Symmetric sequence

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In algebraic topology, an S {\displaystyle \mathbb {S} } {\displaystyle \mathbb {S} }-object (also called a symmetric sequence) is a sequence { X ( n ) } {\displaystyle \{X(n)\}} {\displaystyle \{X(n)\}} of objects such that each X ( n ) {\displaystyle X(n)} {\displaystyle X(n)} comes with an action[note 1] of the symmetric group S n {\displaystyle \mathbb {S} _{n}} {\displaystyle \mathbb {S} _{n}}.

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

S-module

By S {\displaystyle \mathbb {S} } {\displaystyle \mathbb {S} }-module, we mean an S {\displaystyle \mathbb {S} } {\displaystyle \mathbb {S} }-object in the category V e c t {\displaystyle {\mathsf {Vect}}} {\displaystyle {\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 S {\displaystyle \mathbb {S} } {\displaystyle \mathbb {S} }-module determines a Schur functor on V e c t {\displaystyle {\mathsf {Vect}}} {\displaystyle {\mathsf {Vect}}}.

This definition of S {\displaystyle \mathbb {S} } {\displaystyle \mathbb {S} }-module shares its name with the considerably better-known model for highly structured ring spectra due to Elmendorf, Kriz, Mandell and May.

See also

Notes

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

References