliari and Sandra Mantovani.)
 M. Escardó, R. Flagg, Semantic domains, injective spaces and monads, Electr. Notes in Theor. Comp. Science 20, electronic paper 15 (1999).
 D. Hofmann, A four for the price of one duality principle for distributive topological spaces, preprint, arXiv:math.GN/1102.2605.
 G. Richter, Exponentiability for maps means fibrewise core-compactness, J. Pure Appl. Algebra 187 (2004), 295-303.