For some reason I convinced myself that a simplicial set (or maybe I mean directly Kan complex) is homotopy equivalent to the nerve of a groupoid if and only if it has no higher homotopy groups.
Is this true? (it seems like a proof could go through the fact mentioned in the title of my question)
And a related question: can a space with no higher homotopy groups be described (up to homotopy) as a CW complex with only 1-cells? (I expect not, is there a simple counterexample?)
The reason I ask is that it seems to me that CW complexes with only 1-cells seem special, in the sense that they do not have higher homotopy groups (right?) while something like $S^2$ has only 2-cells but has non-vanshing $\pi_3$.