4 - Stable homotopy theory via ∞-categories

Summary

The task before us is to investigate stable homotopy theory — and stable homotopy theories more generally — through the lens of ∞-category theory. Of necessity, this chapter is somewhat ahistorical; we refer the reader to the historical discussions out- lined in Chapter 3 for background on the development of modern categories of spectra. However, the reader who is familiar with that story will have a keen appreciation for the foundational problems that become much cleaner in this framework.

Let us assume familiarity with elementary ∞-category theory as presented by Jacob Lurie in [169] — most particularly, the theory of limits, colimits, adjunctions, and presentability. In particular, Chapter 5 of [169] will be frequently cited, but this is the upper limit: nothing of the later chapters or of any more advanced text will be needed here. We have tried to be systematic in our citations.

Our exposition is largely a gentle introduction to some of the material in [168] and subsequent papers, and of course much of our understanding of spectra was informed by this remarkable and beautiful text. We hope that this presentation will appeal to mathematicians both within and without homotopy theory.

I offer my sincere thanks to Andrew Blumberg for his enormous assistance in making my writing palatable.