  We explore various aspects of 2-form topological gauge theories in (3+1)d.
These theories can be constructed as sigma models with target space the second
classifying space $B^2G$ of the symmetry group $G$, and they are classified by
cohomology classes of $B^2G$. Discrete topological gauge theories can typically
be embedded into continuous quantum field theories. In the 2-form case, the
continuous theory is shown to be a strict 2-group gauge theory. This embedding
is studied by carefully constructing the space of $q$-form connections using
the technology of Deligne-Beilinson cohomology. The same techniques can then be
used to study more general models built from Postnikov towers. For finite
symmetry groups, 2-form topological theories have a natural lattice
interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d
that is exactly solvable. This construction relies on the introduction of a
cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified
with the simplicial cocycles of $B^2G$ as provided by the so-called
$W$-construction of Eilenberg-MacLane spaces. We show algebraically and
geometrically how a 2-form 4-cocycle reduces to the associator and the braiding
isomorphisms of a premodular category of $G$-graded vector spaces. This is used
to show the correspondence between our 2-form gauge model and the Walker-Wang
model.
