This book provides an accessible introduction to stable homotopy theory using the language of ∞-categories. It develops the technical foundations of the subject, including stability and the ∞-category of spectra, the theory of ∞-operads, and the basic theory of ring spectra.
The material begins with a model-independent introduction to ∞-categories. Part I develops stable homotopy theory and its principal examples, Part II constructs the higher algebra used throughout, and Part III collects the ∞-categorical background needed as a reference.
Contents
Introduction
An overview of the book's aims, intended audience, organization, and model-independent approach.
Foundations
Introduction to ∞-categories
Model-independent foundations for ∞-category theory.
Part I: Stable homotopy theory
Animae as spaces
Topological spaces and animae through Grothendieck's homotopy hypothesis.
Homology and cohomology
Generalized homology and cohomology theories, spectra, and Brown representability.
Spectra and stable ∞-categories
Spectra, stabilization, and stable ∞-categories.
The recognition principle
Recognition principles for loop animae and connective spectra.
Homological algebra
Derived categories, t-structures, derived functors, Ext, Tor, and ordinary homology.
Localization and completion
Bousfield localization, prime localization and completion, and arithmetic fracture.
Ring spectra and modules
Ring spectra, module categories, Eilenberg--MacLane algebras, and localization.
Complex K-theory
Classical and spectral complex K-theory.
Thom spectra and bordism theory
Thom spectra, bordism, orientations, and the Pontryagin--Thom theorem.
Duality phenomena
Dualizability, Spanier--Whitehead duality, Atiyah duality, and Poincaré duality.
Part II: Operads and higher algebra
Operads: the heuristics
From classical operads to the span-based theory of ∞-operads.
Span categories
Construction and basic properties of span categories.
Basics on ∞-operads
∞-operads, symmetric monoidal ∞-categories, algebras, and monoidal localizations.
Cartesian and cocartesian monoidal structures
Cartesian and cocartesian monoidal structures via unfurling.
Day convolution and the tensor product of spectra
Day convolution and its applications to rings, spectra, and infinite loop spaces.
The envelope of an ∞-operad
Operadic envelopes, free cocartesian completion, and comparison with Lurie's model.
Operadic Day convolution and stabilization
Operadic Day convolution, stabilization, and their universal properties.
Algebras and modules
Algebras, modules, relative tensor products, monadicity, and Morita theory.
Monoidal Dwyer–Kan localizations and derived categories
Monoidal Dwyer--Kan localization and symmetric monoidal derived categories.
Part III: Background on ∞-category theory
Adjunctions
Adjunctions, Kan extensions, cofinality, and localization.
Presentable ∞-categories
Presentable ∞-categories, compact generation, tensor products, and monoidal structures.
Cocartesian fibrations and (un)straightening
Cartesian and cocartesian fibrations, straightening, and descent.
Complete Segal animae
Complete Segal animae as a model for ∞-categories.
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