  We develop a discrete theory of vector bundles with connection that is
natural with respect to appropriate mappings of the base space. The main
objects are discrete vector bundle valued cochains. The central operators are a
discrete exterior covariant derivative and a combinatorial wedge product. We
demonstrate the key properties of these operators and show that they are
natural with respect to the mappings referred to above. We give a new
interpretation in terms of a double averaging of anti-symmetrized cup product
which serves as our discrete wedge product. We also formulate a well-behaved
definition of metric compatible discrete connections. We characterize when a
discrete vector bundle with connection is trivializable or has a trivial lower
rank subbundle. This machinery is used to define discrete curvature as linear
maps and we show that our formulation satisfies a discrete Bianchi identity.
