S-object

In algebraic topology, an ${\displaystyle \mathbb {S} }$-object (also called a symmetric sequence) is a sequence ${\displaystyle \{X(n)\}}$ of objects such that each ${\displaystyle X(n)}$ comes with an action of the symmetric group ${\displaystyle \mathbb {S} _{n}}$.

The category of combinatorial species is equivalent to the category of finite ${\displaystyle \mathbb {S} }$-sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.)