On page 28 of the COCTALOS lecture notes, it is written that a simplicial set is the nerve of a groupoid if and only if it is a cyclic set in the sense of Connes (I am understanding a presheaf on $\Lambda$ here).
However, I don't see how to prove this equivalence. In particular, I can't find a cyclic automorphism of the set of $n-$composable arrows in a groupoid.
Can anyone help me find this automorphism? I think (hope) that I could take it from there.
For a groupoid ${\mathscr G}$ and $X_0\xrightarrow{f_1} X_1\to\ldots\to X_{n-1}\xrightarrow{f_n} X_n\in {\mathbf N}({\mathscr G})_n$ an $n$-simplex in its nerve, you also have $$X_n\xrightarrow{(f_n\dots f_1)^{-1}} X_0\xrightarrow{f_1} X_1\to\ldots\to X_{n-2}\xrightarrow{f_{n-1}} X_{n-1}\in {\mathbf N}({\mathscr G})_n.$$ Moreover, this association defines an automorphism of order $n$ on ${\mathbf N}({\mathscr G})_n$. Do you need more details?