INTRODUCTION

About twenty-five years ago May, Quinn and Ray introduced the concept of E∞-ring spectra [16, Chapter IV]. The definition was motiviated by the fact that there is no way to construct an internal smash product on the category of ordinary spectra and functions between them in such a way that the smash product would equip this category of spectra with a symmetric monoidal product. Of course there is the well-defined smash product on the homotopy category of spectra. An E∞-structure on a commutative “homotopy ring spectrum” R or on a module M over it essentially guarantees that the homotopy multiplications R ∧ R → R and R ∧ M → M satisfy “all relevant algebraic relations”. For example, E∞-structures allow to define the smash product of two E∞-module spectra over an E∞-ring spectrum which then again is an E∞-module spectrum over the E∞-ring spectrum which then again will be an E∞-module spectrum over the E∞-ring spectrum.

Recently however several people suceeded in defining a symmetric monoidal smash product on certain categories of spectra. Of course, for doing so one needs to put some extra structure on the spectra.