  We define and study the regularity of distance maps on geodesically complete
spaces with curvature bounded above. We prove that such a regular map is
locally a Hurewicz fibration. This regularity can be regarded as a dual concept
of Perelman's regularity in the geometry of Alexandrov spaces with curvature
bounded below. As a corollary we obtain a sphere theorem for geodesically
complete CAT(1) spaces.
