This book, "Simplicial Homotopy Theory" by Paul G. Goerss and John F. Jardine, is a comprehensive treatment of simplicial homotopy theory, a fundamental area in algebraic topology. The authors provide a modern account of the basic theory, filling a gap in the literature that has existed for over twenty-five years. The book is structured into ten chapters, each focusing on different aspects of simplicial homotopy theory. - **Chapter I** introduces the fundamental concepts of simplicial sets, including definitions, examples, and the simplicial set category. - **Chapter II** delves into model categories, providing a detailed treatment of the foundational aspects of abstract homotopy theory. - **Chapter III** covers classical results and constructions, such as the fundamental groupoid, simplicial abelian groups, the Hurewicz map, the Ex∞ functor, and the Kan suspension. - **Chapter IV** discusses bisimplicial sets and bisimplicial abelian groups, including the generalized Eilenberg-Zilber theorem and various closed model structures. - **Chapter V** explores simplicial groups, skeleta, and the equivalence between simplicial groups and simplicial sets with one vertex. - **Chapter VI** focuses on towers of fibrations, nilpotent spaces, and the homotopy spectral sequence. - **Chapter VII** treats Reedy model categories and their applications to cosimplicial spaces. - **Chapter VIII** examines cosimplicial spaces, homotopy inverse limits, completions, and obstruction theory. - **Chapter IX** discusses simplicial functors and homotopy coherence, including the Dwyer-Kan theorem and realization theorems. - **Chapter X** covers localization, including Bousfield localization and a model for the stable homotopy category. The book is intended for graduate students and researchers interested in applying simplicial techniques in various fields of mathematics, such as homological algebra, algebraic geometry, number theory, and algebraic K-theory. It emphasizes both theoretical foundations and practical applications, making it a valuable resource for both beginners and experts in the field.This book, "Simplicial Homotopy Theory" by Paul G. Goerss and John F. Jardine, is a comprehensive treatment of simplicial homotopy theory, a fundamental area in algebraic topology. The authors provide a modern account of the basic theory, filling a gap in the literature that has existed for over twenty-five years. The book is structured into ten chapters, each focusing on different aspects of simplicial homotopy theory. - **Chapter I** introduces the fundamental concepts of simplicial sets, including definitions, examples, and the simplicial set category. - **Chapter II** delves into model categories, providing a detailed treatment of the foundational aspects of abstract homotopy theory. - **Chapter III** covers classical results and constructions, such as the fundamental groupoid, simplicial abelian groups, the Hurewicz map, the Ex∞ functor, and the Kan suspension. - **Chapter IV** discusses bisimplicial sets and bisimplicial abelian groups, including the generalized Eilenberg-Zilber theorem and various closed model structures. - **Chapter V** explores simplicial groups, skeleta, and the equivalence between simplicial groups and simplicial sets with one vertex. - **Chapter VI** focuses on towers of fibrations, nilpotent spaces, and the homotopy spectral sequence. - **Chapter VII** treats Reedy model categories and their applications to cosimplicial spaces. - **Chapter VIII** examines cosimplicial spaces, homotopy inverse limits, completions, and obstruction theory. - **Chapter IX** discusses simplicial functors and homotopy coherence, including the Dwyer-Kan theorem and realization theorems. - **Chapter X** covers localization, including Bousfield localization and a model for the stable homotopy category. The book is intended for graduate students and researchers interested in applying simplicial techniques in various fields of mathematics, such as homological algebra, algebraic geometry, number theory, and algebraic K-theory. It emphasizes both theoretical foundations and practical applications, making it a valuable resource for both beginners and experts in the field.