标题： 量子可观测量代数与抽象微分几何：交换代数局部化的图的到理论动力学 显示英文标题

标题： Quantum Observables Algebras and Abstract Differential Geometry: The Topos-Theoretic Dynamics of Diagrams of Commutative Algebraic Localizations

摘要： 我们构建了一个在带有Grothendieck拓扑的基范畴上量子可观测代数的层论表示，该表示由交换可观测代数的满态射族组成，在测量情况下扮演局部算术的角色。 这一构造使得可以将抽象微分几何（ADG）的方法，按照Mallios的方式，在到拓扑理论环境中进行适应，从而扩展量子领域中的“微分机制”。 在交换代数局部化图中的信息粘合过程生成了动力学，涉及从经典领域到量子领域的转变，该转变以函子性量子联络的上同调形式表述，随后通过该联络的相关曲率进行检测。 显示英文摘要

摘要： We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), a la Mallios, in a topos-theoretic environment, and hence, the extension of the "mechanism of differentials" in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.