In algebraic topology, an **-object** (also called a **symmetric sequence**) is a sequence of objects such that each comes with an action^{[note 1]} of the symmetric group .

The category of combinatorial species is equivalent to the category of finite -sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.)^{[1]}

## -module

By *-module*, we mean an -object in the category of finite-dimensional vector spaces over a field *k* of characteristic zero (the symmetric groups act from the right by convention). Then each -module determines a Schur functor on .

## See also

## Notes

**^**An action of a group*G*on an object*X*in a category*C*is a functor from*G*viewed as a category with a single object to*C*that maps the single object to*X*. Note this functor then induces a group homomorphism ; cf. Automorphism group#In category theory.

## References

**^**Getzler & Jones 1994, § 1

- 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.