  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.
