1
vote
3answers
213 views

Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$ On ...
2
votes
3answers
240 views

How to detect if a simplicial set is the nerve of a groupoid?

I have the following question. Suppose I have a simplicial set. Is there a way to detect if it actually is isomorphic to a nerve of a groupoid? I've seen the fact that if you have a nerve ...
3
votes
2answers
312 views

Cartesian cubes and groupoids

Given a groupoid $G,$ one can consider the canonical epimorphism $$G_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram ...
3
votes
2answers
555 views

Examples of Simplicial Groupoids in Nature

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? ...