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    • Source: S-object

      • In algebraic topology, an




      S



      {\displaystyle \mathbb {S} }

      -object (also called a symmetric sequence) is a sequence



      {
      X
      (
      n
      )
      }


      {\displaystyle \{X(n)\}}

      of objects such that each



      X
      (
      n
      )


      {\displaystyle X(n)}

      comes with an action of the symmetric group





      S


      n




      {\displaystyle \mathbb {S} _{n}}

      .
      The category of combinatorial species is equivalent to the category of finite




      S



      {\displaystyle \mathbb {S} }

      -sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.)


      S-module


      By




      S



      {\displaystyle \mathbb {S} }

      -module, we mean an




      S



      {\displaystyle \mathbb {S} }

      -object in the category





      V
      e
      c
      t




      {\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




      S



      {\displaystyle \mathbb {S} }

      -module determines a Schur functor on





      V
      e
      c
      t




      {\displaystyle {\mathsf {Vect}}}

      .
      This definition of




      S



      {\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.


      See also


      Highly structured ring spectrum


      Notes




      References



      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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