  In the framework of locally covariant quantum field theory, a theory is
described as a functor from a category of spacetimes to a category of
*-algebras. It is proposed that the global gauge group of such a theory can be
identified as the group of automorphisms of the defining functor. Consequently,
multiplets of fields may be identified at the functorial level. It is shown
that locally covariant theories that obey standard assumptions in Minkowski
space, including energy compactness, have no proper endomorphisms (i.e., all
endomorphisms are automorphisms) and have a compact automorphism group.
Further, it is shown how the endomorphisms and automorphisms of a locally
covariant theory may, in principle, be classified in any single spacetime. As
an example, the endomorphisms and automorphisms of a system of finitely many
free scalar fields are completely classified.
