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 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