S-object

In algebraic topology, an ${\displaystyle \mathbb {S} }$ -object (also called a symmetric sequence) is a sequence ${\displaystyle \{X(n)\}}$ of objects such that each ${\displaystyle X(n)}$ comes with an action^{[note 1]} of the symmetric group ${\displaystyle \mathbb {S} _{n}}$ .

The category of combinatorial species is equivalent to the category of finite ${\displaystyle \mathbb {S} }$ -sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.)^[1]

S-module

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

This definition of ${\displaystyle \mathbb {S} }$ -module shares its name with the considerably better-known model for highly structured ring spectra due to Elmendorf, Kriz, Mandell and May.^{[clarification needed]}

See also

• Highly structured ring spectrum

Notes

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

References

1. 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.