Stratified path spaces and fibrations

The main objects of study are the homotopically stratified metric spaces in troduced by Quinn. Closed unions of strata are shown to be stratified forwa rd tame. Stratified fibrations between spaces with stratifications are intr oduced. Paths that lie in a single stratum, except possibly at their initia l points, form a space with a natural stratification, and the evaluation ma p from that space of paths is shown to be a stratified fibration. Applicati ons to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.