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
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
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 ; 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". www.numdam.org. Séminaire Nicolas Bourbaki. MR 1423619. Zbl 0866.18007. Retrieved 2018-09-27.