In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group Σ n {\displaystyle \Sigma _{n}} on X n {\displaystyle X_{n}} such that the composition of structure maps
S 1 ∧ ⋯ ∧ S 1 ∧ X n → S 1 ∧ ⋯ ∧ S 1 ∧ X n + 1 → ⋯ → S 1 ∧ X n + p − 1 → X n + p {\displaystyle 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 Σ p × Σ n {\displaystyle \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 S p Σ {\displaystyle {\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 S p Σ {\displaystyle {\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 the 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.