S-object
In algebraic topology, an -object (also called a symmetric sequence) is a sequence of objects such that each comes with an action[note 1] of the symmetric group .
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.)[1]
S-module
[edit | edit source]By -module, we mean an -object in the category of finite-dimensional vector spaces over a field k of characteristic zero (the symmetric groups act from the right by convention). Then each -module determines a Schur functor on .
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.[clarification needed]
See also
[edit | edit source]Notes
[edit | edit source]- ^ 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 ; cf. Automorphism group#In category theory.
References
[edit | edit source]- ^ Getzler & Jones 1994, § 1
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