Symmetric spectrum

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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.