Symmetric spectrum

Template:No footnotes In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group ${\mathrm{\Sigma }}_n$ 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 ${\mathrm{\Sigma }}_p\times {\mathrm{\Sigma }}_n$. 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 $𝒮p^{\mathrm{\Sigma }}$ 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 $𝒮p^{\mathrm{\Sigma }}$; 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.