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 [math]\displaystyle{ \Sigma_n }[/math] on [math]\displaystyle{ X_n }[/math] such that the composition of structure maps
- [math]\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} }[/math]
is equivariant with respect to [math]\displaystyle{ \Sigma_p \times \Sigma_n }[/math]. 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 [math]\displaystyle{ \mathcal{S}p^\Sigma }[/math] 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 [math]\displaystyle{ \mathcal{S}p^\Sigma }[/math]; 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.
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.
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