only a few of the arrow labels have been given. Others are easy to provide (but moderately horrible to typeset in a sensible way!). Those from $\overline{0}$ to $\overline{1}$ are $\overline{012}$, $\overline{011}$ and $\overline{013}$; those from $\overline{1}$ to $\overline{2}$ are $\overline{122}$ and $\overline{123}$, and finally from $\overline{0}$ to $\overline{3}$, we have $\overline{033}$.

The next dimension is only a little more complicated. It has extra degenerate arrows such as $\overline{0112}$ and $\overline{0122}$ from $\overline{0}$ to $\overline{1}$ but also between these two vertices has $\overline{0123}$, coming from the non-degenerate three simplex of $\Delta[3]$. The full diagram is easy to draw (and again a bit tricky to typeset in a neat way), and is therefore left ‘as an exercise’.

**Remarks**

\(i) The functor $G$ has a right adjoint $\overline{W}$ and the unit $K\to\overline{W}G(K)$ is a weak equivalence of simplicial sets. This is part of the result that shows that simplicially enriched groupoids model all homotopy types, for which see the original paper, or the book by Goerss and Jardine, [@GoerssJardine].

\(ii) It is tantalising that the definition of $S(\mathbb{A})$ for a category $\mathbb{A}$ and of $G(K)$ for a simplicial set $K$ are very similar yet very different. Why does the twist occur in 