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[note 1] 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.)[1]

$\mathbb {S}$ -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 ${\mathsf {Vect}}$ .

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 $G\to \operatorname {Aut} (X)$ ; cf. Automorphism group#In category theory.

References

Getzler & Jones, § 1

Jones, J. D. S.; Getzler, Ezra (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.

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[1] 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 $G\to \operatorname {Aut} (X)$ ; cf. Automorphism group#In category theory.

[2] Getzler & Jones, § 1

S-object

S {\displaystyle \mathbb {S} } -module

See also

Notes

References

$\mathbb {S}$ -module