  Conventional approaches to lattice gauge theories do not properly consider
the topology of spacetime or of its fields. In this paper, we develop a
formulation which tries to remedy this defect. It starts from a cubical
decomposition of the supporting manifold (compactified spacetime or spatial
slice) interpreting it as a finite topological approximation in the sense of
Sorkin. This finite space is entirely described by the algebra of cochains with
the cup product. The methods of Connes and Lott are then used to develop gauge
theories on this algebra and to derive Wilson's actions for the gauge and Dirac
fields therefrom which can now be given geometrical meaning. We also describe
very natural candidates for the QCD theta term and Chern-Simons action
suggested by this algebraic formulation. Some of these formulations are simpler
than currently available alternatives. The paper treats both the functional
integral and Hamiltonian approaches.
