Abstract: This talk discusses the development of a structure-preserving discretization of the exterior calculus of differential forms with values in a vector bundle over a combinatorial manifold equipped with a connection. Compared to their scalar-based counterparts which admit a well-established discretization via cochains, bundle-valued forms, (e.g., with values in the group of rotation matrices) present numerous difficulties when one tries to properly define a discrete counterpart to them and to the exterior covariant derivative operator acting on them. We show however that the use of specifically selected local frame fields allows for the construction of a discrete exterior covariant derivative of bundle-valued forms that not only satisfies the well-known Bianchi identities in this discrete realm, but also converges to its smooth equivalent under mesh refinement.