  We consider exactly solvable models in (3+1)d whose ground states are
described by topological lattice gauge theories. Using simplicial arguments, we
emphasize how the consistency condition of the unitary map performing a local
change of triangulation is equivalent to the coherence relation of the
pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of
such 2-category, we obtain a cohomological model whose underlying 1-category is
a 2-group. Topological models from 2-groups together with their lattice
realization are then studied from a higher gauge theory point of view. Symmetry
protected topological phases protected by higher symmetry structures are
explicitly constructed, and the gauging procedure which yields the
corresponding topological gauge theories is discussed in detail. We finally
study the correspondence between symmetry protected topological phases and 't
Hooft anomalies in the context of these higher group symmetries.
