As a less trivial instance of the correspondence between transitive Lie groupoids and principal bundles, consider the fundamental groupoid Π Π 1 ( M ) of a (connected) smooth manifold M . This is naturally a topological groupoid, which is moreover transitive; one can see that Π Π 1 ( M ) is isomorphic to the gauge groupoid of the universal cover of M . Accordingly, Π Π 1 ( M ) inherits a smooth structure which makes it into a Lie groupoid.