Spectral C*-categories and Fell bundles with path-lifting

Abstract

Following Crane's suggestion that categorification should be of fundamental importance in quantising gravity, we show that finite dimensional even $S^o$-real spectral triples over $\bbc$ are already nothing more than full C*-categories together with a self-adjoint section of their domain and range maps, while the latter are equivalent to unital saturated Fell bundles over pair groupoids equipped with a path-lifting operator given by a normaliser. Interpretations can be made in the direction of quantum Higgs gravity. These geometries are automatically quantum geometries and we reconstruct the classical limit, that is, general relativity on a Riemannian spin manifold.