S-module

In algebraic topology, an $\mathbb$-object (also called a symmetric sequence) is a sequence $\$ of objects such that each $X(n)$ comes with an actionAn action of a group ''G'' on an object ''X'' in a category ''C'' is a functor from ''G'' viewed as a category with a single object to ''C'' that maps the single object to ''X''. Note this functor then induces a group homomorphism $G \to \operatorname(X)$; cf. Automorphism group#In category theory. of the symmetric group $\mathbb_n$.

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

S-module

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

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

See also

*Highly structured ring spectrum

Notes

References

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Algebraic topology