In a paper in Fund Math 130.2 (1988): 125-136. http://eudml.org/doc/211719 Dyer and Eilenberg give an account of the local-global theorem for fibrations by proving a "Schedule Theorem" that, given a numerable cover $\mathcal U $ of a space $X$, "it is possible to continuously decompose each path in $X$ into subpaths such that each subpath lies in a prescribed set of the cover". From this they easily deduce the well known globalisation theorem, first stated by Hurewicz and later developed by Dold and others, which is a foundation for the theory of fibrations.
Question: Is it possible to extend this theorem to higher dimensions replacing paths by cubes $I^n$ and by constructing a corresponding space of schedules of cubes? In particular for $n=2$?
This is phrased for cubes rather than simplices since cubes have a clear notion of multiple composition.
