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 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.