  We show that (3+1)-dimensional topological phases of matter generically
support loop excitations with topological degeneracy. The loops carry "Cheshire
charge": topological charge that is not the integral of a locally-defined
topological charge density. Cheshire charge has previously been discussed in
non-Abelian gauge theories, but we show that it is a generic feature of all
(3+1)-D topological phases (even those constructed from an Abelian gauge
group). Indeed, Cheshire charge is closely related to non-trivial three-loop
braiding. We use a dimensional reduction argument to compute the topological
degeneracy of loop excitations in the (3+1)-dimensional topological phases
associated with Dijkgraaf-Witten gauge theories. We explicitly construct
membrane operators associated with such excitations in soluble microscopic
lattice models in ${\mathbb{Z}_2}\times{\mathbb{Z}_2}$ Dijkgraaf-Witten phases
and generalize this construction to arbitrary membrane-net models. We explain
why these loop excitations are the objects in the braided fusion 2-category
$Z(\mathbf{2Vect}_G^{\omega})$, thereby supporting the hypothesis that
2-categories are the correct mathematical framework for (3+1)-dimensional
topological phases.
