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.[clarification needed]

See also

Notes