Symmetric spectrum

{{No footnotes|date=February 2024}}

In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group $\backslash Sigma\_n$ on $X\_n$ such that the composition of structure maps

:$S^1\; \backslash wedge\; \backslash dots\; \backslash wedge\; S^1\; \backslash wedge\; X\_n\; \backslash to\; S^1\; \backslash wedge\; \backslash dots\; \backslash wedge\; S^1\; \backslash wedge\; X\_\{n+1\}\; \backslash to\; \backslash dots\; \backslash to\; S^1\; \backslash wedge\; X\_\{n+p-1\}\; \backslash to\; X\_\{n+p\}$

is equivariant with respect to $\backslash Sigma\_p\; \backslash times\; \backslash Sigma\_n$. 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 $\backslash mathcal\{S\}p^\backslash Sigma$ 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 $\backslash mathcal\{S\}p^\backslash Sigma$; 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.