  In a previous paper [21] the author studied the homotopy lifting property in
the category dTop of directed spaces in the sense of M. Grandis [12], [13],
[14]. The present paper, which is a continuation of aforementioned article,
introduces and studies the directed weak homotopy property (dWCHP) and directed
weak fibrations, extending to the category dTop the well known Dold's (or weak)
- fibrations [6]. The dWCHP is characterized in several ways. Then the notion
of a directed fibre homotopy equivalence (dFHE) between directed weak
fibrations is studied. Some examples and counterexamples are given.
