on a simplicial set and admissible trivializations, the transition function $v_{\sigma,\tau}$, where $\tau$ is a face of $\sigma$, satisfies;

i\) $\sigma$ is nondegenerate: if $\gamma={d}_j\sigma$ and $\tau={d}_i\gamma$ then $$v_{\sigma,\tau}=(v_{\sigma,\gamma}\circ{{\varepsilon}^i}).v_{\gamma,\tau}.$$

This is called the cocycle condition.

ii\) $\sigma$ is degenerate: If $\sigma={s}_j{\sigma}^{\prime}$ and $\tau={d}_i\sigma$ then when $i<j$ for $\tau={s}_{j-1}{\tau}^{\prime}$ one gets ${\tau}^{\prime}={d}_i{ \sigma}^{\prime}$ and when $i>j+1$ for $\tau={s}_j{\tau}^{\prime}$ one gets ${\tau}^{\prime}={d}_{i-1}{ \sigma}^{\prime}$. For the other cases, $i=j$ or $i=j+1$, $\tau={\sigma}^{\prime}$. Then the transition functions satisfy: $$v_{\sigma,\tau}= \left\{ \begin{array}{r@{\quad: \quad}l}v_{{\sigma}^{\prime},{\tau}^{\prime}}\circ{{\eta}^{j-1}} & i<j  \\
1 &i=j,i=j+1\\
v_{{\sigma}^{\prime},{\tau}^{\prime}}\circ{{\eta}^j}
& i>{j+1}. \end{array} \right.$$

iii\) If $\tau$ is a composition of face operators of $\sigma$, e.g., $\tau={\tilde{d}}^{p-(i-1)}\sigma$, $i=1,...,p$, where ${\tilde{d}}^{p-(i-1)}=d_i\circ{...}\circ{d_p}$ then $$v_{\sigma,\tau}=(v_{\sigma,{\tilde{d}}^1\sigma}\circ{{({\varepsilon}^i)}^{p-i}}).
(v_{{\tilde{d}}^1\sigma,{\tilde{d}}^2\sigma}\circ{{({\varepsilon}^i)}^{p-i-1}})...
(v_{{\tilde{d}}^{p-(i+1)}\sigma,{\tilde{d}}^{p-i}\sigma}\circ{{\varepsilon}^i}).v_{{\tilde{d}}^{p-i}\sigma,\tau}.$$

\[6.7\]

Assume that we have a bundle over $|S|$. Then

1\) There exists admissible trivializations such that the transition function is given by $$v_{\sigma,{d}_i\sigma}=1\text{    }\text{if}\text{   }i<p.$$

2\) For $\tau={\tilde{d}}^{p-(i-1)}\sigma$, $i=1,...,p$, we get $v_{\sigma,\tau}$ as product of some transition functions: $$v_{\sigma,\tau}=(v_{\sigma}\circ{{({\varepsilon}^i)}^{p-i}}).
(v_{{\tilde{d}}^1\sigma}\circ{{({\varepsilon}^i)}^{p-i-1}}).(v_{{\tilde{d}}^2\sigma}\circ{{({\varepsilon}^i)}^{p-i-2}})...
(v_{{\tilde{d}}^{p-(i+1)}\sigma}\circ{{({\varepsilon}^i)}^1}).(v_{{\tilde{d}}^{p-i}\sigma}).$$

3\) The transition functions $v_{\sigma,\tau}$ satisfy the compatibility conditions: $$v_{\sigma}\circ{{\varepsilon}^i}= \left\{ \begin{array}{r@{\quad: \quad}l}v_{{d}_i\sigma}
 & i<p-1  \\ v_{{d}_{p-1}\sigma}.{v_{{d}_p\sigma}}^{-1}
 & i={p-1} \end{array} \right.$$

4\) For a degenerate $\sigma$, we have $$v_{\sigma}\circ{{\eta}^j}=v_{{s}_j\sigma}$$ $\forall {j}$.

\[6.8\]

Given a bundle, one can find admissible trivializations such that the transition functions are determined by functions $v_\sigma:{\Delta}^{p-1}\to{G}$ 