Simplicial homotopy theory

More about the book

Since the modern era of algebraic topology began, simplicial methods have been crucial for computation and theory. With Quillen's concept of closed model categories, particularly simplicial model categories, these methods have become essential for understanding non-abelian homological algebra and addressing various homotopy-theoretical issues, including those in algebraic K-theory. This book offers a contemporary exposition of these concepts, focusing on model category theoretical techniques. It covers the homotopy theory of simplicial sets and foundational topics such as simplicial groups, Postnikov towers, and bisimplicial sets. Advanced topics include homotopy limits and colimits, localization concerning maps and homology theories, cosimplicial spaces, and homotopy coherence. The text features numerous results and ideas familiar to experts but previously uncollected in literature. Aimed at second-year graduate students and beyond, it introduces essential tools of modern homotopy theory without requiring extensive background in topology. Reviews highlight its clarity, usefulness, and the gap it fills in existing literature, making it a valuable resource for students and researchers interested in simplicial techniques and (simplicial) closed model categories.