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 action 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.)

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.