Cotelescopes and approximate lifting properties in shape theory

In this note the cotelescope construction of fibrant extension (Cathey, 1979) of compacta is used to study approximate lifting properties of some shape morphisms (shape fibrations (Mardešić and Rushing, 1978) and shape absolutely soft maps (Zerkalov, 1987)). It is proved that for a shape fibration (respectively for a shape absolutely soft map), there exists a fibrant extension which is a Hurewicz fibration (respectively absolutely soft map (Shchepin, 1984)). Applying this result, it is proved that a map of compacta whose cotelescope fibrant extension is an absolutely soft map is itself a shape absolutely soft map. Moreover, it is proved that any hereditary shape equivalence of compacta is a shape absolutely soft map.