Hopf algebras often play role of the symmetries in noncommutative geometry. However, there is no notion of an open set, so it is hard to make sense of local trivializations. One of the possible replacements for open sets are the localization functors. For principal bundles one of the possible frameworks is the framework of Hopf-Galois extensions. In our work, using localizations one introduces finer class of the locally trivial Hopf-Galois extensions, and, more importantly, the global objects which generalize Hopf-Galois pictrue which in a cover by localizations reduce to it.
Classification of bundles then reduces to a Cech like cocycles with coefficients in a Hopf algebra. The examples of bundles on quantum flag varieties which I considered earlier without Cech picture fit into this context (with applications like quantum group coherent states). We can introduce also the notion of connections on such bundles, but this theory is not much investigated yet.