Let be separable -algebras. A morphism is semiprojective if for any separable -algebra , any increasing sequence of ideals with and any morphism , there exist and an “-relative lift” in the sense that the composition equals the composition , where is the epimorphism induced by the inclusion of ideals.
A separable -algebra is semiprojective if the identity is a semiprojective morphism. In particular, every projective separable -algebra is semiprojective. They are viewed as a generalization of (continuous function algebras) of ANR?s for metric spaces.
This notion is used in the strong shape theory for separable -algebras.