Motivated by statistical practice, category theory terminology is used to introduce Borel data structures and study exchangeability in an abstract framework. A generalization of de Finetti's theorem is shown and natural transformations are used to present functional representation theorems (FRTs). Proofs of the latter are based on a classical result by D.N.Hoover providing a functional representation for exchangeable arrays indexed by finite tuples of integers, together with an universality result for Borel data structures. A special class of Borel data structures are array-type data structures, which are introduced using the novel concept of an indexing system. Studying natural transformations mapping into arrays gives explicit versions of FRTs, which in examples coincide with well-known Aldous-Hoover-Kallenberg-type FRTs for (jointly) exchangeable arrays. The abstract "index arithmetic" presented unifies and generalizes technical arguments commonly encountered in the literature on exchangeability theory. Finally, the category theory approach is used to outline how an abstract notion of seperate exchangeability can be derived, again motivated from statistical practice.

翻译：以统计实践为动力,使用类别理论术语,在抽象框架内引入波罗尔数据结构和研究可交换性。展示了德费内蒂理论的概括性,并使用自然变迁来展示功能性理论(FRTs),后者的证据以D.N.Hoover的经典结果为基础,为用数量有限的整数指数化的可交换阵列提供了功能性代表,同时为波雷尔数据结构提供了普遍性结果。一个特殊类别波雷尔数据结构是阵列型数据结构,采用新颖的指数化系统概念。对阵列的自然变迁图解提供了明确的FRTs版本,这些例子与众所周知的Aldous-Houver-Kallenberg型FRTs(共同)可交换阵列的典型一致。抽象的“指数算术”将关于可交换性理论的文献中常见的技术论点统一化和概括化。最后,使用类别理论方法概述了如何从统计实践中推导出关于可交换性的抽象概念。