Contents

# Contents

## 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.

## References

Last revised on March 24, 2023 at 16:07:21. See the history of this page for a list of all contributions to it.