S-object

In algebraic topology, an $\mathbb{𝕊}$-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 ${\mathbb{𝕊}}_n$. The category of combinatorial species is equivalent to the category of finite $\mathbb{𝕊}$-sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.)

S-module

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

This definition of $\mathbb{𝕊}$-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

References

• Getzler, Ezra; Jones, J. D. S. (1994-03-08). "Operads, homotopy algebra and iterated integrals for double loop spaces". arXiv:hep-th/9403055.
• Loday, Jean-Louis (1996). "La renaissance des opérades" (in en). http://www.numdam.org/item/SB_1994-1995__37__47_0.