5 - Modern Categories of Spectra

Summary

The aim of this chapter is to investigate symmetric monoidal products on our categories of spectra and the stable homotopy category. After motivating this monoidal product in terms of the smash product on spaces and the Spanier–Whitehead category, we show that symmetric spectra and orthogonal spectra are symmetric monoidal model categories. As a consequence, the stable homotopy category is a closed symmetric monoidal category, and this monoidal structure is compatible with the triangulated structure. Using this monoidal product, we can give a modern interpretation of Spanier–Whitehead duality and discuss model categories of ring spectra, modules over ring spectra and commutative ring spectra. We end the chapter with an overview of some of the fundamental properties of spectra and the stable homotopy category, demonstrating that they are central to the study of stable homotopy theory. First, we show that the positive stable model structure on symmetric spectra is initial amongst stable simplicial monoidal model categories. Second, we show that the homotopy category of any stable model category has an “action” of the stable homotopy category.