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^{[note 1]} 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.)^[1]

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.^{[clarification needed]}

• Highly structured ring spectrum

• 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.