Quantization of branched coverings

We identify branched coverings (continuous open surjections p : Y → X of Hausdorff spaces with uniformly boundedly many preimages) with Hilbert C *-modules C ( Y ) over C ( X ) and with faithful unital positive conditional expectations E : C ( Y ) → C ( X ) that are topologically of finite index. The case of nonbranched coverings corresponds to finitely generated projective modules and expectations that are (algebraically) of finite index. This enables us to define noncommutative analogs of (branched) coverings.