  Motivated by topological quantum field theory, we investigate the geometric
aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and
their 2-characters. We show how the basic ideas of geometric quantization are
`categorified' in this context: just as representations of groups correspond to
equivariant line bundles, 2-representations of groups correspond to equivariant
gerbes. We also show how the 2-character of a 2-representation can be made
functorial with respect to morphisms of 2-representations. Under the geometric
correspondence, the 2-character of a 2-representation corresponds to the
geometric character of its associated equivariant gerbe. This enables us to
show that the complexified 2-character is a unitarily fully faithful functor
from the complexified homotopy category of unitary 2-representations to the
category of unitary equivariant vector bundles over the group.
