Polygraphs: From Rewriting to Higher Categories

Dimitri Ara, Albert Burroni, Yves Guiraud, Philippe Malbos, François Métayer, Samuel Mimram

• Publisher: Cambridge University Press
• Year: 2025
• eBook ISBN: 9781009498975
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

This is the first book to revisit the theory of rewriting in the context of strict higher categories, through the unified approach provided by polygraphs, and put it in the context of homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. Illustrated with algorithmic computations on algebraic structures, the only prerequisite in this section is basic category theory. The theory is introduced step-by-step, with detailed proofs. The second half introduces and studies the general notion of n-polygraph, before addressing the homotopy theory of these polygraphs. It constructs the folk model structure on the category on strict higher categories and exhibits polygraphs as cofibrant objects. This allows the formulation of higher-dimensional generalizations of the coherence results developed in the first half. Graduate students and researchers in mathematics and computer science will find this work invaluable.

Table of contents

• Cover
• Series information
• Title page
• Imprints page
• Contents
• Preface
• Part I Fundamentals of Rewriting
• Part II Coherent Presentations
• Part III Diagram Rewriting
• Part IV Polygraphs
• Part V Homotopy Theory of Polygraphs
• Appendices
• References
• Index of Symbols
• Subject Index