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 actionAn 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. 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.){{harvnb|Getzler|Jones|1994|loc=§ 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.{{clarify|what’s “considerably better-known model” here?|date=July 2024}}

Notes

References

{{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}}

Category:Algebraic topology

S-object

S-module

See also

Notes

References