This article delves into the realm of abstract rewriting systems and their significance in understanding the fundamental concepts of category theory. The authors, Dimitri Ara, Albert Burroni, Yves Guiraud, Philippe Malbos, Francois Métayer, Samuel Mimram, and many others, collaborated to create a comprehensive guide that simplifies complex ideas by utilizing relatable analogies and metaphors.
Category Theory
Category theory is a mathematical discipline that examines the relationships between mathematical structures, such as groups, rings, and vector spaces. It provides a framework for organizing these structures into categories, which are collections of objects and arrows (or morphisms) between them. The authors explain how abstract rewriting systems play a crucial role in categorical logic by enabling the study of various algebraic structures through the lens of rewriting theory.
Abstract Rewriting Systems
An abstract rewriting system is a mathematical structure that consists of a set of objects, a set of arrows between them, and a set of rewrite rules that specify how these arrows can be transformed into new arrows. The authors illustrate how abstract rewriting systems are used to define categories by providing examples of categories based on different types of rewriting systems, such as 1-polygraphs (which are like diagrams with one input and one output) and 2-polygraphs (which are more complex diagrams with multiple inputs and outputs).
Generating Categories and Groupoids
The article delves into the process of generating categories and groupoids, which are essential in understanding how abstract rewriting systems operate. The authors explain how coherent confluence in one-dimensional polygraphs (1-polygraphs) enables the construction of presentable categories, while two-dimensional polygraphs (2-polygraphs) allow for the generation of groupoids.
Operations on Presentations
The authors discuss how operations on presentations, such as limits and colimits, can be used to construct new categories and groupoids. They illustrate these concepts through examples of presentations in 1-polygraphs and 2-polygraphs.
String Rewriting and 2-Polygraphs
The authors examine string rewriting systems, which are a type of abstract rewriting system that operates on strings instead of diagrams. They show how string rewriting systems can be used to study the properties of 2-polygraphs and their role in categorical logic.
Conclusion
In conclusion, this article provides a comprehensive overview of abstract rewriting systems and their significance in category theory. By utilizing relatable analogies and metaphors, the authors demystify complex concepts and offer insights into the intricate world of algebraic structures. The article serves as a valuable resource for researchers and students interested in category theory and its applications in mathematics and computer science.
