Symmetric spectrum explained
on
such that the composition of structure maps
S1\wedge...\wedgeS1\wedgeXn\toS1\wedge...\wedgeS1\wedgeXn+1\to...\toS1\wedgeXn+p-1\toXn+p
is equivariant with respect to
. 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
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
; 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
