nLab symmetric set

Idea

A symmetric set (or symmetric simplicial set) is a simplicial set, $X$, equipped with additional transposition maps $t^n_i: X_n \to X_n$ for $i=0,\ldots,n-1$. These transition maps generate an action of the symmetric group $\Sigma_{n+1}$ on $X_n$ 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).

An analogy to keep in mind is

symmetric set : simplicial set :: groupoid : category.

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 $[n]^{-1}[n]$ 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 $!Smp \to Gpd$ from symmetric sets to groupoids is left adjoint to a natural functor $Gpd \to !Smp$, 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.

Related concepts

• symmetric comonad

• cyclic set, skew-simplicial set

References

• William Lawvere, Toposes generated by codiscrete objects in combinatorial topology and functional analysis, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, pdf.

• Marco Grandis, Finite sets and symmetric simplicial sets PDF

• Marco Grandis, Higher fundamental functors for simplicial sets arXiv

• Jiří Rosický, Walter Tholen, Left-determined model categories and universal homotopy theories, Trans. Amer. Math. Soc. 355 (2003), 3611-3623.

• Jiří Rosický, Walter Tholen, 2008, ‘Erratum to “Left-determined model categories and universal homotopy theories”’, Transactions of the American Mathematical Society, vol. 360, no. 11, pp. 6179-6179: doi:10.1090/s0002-9947-08-04727-2

• Denis-Charles Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Astérisque 308, 2006 (doi:10.24033/ast.715, NUMDAM)