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

我们为非交换空间提出了一个点的函子概念，以双模类别为价值，并赋予了一个由厄密结构和高度函数的概念决定的动作泛函，模拟点的经典函子作为物理西格玛模型的解释. 我们讨论了这种高度函数的不同选择，基于不同的体积和轨迹概念，包括基于 Hattori-Stallings 等级的一种。我们表明，高度函数决定了与我们的点函子相关的可观测代数的动态时间演化。我们特别关注非对易算术曲线的情况，其中相关代数是矩阵代数在数域上的除法代数的总和，并且我们讨论了更高维的非对易算术空间的更一般概念，