$c_\sigma$ is the $(r-1)$-cube given by $0\le{s_{a_1}}\le{1},...,0\le{s_{a_{r-1}}}\le{1}$, where $\sigma=<a_0,...,a_r>\in{K}$ with the compatibility conditions

1\. Cocycle condition $$V_\sigma(s_{a_1},...,
s_{a_p}=1,
...,s_{a_{r-1}})=
V_{<a_0,...,a_p>}     (s_{a_1},...,s_{a_{p-1}}                                      ).
V_{<a_p,...,a_r>})(s_{a_{p+1}},...,s_{a_{r-1}}).$$

2\. Compatibility condition

$$V_\sigma(s_{a_1},..., s_{a_p}=0 ,...,s_{a_{r-1}})=
V_{<a_0,...,{\hat{a}}_p,...,a_{r-1}>}
(s_{a_1},...,{\hat{s}}_{a_p},...,s_{a_{r-1}}).$$

Now, suppose we have compatible transition functions $\{v_\sigma\}$ for a principal $G$-bundle $E\to{|K|}$ with triangulated base. Then for $\sigma=<a_0,...,a_r>$, the p.t.f. $V_\sigma:c_\sigma\to{G}$ is given by the parallel transport $E_{a_0}\to{E_{a_r}}$ along paths determined as follows:

Let $\sigma=<a_0,...,a_r>\in{K^s}$ and $s=(s_{a_0},...,s_{a_{r-1}})\in{c_\sigma}$.

We pick $r-1$-points as $P_1,...,P_{r-1}$ so that $P_1$ is on the line segment from $a_0$ to $a_1$, that is, $$P_1=(1-s_{a_1})a_0+s_{a_1}a_1=((1-s_{a_1},s_{a_1}),<a_0,a_1>)\in{|K|}.$$ Similarly, $P_2$ is on the line segment from $P_1$ to $a_2$, $P_2=(1-s_{a_2})P_1+s_{a_2}a_2$. Then $$P_2=((1-s_{a_2})(1-s_{a_1}),(1-s_{a_2})s_{a_1},s_{a_2},<a_0,a_1,a_2>).$$ By continuing in the same way, we get $$P_{r-1}=(1-s_{a_{r-1}})P_{r-2}+s_{a_{r-1}}a_{r-1}.$$ Let $\alpha$ be the piecewise linear path from $a_0$ through $P_1,...,P_{r-1}$ to $a_r$. In other words, $\alpha$ is determined uniquely up to parametrization by $r-1$ numbers $s_{a_1},...,s_{a_{r-1}}$. For $P_{r-1}=(t,d_r\sigma)\in{\Delta^{r-1}}\times{K_{r-1}}$, $d_r\sigma=<a_1,...,a_{r-1}>$, the element$$V_{\sigma}(s_1,...,s_{r-1})=v_{\sigma}(t)\in{G}$$ is to be interpreted as the parallel transport along $\alpha$.

![image](figs.2){width="5cm"}\

**[The Classifying Map]{}\[eight\]**
====================================

[**[The construction of The Classifying Map]{}**]{}

For a given set of compatible transition functions (c.t.f.) $\{v_\sigma\}$ satisfying Proposition \[6.9\] we have seen in Proposition \[6.10\] that there is an associated $G$- bundle $F$ over $|S_.|$. Recall that the composite map $\text{proj}\circ{L}:\|\text{
