Higher lattice gauge theory from representations of 2-groups and 3+1D topological phases

We construct a higher lattice gauge theory based on the representation of 2-groups described by a category of crossed modules on a lattice model described by path 2-groupoids. Using these lattice gauge representations, an exactly solvable Hamiltonian for topological phases in 3+1 dimensions is constructed. We show that the ground states of this model are topological observables.