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In algebraic topology, a '''symmetric spectrum''' ''X'' is a [[spectrum (topology)|spectrum]] of pointed [[simplicial set]]s that comes with an action of the [[symmetric group]] <math>\Sigma_n</math> on <math>X_n</math> such that the composition of structure maps
:<math>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}</math>
is equivariant with respect to <math>\Sigma_p \times \Sigma_n</math>. 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 <math>\mathcal{S}p^\Sigma</math> of symmetric spectra is that it has a closed [[symmetric monoidal category|symmetric]] [[Monoidal category|monoidal structure]] (with respect to [[smash product]]). It is also a [[simplicial model category]]. A '''symmetric ring spectrum''' is a monoid in <math>\mathcal{S}p^\Sigma</math>; 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 ==
{{reflist}}
*[http://www.math.uchicago.edu/~mitya/langlands/spectra/iss1.pdf Introduction to symmetric spectra I]
*M. Hovey, B. Shipley, and J. Smith, “Symmetric spectra”, Journal of the AMS 13 (1999), no. 1, 149 – 208.

[[Category:Algebraic topology]]
[[Category:Simplicial sets]]
[[Category:Symmetry]]

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