Symmetric spectra

Steve Lack – 4 July 2001

The aim of this talk was to describe the paper M. Hovey, B. Shipley, & J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13:149-208, 1999. I started by briefly reviewing spectra, as ``representing objects'' for generalized cohomology theories. More precisely, spectra are the objects of the stable homotopy category, first defined by Boardman, which contains spaces as a full subcategory; generalized cohomology theories can then be described by passing from spaces into the stable homotopy category, and then homming into some spectrum. Then I described the Quillen model structure on the category of spectra defined in A.K. Bousfield and E.M. Friedlander, Homotopy theory of Gamma-spaces, spectra, and bisimplicial sets, in Geometric Applications of Homotopy Theory II, Lecture Notes in Mathematics 658, Springer. The resulting homotopy category is the stable homotopy category; this provides a simpler description than that of Boardman. The stable homotopy has all sorts of nice categorical structure (Margolis has conjectured that it can be characterized by five simple axioms), in particular it is symmetric monoidal. The symmetric monoidal structure, however, is not induced by a symmetric monoidal structure on spectra, which makes calculation rather difficult. The idea of Hovey, Shipley, and Smith is to replace the category of spectra by a different category - the category of symmetric spectra - which is symmetric monoidal, and which has a model structure for which the homotopy category is still the stable homotopy category, and moreover the symmetric monoidal structure on the stable homotopy category is induced by that on the category of symmetric spectra. The symmetric monoidal category of symmetric spectra is easily described; the model structure requires more work. A symmetric sequence is defined to be a sequence Xn of pointed simplicial sets, indexed by the natural numbers, with for each n an action of the symmetric group Sn on Xn. The category of symmetric sequences is symmetric monoidal closed: the tensor product is defined using convolution in the sense of B.J. Day, On closed categories of functors, in Lecture Notes in Mathematics 1970, Springer. The sequence (S0,S1,S2,...) has a natural structure of commutative monoid in the symmetric monoidal closed category of symmetric sequences, and a symmetric spectrum is a symmetric sequence equipped with an action of this commutative monoid. More concretely, this amounts to a symmetric sequence equipped with suitably equivariant maps S1xXn-->Xn+1 for all n. The symmetric monoidal structure on the category of symmetric spectra is a formal consequence of the description of symmetric spectra in terms of actions of a commutative monoid. The weak equivalences for the relevant model strucure are called the stable equivalences. Although it is possible to define stable homotopy groups of symmetric spectra, they are not sufficient to determine the stable equivalences. Rather, the stable equivalences are the maps which are inverted by a certain class of cohomology theories.