Let $G$ be a connected groupoid. Is the nerve $BG$ a $K(\pi, 1)$, and if so, is there a groupoid homomorphism $f:G\to \pi$ that induces the homotopy equivalence?

Another way of putting this is that there is a notion of $K(G,1)$ for $G$ a groupoid, in fact this is just the classifying space $BG$ of the groupoid, i.e. the realisation of the simplicial nerve of $G$. As pointed out by Segal, the nerve construction on categories (or groupoids) takes equivalences to homotopy equivalences. 

