Symmetric spectrum

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.