Remarks on Morphisms of Spectral Geometries

Abstract: Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain Connes' spectral triples, in this very preliminary review, we first present a description of the relevant categories of geometrical spaces, namely compact Hausdorff smooth finite-dimensional orientable Riemannian manifolds (or more generally Hermitian bundles of Clifford modules over them); we give some tentative definitions of the relevant categories of algebraic structures, namely "propagators" and "spectral correspondences" of commutative Riemannian spectral triples; and we provide a construction of functors that associate a naive morphism of spectral triples to every smooth (totally geodesic) map. The full construction of spectrum functors (reconstruction theorem for morphisms) and a proof of duality between the previous "geometrical' and "algebraic" categories are postponed to subsequent works, but we provide here some hints in this direction. We also show how the previous categories of "propagators" of commutative C*-algebras embed in the mildly non-commutative environments of categories of suitable Hilbert C*-bimodules, factorizable over commutative C*-algebras, with composition given by internal tensor product.