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 $\mathcal{N}$ of a groupoid $\mathcal{G}$ then all homotopy groups $\pi\_n(\mathcal{N})$ for $n\geq 2$ must vanish. Is the converse true? I don't think so, but I don't know how to check that.


Though I would like some other way of detecting nerves of groupoids. The reason is, I have a simplicial set, and I want to make a conclusion about its homotopy groups using the fact that it is a nerve of a groupoid (I don't know this fact yet).


I am a complete novice in the field, so my question might be very easy.


Thank you very much for your help!



How to detect if a simplicial set is the nerve of a groupoid?
The answer is sort of well known. A simplicial set $X$ is the nerve of a groupoid if and only if any $n$-horn has a unique filler for all $n\geq 2$. The horn $\Lambda^k[n]$ is the simplicial subset of $\Delta[n]$ obtained by removing the non-degenerate $n$-simplex of $\Delta[n]$ and it's $k^{\text{th}}$ face. The previous statement means that any map $\Lambda^k[n]\rightarrow X$ extends to $\Delta[n]$ in a unique way, for all $n\geq 2$ and $0\leq k\leq n$. Alternatively, a simplicial set $X$ is the nerve of a groupoid if and only if it is a Kan complex and $\pi\_n(X)=0$ for all $n\geq 2$. The nerve of the category with only one object and endomorphism monoid $\mathbb N$ has trivial homotopy groups in dimensions $\geq 2$ but it is not the nerve of a groupoid (it is not a Kan complex).


