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 on 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 is equivariant with respect to Σ p × Σ 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 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 ; 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.
