# Axiomatic Stable Homotopy Theory

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*Mark Hovey; John H. Palmieri; Neil P. Strickland*

This book gives an axiomatic presentation of stable homotopy
theory. It starts with axioms defining a “stable homotopy category”; using these axioms, one can make various constructions—cellular
towers, Bousfield localization, and Brown representability, to name a
few. Much of the book is devoted to these constructions and to the
study of the global structure of stable homotopy categories.

Next, a number of examples of such categories are presented. Some of
these arise in topology (the ordinary stable homotopy category of
spectra, categories of equivariant spectra, and Bousfield
localizations of these), and others in algebra (coming from the
representation theory of groups or of Lie algebras, as well as the
derived category of a commutative ring). Hence one can apply many of
the tools of stable homotopy theory to these algebraic situations.

Features:

- Provides a reference for standard results and constructions in stable homotopy theory.
- Discusses applications of those results to algebraic settings, such as group theory and commutative algebra.
- Provides a unified treatment of several different situations in stable homotopy, including equivariant stable homotopy and localizations of the stable homotopy category.
- Provides a context for nilpotence and thick subcategory theorems, such as the nilpotence theorem of Devinatz-Hopkins-Smith and the thick subcategory theorem of Hopkins-Smith in stable homotopy theory, and the thick subcategory theorem of Benson-Carlson-Rickard in representation theory.

This book presents stable homotopy theory as a branch of mathematics in its own right with applications in other fields of mathematics. It is a first step toward making stable homotopy theory a tool useful in many disciplines of mathematics.

#### Table of Contents

# Table of Contents

## Axiomatic Stable Homotopy Theory

- Contents vii8 free
- 1. Introduction and definitions 112 free
- 2. Smallness, limits and constructibility 1223
- 3. Bousfield localization 2738
- 3.1. Localization and colocalization functors 2738
- 3.2. Existence of localization functors 3445
- 3.3. Smashing and finite localizations 3546
- 3.4. Geometric morphisms 3950
- 3.5. Properties of localized subcategories 4051
- 3.6. The Bousfield lattice 4455
- 3.7. Rings, fields and minimal Bousfield classes 4859
- 3.8. Bousfield classes of smashing localizations 5061

- 4. Brown representability 5364
- 5. Nilpotence and thick subcategories 6273
- 6. Noetherian stable homotopy categories 6677
- 7. Connective stable homotopy theory 7586
- 8. Semisimple stable homotopy theory 7788
- 9. Examples of stable homotopy categories 7990
- 10. Future directions 99110
- Appendix A. Background from category theory 102113
- References 109120
- Index 112123 free