  We define two dual tensor network representations of the (3+1)d toric code
ground state subspace. These two representations, which are obtained by
initially imposing either family of stabilizer constraints, are characterized
by different virtual symmetries generated by string-like and membrane-like
operators, respectively. We discuss the topological properties of the model
from the point of view of these virtual symmetries, emphasizing the differences
between both representations. In particular, we argue that, depending on the
representation, the phase diagram of boundary entanglement degrees of freedom
is naturally associated with that of a (2+1)d Hamiltonian displaying either a
global or a gauge $\mathbb Z_2$-symmetry.
