functions defined over the faces of a simplex. These they call “parallel transport functions” since they are determined by parallel transport for the connection. In this section we shall introduce similar “compatible transition functions” for $K$ replaced by a simplicial set $S$ and in the following section we shall use these to construct a classifying map on the star complex $\bar{P_.}S_.$. First we consider $G$-bundles over simplicial sets.

\[6.2\] A bundle over $|S|$ is a sequence of bundles over ${\Delta}^p\times{\sigma}$ for all ${p}$, where $\sigma\in{S_p}$ and with commutative diagrams; $$\begin{aligned}
\xymatrix{
F_{{d}_j\sigma} \ar[d] \ar[r]^-{{\bar{\varepsilon}}^j}
&{F_{\sigma}}   \ar[d] \\
{\Delta}^{p-1}\times{{d}_j\sigma} \ar[r]^-{{\varepsilon}^j}
& {\Delta}^p\times{\sigma}
}\end{aligned}$$ and $$\begin{aligned}
\xymatrix{
F_{{s}_j\sigma} \ar[d] \ar[r]^-{{\bar{\eta}}^j}
&{F_{\sigma}}   \ar[d] \\
{\Delta}^{p+1}\times{{s}_j\sigma} \ar[r]^-{{\eta}^j}
& {\Delta}^p\times{\sigma}
}\end{aligned}$$ with the compatibility conditions: $$\begin{aligned}
{\bar{\varepsilon}}^j{\bar{\varepsilon}}^i= \left\{ \begin{array}{r@{\quad: \quad}l}{\bar{\varepsilon}}^i {\bar{\varepsilon}}^{j-1} & i<j  \\
{\bar{\varepsilon}}^{i+1}
{\bar{\varepsilon}}^j & i\geq{j}, \end{array} \right.\end{aligned}$$

$$\begin{aligned}
{\bar{\eta}}^j{\bar{\eta}}^i= \left\{ \begin{array}{r@{\quad: \quad}l}{\bar{\eta}}^i {\bar{\eta}}^{j+1} & i\leq{j} \\
{\bar{\eta}}^{i-1}
{\bar{\eta}}^j & i>j, \end{array} \right.\end{aligned}$$

and $$\begin{aligned}
{\bar{\eta}}^j{\bar{\varepsilon}}^i= \left\{ \begin{array}{r@{\quad: \quad}l}{\bar{\varepsilon}}^i {\bar{\eta}}^{j-1} & i<j  \\
1 &i=j,i=j+1\\
{\bar{\varepsilon}}^{i-1}
{\bar{\eta}}^i & i>{j+1}. \end{array} \right.\end{aligned}$$ 0.3 cm Given a $G$-bundle $F\to{|S|}$, $G$ a Lie group, since ${\Delta}^p$ is contractible, we can choose a trivialization ${\varphi}_{\sigma}:F_{\sigma}\to{{\Delta}^p}\times{\sigma}\times{G}$ for a non-degenerate $\sigma\in{S_p}$. If $\sigma$ is degenerate, that is, there exists 