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 $𝖵 𝖾 𝖼 𝗍$ 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 $𝖵 𝖾 𝖼 𝗍$ .

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.^{[clarification needed]}

Notes

↑ An 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 $G \to Aut (X)$ ; cf. Automorphism group#In category theory.

References

↑ Getzler & Jones 1994, § 1

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". www.numdam.org. Séminaire Nicolas Bourbaki. MR 1423619. Zbl 0866.18007. Retrieved 2018-09-27.

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[1] An 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 $G \to Aut (X)$ ; cf. Automorphism group#In category theory.

[2] Getzler & Jones 1994, § 1

S-object

S-module

See also

Notes

References