Categorical Homotopy Theory: A Graduate-Level Mathematics Textbook by Emily Riehl

Introduction

Categorical homotopy theory stands at the crossroads of two of the most profound areas of modern mathematics: category theory and homotopy theory. For graduate students and researchers seeking a rigorous yet accessible entry point, Categorical Homotopy Theory by Emily Riehl offers a masterful synthesis of these fields. Published by Cambridge University Press as part of the prestigious New Mathematical Monographs series, this hardcover volume is an indispensable resource for anyone aiming to understand the deep structural connections between algebraic topology and categorical foundations.

Book Overview

This book provides a comprehensive introduction to the categorical perspective on homotopy theory, starting from the basics and building toward advanced topics. Emily Riehl, a celebrated mathematician and expositor, guides readers through model categories, simplicial sets, and derived functors with exceptional clarity. The text is designed to be self-contained, making it suitable for both classroom use and independent study. It bridges the gap between classical homotopy theory and modern categorical machinery, equipping readers with the tools to explore contemporary research in algebraic topology, higher category theory, and related disciplines.

Key Highlights

• Authoritative Coverage: Written by Emily Riehl, a leading expert in category theory and homotopy theory, known for her clear and engaging teaching style.
• Self-Contained Approach: Assumes only basic knowledge of category theory and algebraic topology, gradually introducing advanced concepts.
• Modern Perspective: Integrates the latest developments in the field, including Quillen model categories and simplicial homotopy theory.
• Rigorous Yet Readable: Balances formal proofs with intuitive explanations, making complex ideas accessible.
• Comprehensive Exercises: Includes numerous problems at the end of each chapter to reinforce learning and encourage deeper exploration.

Inside the Book

The book is organized into twelve chapters, each building on the previous one. Early chapters cover the fundamentals of model categories, homotopy limits and colimits, and simplicial sets. Later chapters delve into derived functors, homotopy coherence, and the theory of ∞-categories. The appendix provides a quick refresher on essential category theory. Throughout, Riehl uses a wealth of examples and diagrams to illuminate abstract concepts, ensuring that readers can connect theory to practice.

Key Topics

• Model categories and their homotopy categories
• Simplicial sets and simplicial homotopy theory
• Homotopy limits and colimits
• Derived functors and derived categories
• Homotopy coherence and mapping spaces
• ∞-categories and their relation to model categories
• Kan extensions and their homotopical analogues
• Monoidal model categories and enriched homotopy theory

Reader Benefits

By studying this book, readers will develop a solid foundation in categorical homotopy theory, enabling them to read and contribute to current research. The logical progression from basic to advanced topics ensures a steady learning curve. The exercises are carefully crafted to build problem-solving skills and deepen conceptual understanding. Moreover, the book's emphasis on categorical thinking fosters a versatile mathematical mindset applicable beyond homotopy theory.

Learning Outcomes

Upon completing this book, readers will be able to:

• Define and work with model categories, including constructing homotopy categories.
• Understand and manipulate simplicial sets and their geometric realizations.
• Compute homotopy limits and colimits in various contexts.
• Apply derived functors to problems in algebra and topology.
• Navigate the basics of ∞-category theory and its applications.
• Read advanced research papers that rely on categorical homotopy theory.

Who Should Read

This book is ideal for graduate students in mathematics who have completed introductory courses in category theory and algebraic topology. Researchers in algebraic topology, representation theory, algebraic geometry, and mathematical physics will also find it invaluable. Instructors looking for a modern textbook for a course on homotopy theory or categorical methods will appreciate its structured presentation. Additionally, advanced undergraduates with a strong background in abstract mathematics can benefit from its clear exposition.

About the Author

Emily Riehl is a Professor of Mathematics at Johns Hopkins University and a prolific researcher in category theory, homotopy theory, and higher-dimensional algebra. She is the author of several acclaimed textbooks, including Category Theory in Context and Elements of ∞-Category Theory. Known for her lucid writing and dedication to mathematical pedagogy, Riehl has received numerous awards for her teaching and research, including the American Mathematical Society's Levi L. Conant Prize. Her work has significantly shaped the modern landscape of categorical mathematics.

About the Publisher

Cambridge University Press is one of the world's oldest and most respected academic publishers, with a history dating back to 1534. The New Mathematical Monographs series, of which this book is a part, showcases cutting-edge research and authoritative textbooks in pure and applied mathematics. Cambridge's rigorous editorial standards ensure that each volume meets the highest scholarly criteria, making it a trusted source for mathematicians worldwide.

Conclusion

Categorical Homotopy Theory by Emily Riehl is a definitive guide to a vibrant and essential area of mathematics. Whether you are a student embarking on advanced study or a researcher seeking to expand your toolkit, this book provides the clarity, depth, and inspiration you need. Order your hardcover copy from Bookshops.in today and add this masterpiece to your mathematical library.