S-object
In algebraic topology, an [math]\displaystyle{ \mathbb{S} }[/math]-object (also called a symmetric sequence) is a sequence [math]\displaystyle{ \{ X(n) \} }[/math] of objects such that each [math]\displaystyle{ X(n) }[/math] comes with an action[note 1] of the symmetric group [math]\displaystyle{ \mathbb{S}_n }[/math]. The category of combinatorial species is equivalent to the category of finite [math]\displaystyle{ \mathbb{S} }[/math]-sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.)[1]
S-module
By [math]\displaystyle{ \mathbb{S} }[/math]-module, we mean an [math]\displaystyle{ \mathbb{S} }[/math]-object in the category [math]\displaystyle{ \mathsf{Vect} }[/math] of finite-dimensional vector spaces over a field k of characteristic zero (the symmetric groups act from the right by convention). Then each [math]\displaystyle{ \mathbb{S} }[/math]-module determines a Schur functor on [math]\displaystyle{ \mathsf{Vect} }[/math].
This definition of [math]\displaystyle{ \mathbb{S} }[/math]-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
- ↑ 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 [math]\displaystyle{ G \to \operatorname{Aut}(X) }[/math]; 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" (in en). http://www.numdam.org/item/SB_1994-1995__37__47_0.
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