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S-object

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Source updated July 31, 2024

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 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.)

01S-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 ${\mathsf {Vect}}$ .

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

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S-object

01S-module

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Related topics

Highly structured ring spectrum