Polygraphs: From Rewriting to Higher Categories

by Dimitri Ara, Albert Burroni, Yves Guiraud, Philippe Malbos, François Métayer

Overview

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.

Product Details

ISBN-13:	9781009498982
Publisher:	Cambridge University Press
Publication date:	04/03/2025
Series:	London Mathematical Society Lecture Note Series , #495
Pages:	666
Product dimensions:	5.94(w) x 8.94(h) x 1.22(d)

About the Author

Dimitri Ara is Associate Professor at Aix-Marseille Université.

Albert Burroni is Associate Researcher at Université Paris Cité.

Yves Guiraud is Researcher at Université Paris Cité.

Philippe Malbos is Professor at Université Claude Bernard Lyon 1.

Francois Métayer is Associate Professor Emeritus at Université Paris Cité.

Samuel Mimram is Professor at the LIX laboratory of École Polytechnique.

Table of Contents

Part I. Fundamentals of Rewriting: 1. Abstract rewriting and one-dimensional polygraphs; 2. Two-dimensional polygraphs; 3. Operations on presentations; 4. String rewriting and 2-polygraphs; 5. Tietze transformations and completion; 6. Linear rewriting; Part II. Coherent Presentations: 7. Coherence by convergence; 8. Categories of finite derivation type; 9. Homological syzygies and confluence; Part III. Diagram Rewriting: 10. Three-dimensional polygraphs; 11. Termination of 3-polygraphs; 12. Coherent presentations of 2-categories; 13. Term rewriting systems; Part IV. Polygraphs: 14. Higher categories; 15. Polygraphs; 16. Properties of the category of 𝑛-polygraphs; 17. A catalogue of 𝑛-polygraphs; 18. Generalized polygraphs; Part V. Homotopy Theory of Polygraphs; 19. Polygraphic resolutions; 20. Towards the folk model structure; 21. The folk model structure; 22. Homology of 𝜔-categories; 23. Resolutions by (𝜔, 1)-polygraphs; Appendix A. A catalogue of 2-polygraphs; Appendix B. Examples of coherent presentations of monoids; Appendix C. A catalogue of 3-polygraphs; Appendix D. A syntactic description of free 𝑛-categories; Appendix E. Complexes and homology; Appendix F. Homology of categories; Appendix G. Locally presentable categories; Appendix H. Model categories; References; Index of notations; Index of terminology.Part I. Fundamentals of Rewriting: 1. Abstract rewriting and one-dimensional polygraphs; 2. Two-dimensional polygraphs; 3. Operations on presentations; 4. String rewriting and 2-polygraphs; 5. Tietze transformations and completion; 6. Linear rewriting; Part II. Coherent Presentations: 7. Coherence by convergence; 8. Categories of finite derivation type; 9. Homological syzygies and confluence; Part III. Diagram Rewriting: 10. Three-dimensional polygraphs; 11. Termination of 3-polygraphs; 12. Coherent presentations of 2-categories; 13. Term rewriting systems; Part IV. Polygraphs: 14. Higher categories; 15. Polygraphs; 16. Properties of the category of 𝑛-polygraphs; 17. A catalogue of 𝑛-polygraphs; 18. Generalized polygraphs; Part V. Homotopy Theory of Polygraphs: 19. Polygraphic resolutions; 20. Towards the folk model structure; 21. The folk model structure; 22. Homology of 𝜔-categories; 23. Resolutions by (𝜔, 1)-polygraphs; Appendix A. A catalogue of 2-polygraphs; Appendix B. Examples of coherent presentations of monoids; Appendix C. A catalogue of 3-polygraphs; Appendix D. A syntactic description of free 𝑛-categories; Appendix E. Complexes and homology; Appendix F. Homology of categories; Appendix G. Locally presentable categories; Appendix H. Model categories; References; Index of notations; Index of terminology.