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.

The main feature of the construction of the nerve is of Dold-Kan type, but using the fundamental crossed complex functor

$$\Pi: (\text{filtered spaces}) \to (\text{crossed complexes}).$$ 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.

However the extra structure that you ask about has not been studied to my knowledge.

Variations on the "cubical site", including symmetries, are in the paper Grandis, M. and Mauri, L. "Cubical sets and their site". Theory Applic. Categories 11 (2003) 185--201.

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    $\begingroup$ Why has this answer been downvoted?? $\endgroup$ Feb 6 '16 at 22:00

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