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.