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 ${\displaystyle \Sigma _{n}}$ on ${\displaystyle X_{n}}$ such that the composition of structure maps

${\displaystyle S^{1}\wedge \dots \wedge S^{1}\wedge X_{n}\to S^{1}\wedge \dots \wedge S^{1}\wedge X_{n+1}\to \dots \to S^{1}\wedge X_{n+p-1}\to X_{n+p}}$

is equivariant with respect to ${\displaystyle \Sigma _{p}\times \Sigma _{n}}$. 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 ${\displaystyle {\mathcal {S}}p^{\Sigma }}$ 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 ${\displaystyle {\mathcal {S}}p^{\Sigma }}$; 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.