(this is a stub yet)
A map between metric compacta is called a shape fibration, provided it is induced by a level map of ANR-sequences satisfying the approximate homotopy lifting property with respect to any metric space and the lifting index, and lifting mesh do not depend on .
If, more generally, and are locally compact metric spaces. A proper map is a shape fibration if for any compact the restriction of to a map is a shape fibration between metric compacta.
If and are locally compact metric ANR’s then the notion of shape fibration coincides with a notion of an approximate fibration.
S. Mardešić, T. B. Rushing, Shape fibrations. I. General Topology Appl. 9 (1978), no. 3, 193–215.
S. Mardešić, J. Segal, (1982) Shape Theory, North Holland.
T. B. Rushing, Cell-like maps, approximate fibrations and shape fibrations