In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group \( \Sigma _{n} \) on \( X_{n} \) such that the composition of structure maps
\( S^{1}\wedge \dots \wedge S^{1}\wedge X_{n}\to S^{1}\wedge \dots \wedge S^{1}\wedge X_{{n+1}}\to \dots \to S^{1}\wedge X_{{n+p-1}}\to X_{{n+p}} \)
is equivariant with respect to \( \Sigma _{p}\times \Sigma _{n} \). A morphism between symmetric spectra is a morphism of spectra that is equivariant with respect to the actions of symmetric groups.
The technical advantage of the category \( {\mathcal {S}}p^{\Sigma } \) of symmetric spectra is that it has a closed symmetric monoidal structure (with respect to smash product). It is also a simplicial model category. A symmetric ring spectrum is a monoid in \( {\mathcal {S}}p^{\Sigma } \); if the monoid is commutative, it's a commutative ring spectrum. The possibility of this definition of "ring spectrum" was one of motivations behind the category.
A similar technical goal is also achieved by May's theory of S-modules, a competing theory.
References
Introduction to symmetric spectra I
M. Hovey, B. Shipley, and J. Smith, “Symmetric spectra”, Journal of the AMS 13 (1999), no. 1, 149 – 208.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
