groupoid.

Define $\sigma_i\overline{x} = \overline{s_{i+1}x} \quad \mbox{ for } i\geq 0$, and, for $i > 0$, $\delta_i\overline{x} = \overline{d_{i +1}x}$, but for $i = 0$, $\delta_0\overline{x} = (\overline{d_1x})(\overline{d_0x})^{-1}$.

These definitions yield a simplicial groupoid as is easily checked and, as is clear, its simplicial set of objects is constant, so it also can be considered as a simplicially enriched groupoid, $G(K)$.

(NB. Beware there are several ‘typos’ in the original paper relating to these formulae for the construction and in some of the related material.)

As before it is instructive to compute some examples and we will look at $G(\Delta[2])$ and $G(\Delta[3]$. ) simplicially enriched groupoid are free groupoids in each simplicial dimension, their structure can be clearly seen from the generating graphs. For instance, $G(\Delta[2])_0$ is the free groupoid on the graph $$\xymatrix{&\overline{1}\ar[dr]^{\overline{12}}&\\
\overline{0}\ar[rr]_{\overline{02}}\ar[ur]^{\overline{01}}&&\overline{2}}$$ whilst $G(\Delta[2])_1$ is the free groupoid on the graph $$\xymatrix{&\overline{1}\ar[dr]^{\overline{122}}&\\
\overline{0}\ar[rr]_{\overline{022}}\ar[ur]<.5ex>^{\overline{011}}\ar[ur]<-.5ex>_{\overline{012}}&&\overline{2}}$$ Here it is worth noting that $\delta_0(\overline{012}) = (\overline{02}).(\overline{12})^{ -1}$. Higher dimensions do not have any non-degenrate generators.

Again with $G(\Delta[3])$, in dimension 0 we have the free groupoid on the directed graph give by the 1-skelton of $\Delta[3]$. In dimension 2, the generating directed graph is $$\xymatrix@+20pt{&\overline{1}\ar[dr]^{\overline{133}}\ar[dd]<.5ex>\ar[dd]<-.5ex>&\\
\overline{0}\ar[ur]<.7ex>\ar[ur]\ar[ur]<-.7ex>\ar'[r][rr]\ar[dr]<.5ex>^{\overline{023}}\ar[dr]<-.5ex>_{\overline{022}}&&\overline{3}\\
&\overline{2} \ar[ur]_{\overline{233}}  &   }$$ Here 