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The notion of principal bundle over an affine base is
replaced in the quantum setting by Hopf-Galois extension of the algebra of functions on the base. This approach is not suitable when the base is not affine, because in this case, we need
an affine open cover to properly describe our geometrical object. The algebra of functions must be replaced by the structural sheaf and care must be exterted in gluing and localizing. Once the quantum principal bundle concept is established in this more general setting, we will proceed constructing and defining a first order differential calculus (FODC) on QPB. The example of SL(2) and P1 will be systematically used to illustrate the general concepts.

https://ucdavis.zoom.us/j/94506910050