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^{[note 1]} 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.)^[1]

S-module

By $\mathbb {S}$ -module, we mean an $\mathbb {S}$ -object in the category ${\mathsf {Vect}}$ 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 ${\mathsf {Vect}}$ .

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.^{[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 $G\to \operatorname {Aut} (X)$ ; 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 $G\to \operatorname {Aut} (X)$ ; cf. Automorphism group#In category theory.

[2] Getzler & Jones 1994, § 1

[note 1]

[1]

S-module

See also

Notes

References