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In [[algebraic topology]], an '''<math>\mathbb{S}</math>-object''' (also called a '''symmetric sequence''') is a sequence <math>\{ X(n) \}</math> of objects such that each <math>X(n)</math> comes with an action<ref group=note>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 <math>G \to \operatorname{Aut}(X)</math>; cf. [[Automorphism group#In category theory]].</ref> of the [[symmetric group]] <math>\mathbb{S}_n</math>.<!-- The notion is due to J.P.May. -->

The category of [[combinatorial species]] is equivalent to the category of finite <math>\mathbb{S}</math>-sets (roughly because the [[permutation category]] is equivalent to the category of finite sets and bijections.)<ref>{{harvnb|Getzler|Jones|1994|loc=§ 1}}</ref>

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

This definition of <math>\mathbb{S}</math>-module shares its name with the considerably better-known model for [[Highly structured ring spectrum|highly structured ring spectra]] due to Elmendorf, Kriz, Mandell and May.{{clarify|what’s “considerably better-known model” here?|date=July 2024}}

== See also ==
*[[Highly structured ring spectrum]]

== Notes ==
{{reflist|group=note}}

== References ==
{{reflist}}
*{{cite arXiv|last2=Jones|first2=J. D. S.|last1=Getzler|first1=Ezra|date=1994-03-08|title=Operads, homotopy algebra and iterated integrals for double loop spaces|eprint=hep-th/9403055|language=en}}
*{{Cite web|url=http://www.numdam.org/item/SB_1994-1995__37__47_0|title=La renaissance des opérades|last=Loday|first=Jean-Louis|authorlink=Jean-Louis Loday|year=1996|website=www.numdam.org|series=[[Séminaire Nicolas Bourbaki]]|language=en|mr=1423619|zbl=0866.18007|access-date=2018-09-27}}


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[[Category:Algebraic topology]]