A symmetric set (or symmetric simplicial set) is a simplicial set, , equipped with additional transposition maps for . These transition maps generate an action of the symmetric group on and satisfy certain commutation relations with the face and degeneracy maps. The result is that a symmetric set is a presheaf of sets on the category FinSet of nonempty finite sets (or on its skeleton).
This analogy can be formalized by noticing that the skeletal category of finite sets is simply the full subcategory of Cat whose objects are the localizations which are groupoids. By the universal property of localization, the usual (simplicial) nerve of a groupoid has a canonical symmetric structure.
Grandis proves that the fundamental groupoid functor from symmetric sets to groupoids is left adjoint to a natural functor , the symmetric nerve of a groupoid, and preserves all colimits - a van Kampen theorem. Similar results hold in all higher dimensions.
The notion of cyclic set is intermediate between symmetric sets and simplicial sets. In particular, any symmetric set, such as the nerve of a groupoid, also has a cyclic structure.
Mike: I’ve never seen this called a “symmetric set” only a “symmetric simplicial set,” which additionally has the advantage of being more descriptive. “Symmetric set” sounds to me like it might also refer to a presheaf on the groupoid of finite sets (sometimes called a “symmetric sequence”). Is there an advantage of “symmetric set” over “symmetric simplicial set?”
Zoran Skoda: You are right, topologists more often say symmetric simplicial sets/objects, for example one of the papers devoted to the topic by Marco Grandis. But symmetric sets is also used, see for example the quote in Getzler’s paper on L-infty integration where he quotes Loday and Fiedorowicz, I heard this shorter term many time from conversations with Jibladze and many others. There is an advantage and that is exactly the theory of crossed simplicial groups in the sense of Loday and Fiedorowicz or skewsimplicial groups in the sense of a bit earlier work fo Krasauskas. The point is to treat dihedral, cyclic, symmetric etc. homologies on the same footing, hence symmetric, cyclic, dihedral etc. sets. The term symmetric homology is for example in Loday-Fiedorowicz and then in Pirashvili-Richer and so on. I did not hear somebody saying dihedral simplicial sets, or cyclic simplicial sets, though it would be OK with me. I am happy with any solution for nlab and personally use both expressions in my writings and every day usage.
Mike: I see the point. Also, for instance, a symmetric spectrum is a particular sort of symmetric space, i.e. a symmetric object in spaces in this sense. So I’m okay with “symmetric set” as a name for this page.