  We study fibrations in the category of cubespaces/nilspaces. We show that a
fibration of finite degree $f \colon X\rightarrow Y$ between compact ergodic
gluing cubespaces (in particular nilspaces) factors as a (possibly countable)
tower of compact abelian Lie group principal fiber bundles over $Y$. If the
structure groups of $f$ are connected then the fibers are (uniformly)
isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition
we give conditions under which the fibers of $f$ are isomorphic as
subcubespaces. We introduce regionally proximal equivalence relations relative
to factor maps between minimal topological dynamical systems for an arbitrary
acting group. We prove that any factor map between minimal distal systems is a
fibration and conclude that if such a map is of finite degree then it factors
as a (possibly countable) tower of principal abelian Lie compact group
extensions, thus achieving a refinement of both the Furstenberg's and the
Bronstein-Ellis structure theorems in this setting.
