Stable homotopy theory

This first part develops the foundations of stable homotopy theory in the language of \(\infty \)-categories. Its central object is the \(\infty \)-category \(\Sp \) of spectra: a setting in which every generalized (co)homology theory is representable, in which the stability axiom becomes a structural property rather than an axiom, and in which the constructions of homological algebra may be performed without first deriving them.

Throughout we build on Chapter 1, which precedes both mathematical parts and supplies the model-independent language of \(\infty \)-categories and of animae, the \(\infty \)-categories in which every morphism is invertible. We begin here with the topological preliminaries. Chapter 2 identifies animae with spaces up to weak homotopy equivalence, and transports the classical homotopy pushouts and pullbacks, suspensions and loop spaces into this setting. Chapter 3 treats (co)homology theories axiomatically and introduces spectra together with Brown’s representability theorem; the online version also develops cellular homology from the axioms and proves Brown representability.

Spectra are then studied in their own right. Chapter 4 develops fiber and cofiber sequences, the notion of a stable \(\infty \)-category, and the stabilization \(\Sp (C)\). Chapter 5 proves the recognition principle, identifying connective spectra with commutative groups in \(\An \), so that a connective spectrum becomes a coherent analogue of an abelian group. Chapter 6 develops homological algebra from this viewpoint: derived \(\infty \)-categories, t-structures and their hearts, Ext- and Tor-groups, and derived functors. Chapter 7 treats localization and completion of spectra at a prime.

The remaining chapters treat the principal examples. Chapter 8 develops ring spectra and their modules, taking the coherent algebra of Part II as a black box. Chapter 9 constructs complex K-theory, Chapter 10 the Thom spectra and their bordism theories, and Chapter 11 treats Spanier–Whitehead, Atiyah and Poincaré duality.