It is very well know that groupoids, considered as spaces via the nerve construction, are homotopy 1-types, i.e. aspherical. Here is a sketch of proof: Consider the canonical functor $f:C\rightarrow \pi\_1(C)$ for a category $C$ and its fundamental groupoid $\pi\_1(C)$. The Quillen-fiber based at some object $Y$, i.e. the slice category $Y\backslash f$, is the universal covering category of $C$. If $C=G$ is a groupoid, then $\pi\_1(G)$ is $G$ itself and $f$ is essentially the identity. It is not hard to see that the Quillen-fiber in this case has a terminal object and thus is contractible, which proves the statement.


Is this the standard way to prove that groupoids have vanishing higher homotopy groups?



Standard way to prove that groupoids are homotopy 1-types
A low tech-argument: 


-a groupoid is a disjoint union of connected groupoids


-the nerve construction preserves disjoint unions


-a connected groupoid is equivalent to a group, and thus its nerve is aspherical.


