S-object explained

In algebraic topology, an

-object (also called a symmetric sequence) is a sequence

\{X(n)\}

of objects such that each

X(n)

comes with an action^[1] of the symmetric group

S_n

.

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.)

S-module

By

-module, we mean an

S

-object in the category

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

-module determines a Schur functor on

Vect

.

This definition of

-module shares its name with the considerably better-known model for highly structured ring spectra due to Elmendorf, Kriz, Mandell and May.

See also

• Highly structured ring spectrum

References

• Jones. J. D. S.. Getzler. Ezra. 1994-03-08. Operads, homotopy algebra and iterated integrals for double loop spaces. hep-th/9403055. en.
• Web site: La renaissance des opérades. Loday. Jean-Louis. Jean-Louis Loday. 1996. www.numdam.org. Séminaire Nicolas Bourbaki. en. 1423619. 0866.18007. 2018-09-27.

Notes and References

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.