S-object

In algebraic topology, an $𝕊$-object (also called a symmetric sequence) is a sequence $\{X(n)\}$ of objects such that each $X(n)$ comes with an action of the symmetric group ${𝕊}_n$.

The category of combinatorial species is equivalent to the category of finite $𝕊$-sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.)

S-module

By $𝕊$-module, we mean an $𝕊$-object in the category $𝖵𝖾𝖼𝗍$ of finite-dimensional vector spaces over a field k of characteristic zero (the symmetric groups act from the right by convention). Then each $𝕊$-module determines a Schur functor on $𝖵𝖾𝖼𝗍$.

This definition of $𝕊$-module shares its name with the considerably better-known model for highly structured ring spectra due to Elmendorf, Kriz, Mandell and May.Template:Clarify

See also

• Highly structured ring spectrum

References

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