Mathematics

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative  principal bundle as a quantization of the classical gauge groupoid of  a principal bundle. When the base algebra is in the centre of the  total space algebra, the gauge group of the noncommutative principal  bundle is isomorphic to the group of bisectionsof the bialgebroid. In particular we consider Galois objects  (non-trivial noncommutative bundles over a point in a sense) for which  the bialgebroid isa Hopf algebra. For these we give a crossed module structure for the  bisections and the automorphisms of the bialgebroid. Examples include  Galois objectsof group Hopf algebras and of Taft algebras.