A Remark on Gelfand Duality for Spectral Triples

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable metric category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a quotient of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.