Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{ C}$ to $\mathsf{ Set}$. For instance, graphs are functors to $\mathsf{ Set}$ from the category with two parallel arrows. Such functors are known informally as $\mathsf{ C}$-sets. In this paper, we describe and implement an extension of $\mathsf{ C}$-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 $\mathsf{ C}$-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.
Categories: math.CT cs.DB cs.LO