A Universal Property of the Groups Spin^c and Mp^c
It is well known that spinors on oriented Riemannian manifolds cannot be defined as sections of a vector bundle associated with the frame bundle. For this reason spin and spin^c structures are often introduced. In this paper we prove that spin^c structures have a universal property among all other structures that enable the construction of spinor bundles. We proceed to prove a similar result for metaplectic^c structures on symplectic manifolds.