1 - Introduction

Summary

The modern era in homotopy theory began in the 1960s with the profound realization, first codified by Boardman in his construction of the stable category, that the category of spaces up to stable homotopy equivalence is equipped with a rich algebraic structure, formally similar to the derived category of a commutative ring R. For example, for pointed spaces the natural map from the categorical coproduct to the categorical product becomes more and more connected as the pieces themselves become more and more connected. In the limit, this map becomes a stable equivalence, just as finitely indexed direct sums and direct products coincide for R-modules.

From this perspective, the objects of the stable category are modules over an initial commutative ring object that replaces the integers: the sphere spectrum. However, technical difficulties immediately arose. Whereas the tensor product of R-modules is an easy and familiar construction, the analogous construction of a symmetric monoidal smash product on spectra seemed to involve a huge number of ad hoc choices [1]. As a consequence, the smash product was associative and commutative only up to homotopy. The lack of a good point-set symmetric monoidal product on spectra precluded making full use of the constructions from commutative algebra in this setting — even just defining good categories of modules over a commutative ring spectrum was difficult. In many ways, finding ways to rectify this and to make the guiding metaphor provided by “modules over the sphere spectrum” precise has shaped the last 60 years of homotopy theory.

This book arose from a desire by the editors to have a reference to give to their students who have taken a standard algebraic topology sequence and who want to learn about spectra and structured ring spectra. While there are many excellent texts which introduce students to the basic ideas of homotopy theory and to spectra, there has not been a place for students to engage directly with the ideas needed to connect with commutative ring spectra and work with these objects. This book strives to provide an introduction to this whole circle of ideas, describing the tools that homotopy theorists have developed to build, explore, and use symmetric monoidal categories of spectra that refine the stable homotopy category:

• 1. model category structures on symmetric monoidal categories of spectra,

• 2. stable ∞-categories, and

• 3. operads and operadic algebras.

These three concepts are closely intertwined, and they all engage deeply with a fundamental principle: if the choices for some construction or map are parameterized by a space, then recording that space as part of the data makes the construction more natural.