A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.

中文翻译：

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量子集

量子集被定义为简单的一组非零有限维希尔伯特空间。量子集与二元关系（本质上是Weaver的量子关系）一起构成了匕首紧凑类别。量子集之间的函数是某些二进制关系，可以用这种匕首紧凑结构来表征，并且所得的量子集和函数的类别以非交换数学的方式概括了普通集和函数的类别。特别地，该类别是冯·诺依曼代数的子类别的对偶。全面介绍了量子集的基本属性，同时牢记了非交换字典，并着眼于方便应用。作为激励性的例子，在此框架内得出了量子图着色的概念，