A bisection of a Lie groupoid G ⇉ ⇉ M is a smooth map b : M → → G such that s ∘ ∘ b = i d M and t ∘ ∘ b is a diffeomorphism of M . In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold B ⊆ ⊆ G such that s ∣ ∣ B : B → → M and t ∣ ∣ B : B → → M are diffeomorphisms; the relation between the two definitions is given by B = b ( M ) .