  Just as 3d state sum models, including 3d quantum gravity, can be built using
categories of group representations, "2-categories of 2-group representations"
may provide interesting state sum models for 4d quantum topology, if not
quantum gravity. Here we focus on the "Euclidean 2-group", built from the
rotation group SO(4) and its action on the group of translations of 4d
Euclidean space. We explain its infinite-dimensional unitary representations,
and construct a model based on the resulting representation 2-category. This
model, with clear geometric content and explicit "metric data" on triangulation
edges, shows up naturally in an attempt to write the amplitudes of ordinary
quantum field theory in a background independent way.
