We build upon our recently introduced concept of an update structure to show that it is a generalisation of very-well-behaved lenses, that is, there is a bijection between a strict subset of update structures and vwb lenses in cartesian categories. We show that update structures are also sufficiently general to capture quantum observables, pinpointing the additional assumptions required to make the two coincide. In doing so, we shift the focus from special commutative dagger-Frobenius algebras to interacting (co)magma (co)module pairs, showing that the algebraic properties of the (co)multiplication arise from the module-comodule interaction, rather than direct assumptions about the magma-comagma pair. We then begin to investigate the zoo of possible update structures, introducing the notions of classical security-flagged databases, and databases of quantum systems. This work is of foundational interest as update structures place previously distinct areas of research in a general class of operationally motivated structures, we expect the taming of this class to illuminate novel relationships between separately studied topics in computer science, physics and mathematics.
翻译:我们以我们最近引入的更新结构概念为基础,表明它是对非常友好的透镜的概括化,也就是说,在卡通氏类中,一个严格的更新结构子集与Vwb透镜层之间有一个两侧。我们显示,更新结构也足够笼统,足以捕捉量可观测数据,确定了使两者同时出现所需的额外假设。我们这样做时,将重点从特殊的交流匕首-Frobenius代数转向互动(co)magma (co)module 配对),表明模块-组合相互作用产生的(co)倍增的代数特性,而不是对磁群-comagma双组的直接假设。我们随后开始调查可能更新结构的动物园区,引入了传统安全滞后数据库和量子系统数据库的概念。这项工作具有基本兴趣,因为更新结构以前在一般业务动机结构中独特的研究领域,我们期待这一类的调调,以阐明计算机科学、物理学和数学中分别研究的专题之间的新关系。