  In this paper, we investigate how it is possible to define a new class of
lattice gauge models based on a dualization procedure of a previous
generalization of the Kitaev Quantum Double Models. In the case of this
previous generalization that will be used as a basis, it was defined by adding
new qudits (which were denoted as matter fields in reference to some works) to
the lattice vertices with the intention of, for instance, interpreting its
models as Kitaev Quantum Double Models coupled with Potts ones. Now, with
regard to the generalization that we investigate here, which we want to define
as the dual of this previous one, these new qudits were added to the lattice
faces. And as the coupling between gauge and matter qudits of the previous
generalization was performed by a gauge group action, we show that the dual
behaviour of these two generalizations was achieved by coupling these same
qudits in the second one through a gauge group co-action homomorphism. One of
the most striking dual aspects of these two generalizations is that, in both,
part of the quasiparticles that were inherited from the Kitaev Quantum Double
Models become confined when these action and co-action are non-trivial. But the
big news here is that, in addition to the group homomorphism (that defines this
gauge group co-action) allows us to classify all the different models of this
second generalization, this same group homomorphism also suggests that all
these models can be interpreted as two-dimensional restrictions of the $ 2
$-lattice gauge theories.
