  For a smooth manifold $M$, possibly with boundary and corners, and a Lie
group $G$, we consider a suitable description of gauge fields in terms of
parallel transport, as groupoid homomorphisms from a certain path groupoid in
$M$ to $G$. Using a cotriangulation $\mathscr{C}$ of $M$, and collections of
finite-dimensional families of paths relative to $\mathscr{C}$, we define a
homotopical equivalence relation of parallel transport maps, leading to the
concept of an extended lattice gauge (ELG) field. A lattice gauge field, as
used in Lattice Gauge Theory, is part of the data contained in an ELG field,
but the latter contains further local topological information sufficient to
reconstruct a principal $G$-bundle on $M$ up to equivalence. The space of ELG
fields of a given pair $(M,\mathscr{C})$ is a covering for the space of fields
in Lattice Gauge Theory, whose connected components parametrize equivalence
classes of principal $G$-bundles on $M$. We give a criterion to determine when
ELG fields over different cotriangulations define equivalent bundles.
