Examples of Simplicial Groupoids in Nature

Question

For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition.

Do you have examples simplicial groupoids that occur in nature? Here's what I have got:

1. Given a simplicial group $G$ acting on a simplicial set $X$, the action groupoid $X//G$ is a simplicial groupoid.
2. The fundamental groupoid $\Pi X$ of a bisimplicial set or simplicial space.
3. Given a functor $F:I\to {\mathbf{sSet}}$ from a diagram category $I$ to the category of simplicial sets, one can form what Goerss-Jardine calls the translation category $E_I F$ and what Mac Lane-Moerdijk calls the category of elements $\int_{I^{\mathrm{op}}}\, F$. The nerve of this category calculates the homotopy colimit ${\mathrm{hocolim}}_I \, F$. In the case where the diagram category is a groupoid, then this translation category/category of element is a simplicial groupoid.
4. Given a simplicial set $X$, the loop groupoid $GX$ is a simplicial groupoid.

Answer 1

See some papers following

Ehlers, P.J. and Porter, T. Varieties of simplicial groupoids. I. Crossed complexes. J. Pure Appl. Algebra 120~(3) (1997) 221--233.

(which you may already have).

As another example, given a double groupoid, it's simplicial nerve in one direction is a simplicial groupoid in your sense. There are lots of examples of double groupoids, see for example my arXiv paper 0903.2627.

Answer 2

Any time you have a topological groupoid, you have a simplicial groupoid since geometric realization preserves finite limits. Although this seems like a rather trivial remark this allows for one to study the homotopy theory of orbifolds, and orbicell complexes, since they may be defined as an equivalence class of Lie groupoids. See the following source below for more details.

Orbifolds as Groupoids: an Introduction at http://arxiv.org/abs/math/0203100