Categorical Data Structures for Technical Computing

Many mathematical objects can be represented as functors from finitely-presented categories mathsfC to mathsfSet. For instance, graphs are functors to mathsfSet from the category with two parallel arrows. Such functors are known informally as mathsfC-sets. In this paper, we describe and implement an extension of mathsfC-sets having data attributes with fixed types, such as graphs with labeled vertices or real-valued edge weights. We call such structures "acsets," short for "attributed mathsfC-sets." Derived from previous work on algebraic databases, acsets are a joint generalization of graphs and data frames. They also encompass more elaborate graph-like objects such as wiring diagrams and Petri nets with rate constants. We develop the mathematical theory of acsets and then describe a generic implementation in the Julia programming language, which uses advanced language features to achieve performance comparable with specialized data structures.