代数数据库
Algebraic Databases
AI总结：
本文提出代数数据库框架，用多类代数理论扩展集合值函子模型以支持具体数据、操作、约束与查询，并用 proarrow equipment 统一组织各组件。
摘要：
数十年来，数据库一直从范畴论角度得到研究。数据库模式（schema）旨在组织高层概念实体，通常被建模为一个范畴或草图（sketch）。数据本身常称为实例（instance），通常被建模为一个集合值函子，为每个概念实体分配一个示例集合。尽管这些范畴模型在数学上很优雅，但它们通常难以表示整数或字符串等具体数据。在本工作中，我们提出对集合值函子模型的一种扩展，利用多类代数理论（multisorted algebraic theories，又称 Lawvere theories），以有原则的方式纳入具体数据。这也使约束和查询能够利用数据上的操作，例如数字的乘法或比较，从而帮助弥合传统数据库与编程语言之间的差距。我们还展示了模型的所有组成部分——包括模式、实例、模式变更函子和查询——如何纳入一个称为 proarrow equipment（又称 framed bicategory，框架双范畴）的单一双范畴结构中。
英文摘要：
Databases have been studied category-theoretically for decades. The database schema -- whose purpose is to arrange high-level conceptual entities -- is generally modeled as a category or sketch. The data itself, often called an instance, is generally modeled as a set-valued functor, assigning to each conceptual entity a set of examples. While mathematically elegant, these categorical models have typically struggled with representing concrete data such as integers or strings. In the present work, we propose an extension of the set-valued functor model, making use of multisorted algebraic theories (a.k.a. Lawvere theories) to incorporate concrete data in a principled way. This also allows constraints and queries to make use of operations on data, such as multiplication or comparison of numbers, helping to bridge the gap between traditional databases and programming languages. We also show how all of the components of our model -- including schemas, instances, change-of-schema functors, and queries - fit into a single double categorical structure called a proarrow equipment (a.k.a. framed bicategory).
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