Polygraphs: From Rewriting to Higher Categories
Polygraphs: From Rewriting to Higher Categories
Albert Burroni, Université Paris Cité
Yves Guiraud, Université Paris Cité
Philippe Malbos, Université Claude Bernard Lyon 1
François Métayer, Université Paris Cité
Samuel Mimram, École Polytechnique, Paris
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.
- The first book about polygraphs, providing detailed proofs and presentations
- Introduces the theory step-by-step using low-dimensional illustrations
- Provides a global overview of the topic, bringing together category theory, rewriting and homological algebra
Product details
April 2025Paperback
9781009498982
666 pages
228 × 152 mm
Not yet published - available from March 2025
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.
