Algebraic Databases

TL;DR

This paper extends categorical database models by integrating algebraic theories to handle concrete data like numbers and strings, enabling more expressive constraints and queries within a unified mathematical framework.

Contribution

It introduces a novel extension of categorical database models using algebraic theories to incorporate concrete data and operations, unifying schemas, instances, and queries in a double categorical structure.

Findings

Enhanced modeling of concrete data within categorical databases

Incorporation of data operations like multiplication and comparison

Unified framework for schemas, instances, and queries

Abstract

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).