  In recent years, attempts to generalize lattice gauge theories to model
topological order have been carried out through the so called $2$-gauge
theories. These have opened the door to interesting new models and new
topological phases which are not described by previous schemes of
classification. In this paper we show that we can go beyond the $2$-gauge
construction when considering chain complexes of abelian groups. Based on
elements of homological algebra we are able to greatly simplify already known
constructions for abelian theories under a single all encompassing framework.
Furthermore, this formalism allows us to systematize the computation of the
corresponding topological degeneracies of the ground states and establishes a
connection between them and a known cohomology, which conveniently
characterizes them with a suitable set of quantum numbers.
