  We translate some fundamental properties satisfied by topological principal
bundles into the setting of Hopf-Galois extensions. The properties are:
functoriality, homotopy, and triviality. The main new concept of the paper is
the homotopy equivalence of Hopf-Galois extensions. We work in the particular
case where the subalgebra of coinvariants is central but without any
restriction on the Hopf algebras coacting on the quantum principal bundle. We
examine in detail the case when the Hopf algebra is one of Sweedler or Taft's
finite-dimensional Hopf algebras.
