Functor of points and height functions for noncommutative Arakelov geometry

We propose a notion of functor points for noncommutative spaces, valued in categories bimodules, and endowed with an action functional determined by hermitian structures height functions, modeled on interpretation the classical as physical sigma model. discuss different choices such based notions volumes traces, including one Hattori-Stallings rank. show that function determines dynamical time evolution algebra observables associated to our points. focus particular case arithmetic curves, where relevant algebras are sums matrix over division number fields, we more general spaces higher dimensions, approach suggests Jones index function.