Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.
share|improve this question
add comment

2 Answers 2

up vote 1 down vote accepted

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.

share|improve this answer
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.