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Applied Category Theory in the Wolfram Language using Categorica I: Diagrams, Functors and Fibrations (2403.16269v1)

Published 24 Mar 2024 in math.CT and cs.SC

Abstract: This article serves as a preliminary introduction to the design of a new, open-source applied and computational category theory framework, named Categorica, built on top of the Wolfram Language. Categorica allows one to configure and manipulate abstract quivers, categories, groupoids, diagrams, functors and natural transformations, and to perform a vast array of automated abstract algebraic computations using (arbitrary combinations of) the above structures; to manipulate and abstractly reason about arbitrary universal properties, including products, coproducts, pullbacks, pushouts, limits and colimits; and to manipulate, visualize and compute with strict (symmetric) monoidal categories, including full support for automated string diagram rewriting and diagrammatic theorem-proving. In so doing, Categorica combines the capabilities of an abstract computer algebra framework (thus allowing one to compute directly with epimorphisms, monomorphisms, retractions, sections, spans, cospans, fibrations, etc.) with those of a powerful automated theorem-proving system (thus allowing one to convert universal properties and other abstract constructions into (higher-order) equational logic statements that can be reasoned about and proved using standard automated theorem-proving methods, as well as to prove category-theoretic statements directly using purely diagrammatic methods). In this first of two articles introducing the design of the framework, we shall focus principally upon its handling of quivers, categories, diagrams, groupoids, functors and natural transformations, including demonstrations of both its algebraic manipulation and theorem-proving capabilities in each case.

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Summary

  • The paper introduces Categorica as an innovative framework that integrates abstract category theory with computational algebra in the Wolfram Language.
  • It employs advanced graph and hypergraph rewriting techniques to seamlessly translate between diagrammatic and symbolic representations.
  • The methodology supports practical applications in automated theorem proving, quantum mechanics, and network theory, marking a significant step in computational category theory.

An Overview of "Applied Category Theory in the Wolfram Language using Categorica I: Diagrams, Functors and Fibrations"

The paper "Applied Category Theory in the Wolfram Language using Categorica I: Diagrams, Functors and Fibrations" by Jonathan Gorard provides a systematic exploration of the implementation of applied category theory within the Wolfram Language through a framework called Categorica. It focuses on core category-theoretic constructs including quivers, categories, functors, and natural transformations, as well as demonstrating the framework's capabilities in computational algebraic manipulation and theorem proving.

Core Concepts and Framework Architecture

Categorica is designed to integrate deeply within the Wolfram Language, providing a comprehensive toolset for manipulating abstract mathematical structures such as quivers (directed multigraphs) and various kinds of categories, including strict monoidal categories. It enables users to perform complex abstract algebraic computations, reasoning about universal properties and visualizing structures via string diagram rewriting.

Categorica amalgamates capabilities akin to computer algebra systems with automated theorem proving techniques. It can simulate algebraic operations involving epimorphisms, monomorphisms, sections, fibrations, and more. The framework also facilitates translating abstract constructions into equational logic problems, solvable with standard automated theorem-proving approaches.

Algebraic and Diagrammatic Reasoning

Categorica extensively employs state-of-the-art graph and hypergraph rewriting techniques to transform between diagrammatic and symbolic representations of category-theoretic entities. This approach allows for seamless computational handling and theorem-proving using both graphical and algebraic methods. Gorard illustrates Categorica's functionality by focusing on representations of quivers, categories, functors, and natural transformations.

For instance, the tool can address categorical dualities such as monomorphisms versus epimorphisms, retractions versus sections, and constant versus coconstant morphisms. These functionalities are not only relevant for abstract theorization but are essential for practical applications, particularly in fields like quantum mechanics, network theory, and theoretical computer science.

Practical and Theoretical Implications

Practically, Categorica provides researchers in applied category theory and related disciplines a robust computational toolset integrated within the widely used Mathematica environment. It enhances capabilities for analyzing and processing categorical data structures, potentially leading to advancements in both theoretical and practical domains.

Theoretically, Categorica represents a significant advancement in the computational implementation of category theory. It opens avenues for refining categorical frameworks and improving algorithmic efficiencies, echoes the larger ambitions of category theory to unify mathematical paradigms under relational and process-theoretical perspectives.

Future Directions

Looking forward, Gorard indicates further development trajectories for Categorica, including support for universal constructions like limits and colimits, enhancing capabilities for handling monoidal categories and applications to string diagrammatic theorem-proving. Additionally, the prospect of extending the framework's functionalities to embrace higher categories and adjoint functor relationships suggests ongoing enhancement and expansion of its utility and scope.

Overall, this work exemplifies a forward-thinking approach to integrating category theory with computational tools, poised to influence advancements in mathematical research, theoretical computer science, and beyond.

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Authors (1)

  1. Jonathan Gorard

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