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. 
Paolo Bertozzini (1), Roberto Conti (2), Wicharn Lewkeeratiyutkul (3)   ((1) Thammasat University, Bangkok, Thailand, (2) Un...