Given a Lie groupoid G ⇉ ⇉ M , a principal G -bundle consists of a G -space P and a G -invariant surjective submersion π π : P → → N such that P × × N G → → P × × π π P , ( p , g ) ↦ ↦ ( p , p ⋅ ⋅ g ) is a diffeomorphism. Equivalent (but more involved) definitions can be given using G -valued cocycles or local trivialisations.