In a previous paper the authors
defined the approximate homotopy lifting property and studied its implications. This
property is a generalization of the homotopy lifting property of classical fiber space
theory. Here a necessary and sufficient condition on point-inverses for a map to have
the approximate homotopy lifting property for n-cells is given; and the approximate
homotopy lifting property for n-cells is shown to imply the approximate homotopy
lifting property for all spaces. A corollary is that, in a fairly general context,
any two point-inverses of a Serre (weak) fibration have the same shape. By
combining these results with results of L. Husch, some conditions are obtained
under which a map between manifolds can be approximated by locally trivial
fibrations.
