  Motivated by algebraic quantum field theory, we study presheaves of symmetric
tensor categories defined over the base of a space, intended as a spacetime.
Any section of a presheaf (that is, any "superselection sector", in the
applications that we have in mind) defines a holonomy representation whose
triviality is measured by Cheeger-Chern-Simons characteristic classes, and a
non-abelian unitary cocycle defining a Lie group gerbe. We show that, given an
embedding in a presheaf of full subcategories of the one of Hilbert spaces, the
section category of a presheaf is a Tannaka-type dual of a locally constant
group bundle (the "gauge group"), which may not exist and in general is not
unique. This leads to the notion of gerbe of C*-algebras, defined on the given
base.
