for $\sigma\in{S_p}$ nondegenerate.

\[6.9\]

Suppose given a set of functions $$v_{\sigma}:{\Delta}^{p-1}\to{G}$$ for $\sigma\in{S_p}$ for all ${p}$, satisfying the compatibility conditions $$v_{\sigma}\circ{{\varepsilon}^i}= \left\{ \begin{array}{r@{\quad: \quad}l}v_{{d}_i\sigma}
 & i<p-1  \\ v_{{d}_{p-1}\sigma}.{v_{{d}_p\sigma}}^{-1}
 & i={p-1} \end{array} \right.$$ and $$v_{s_j\sigma}=v_{\sigma}\circ{\eta^j}.$$

Then one can define for each $\sigma\in{S_p}$ and each lower dimensional face $\tau$ of $\sigma$, a function $v_{\sigma,\tau}$ such that i) and ii) in Proposition \[6.6\] hold and such that $$v_{\sigma,\tau}= \left\{ \begin{array}{r@{\quad: \quad}l}v_{\sigma} & i=p  \\
1 & i<p. \end{array} \right.$$

\[6.10\]

Given a set of transition functions $v_{\sigma,\tau}$ satisfying i) and ii) in Proposition \[6.6\], there is a bundle $F$ over $|S|$ and trivializations with transition functions $v_{\sigma,\tau}$.

\[6.11\]

Given a set of functions $v_{\sigma}$ satisfying the compatibility conditions in Proposition \[6.7\], one can construct a bundle $F$ over $|S|$ and the trivializations with the transition functions $v_{\sigma,d_p\sigma}=v_{\sigma}$ and $v_{\sigma,d_i\sigma}=1$ when $i<p$ and $v_{s_i\sigma}=v_{\sigma}\circ{\eta^i}$ for a degenerate $\sigma$.

\[6.12\] A set of functions $\{v_{\sigma}\}_{\sigma\in{S}}$ as in Proposition \[6.9\] are called a set of “compatible transition functions”.

We end this section by comparing these compatible transition functions with the “parallel transport functions” (p.t.f.) of Phillips and Stone [@PS1]. For $S=K^s$ these consist of a set of maps, $V_\sigma:c_\sigma\to{G}$ for each $r$-simplex $\sigma$ of $K$, $r\ge{1}$, 