  We prove a theorem in 3-dimensional topological field theory: a
Reshetikhin-Turaev theory admits a nonzero boundary theory iff it is a
Turaev-Viro theory. The proof immediately implies a characterization of fusion
categories in terms of dualizability. The main theorem applies to physics,
where it implies an obstruction to a gapped 3-dimensional quantum system
admitting a gapped boundary theory. Appendices on bordism multicategories and
on internal duals may be of independent interest.; v2 extensive revision: added
theorem on dualizable 2-categories, material on natural transformations,
reworked theorems and several proofs, and more.
