# Properties of nerve of strict n-groupoid

The nerve of a groupoid is (by construction) a simplicial set and this works for any category. However there is extra structure in the case of a groupoid. Namely the set of $n$-simplices has an action of the symmetric group $S_{n+1}$. Furthermore it appears that these group actions and the simplicial maps combine to make the nerve a $\Delta S$-set (where $\Delta S$ is the crossed simplicial group). In fact I suspect the symmetric groups here can be replaced by the hyperoctahedral groups but this should be checked.

[ Crossed simplicial groups were defined in MR0998125 Fiedorowicz, Zbigniew ; Loday, Jean-Louis Crossed simplicial groups and their associated homology. Trans. Amer. Math. Soc. 326 (1991), no. 1, 57--87. ]

Does this extend to strict $n$-groupoids? The nerve of a strict $n$-category is a simplicial set (by construction) and a strict $n$-groupoid is a strict $n$-category in which all morphism are invertible. So I am asking for actions of symmetric groups (or hyperoctahedral groups) on the simplices in the case all morphisms are invertible.

The reason I am confused is that an $n$-simplex is a map of $n$-categories from the oriental $O(n)$ to the $n$-groupoid and the definition of $O(n)$ seems complicated to me.

The simplicial and cubical nerves of crossed complexes are defined in the book partially titled Nonabelian Algebraic Topology (EMS 2011). Since strict $n$-groupoids are equivalent to $n$-truncated crossed complexes, this gives the construction. The globular case is discussed in this paper.
$$\Pi: (\text{filtered spaces}) \to (\text{crossed complexes}).$$ So if $X_*$ is a filtered space we get a crossed complex $\Pi X_*$. In particular this applies to the skeletal filtration of CW-complexes, and so to $\Delta_*^n$ and $I^n_*$. So the cubical nerve of a crossed complex is defined by $$N^\Box(C)_n = Crs(\Pi(I^n_*), C).$$ I give the cubical nerve here because the $\omega$-groupoid structure this has is a major feature of the above book.