  In the literature, abelian higher gauge symmetry models are shown to be valid
in all finite dimensions and exhibit the characteristic behavior of SPT phases
models. While the ground state degeneracy and the entanglement entropy were
thoroughly studied, the classification of the ground state space still remained
obscure. Anticipating the notation of the current paper, if $\left( C_{\bullet}
, \partial^C_{\bullet} \right)$ is the chain complex associated to the
geometrical content of these models, while $\left( G_{\bullet} ,
\partial^G_{\bullet} \right)$ is its symmetries counterpart, we show that the
ground state space is classified by a $H^0 (C,G) \times H_0 (C,G)$ group, where
$H^0(C,G)$ is the $0$-th cohomology group associated to its structure and $H_0
(C,G)$ is the corresponding $0$-th homology group.
