Cartesian double theories: A double-categorical framework for categorical doctrines

The categorified theories known as "doctrines" specify a category equipped with extra structure, analogous to how ordinary theories specify a set with extra structure. We introduce a new framework for doctrines based on double category theory. A cartesian double theory is defined to be a small double category with finite products and a model of a cartesian double theory to be a finite product-preserving lax functor out of it. Many familiar categorical structures are models of cartesian double theories, including categories, presheaves, monoidal categories, braided and symmetric monoidal categories, 2-groups, multicategories, and cartesian and cocartesian categories. We show that every cartesian double theory has a unital virtual double category of models, with lax maps between models given by cartesian lax natural transformations, bimodules between models given by cartesian modules, and multicells given by multimodulations. In many cases, the virtual double category of models is representable, hence is a genuine double category. Moreover, when restricted to pseudo maps, every cartesian double theory has a virtual equipment of models, hence an equipment of models in the representable case. Compared with 2-monads, double theories have the advantage of being straightforwardly presentable by generators and relations, as we illustrate through a large number of examples.Comment: 98 pages; minor edits and additions in response to feedbac

Categorical Data Structures for Technical Computing

Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{C}$ 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.Comment: 26 pages, 7 figure

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure. Building on recent work in applied category theory, we show how deterministic dynamical systems, discrete and continuous, can be composed in a hierarchical style. In mathematical terms, we reformulate some existing operads of wiring diagrams and introduce new ones, using the general formalism of C-sets (copresheaves). We then establish dynamical systems as algebras of these operads. In a computational vein, we show that Euler's method is functorial for undirected systems, extending a previous result for directed systems. All of the ideas in this paper are implemented as practical software using Catlab and the AlgebraicJulia ecosystem, written in the Julia programming language for scientific computing.Comment: In Proceedings ACT 2021, arXiv:2211.0110

Computational category-theoretic rewriting

We demonstrate how category theory provides specifications that can efficiently be implemented via imperative algorithms and apply this to the field of graph rewriting. By examples, we show how this paradigm of software development makes it easy to quickly write correct and performant code. We provide a modern implementation of graph rewriting techniques at the level of abstraction of finitely-presented C-sets and clarify the connections between C-sets and the typed graphs supported in existing rewriting software. We emphasize that our open-source library is extensible: by taking new categorical constructions (such as slice categories, structured cospans, and distributed graphs) and relating their limits and colimits to those of their underlying categories, users inherit efficient algorithms for pushout complements and (final) pullback complements. This allows one to perform double-, single-, and sesqui-pushout rewriting over a broad class of data structures