$\tau$ such that $\sigma={s}_i\tau$, then the trivialization of $\sigma$ is defined as pullback of the trivialization of $\tau$, that is, ${\varphi}_{\sigma}={{\eta}^i}^*({\varphi}_{\tau})$.

\[6.3\] ( Admissible Trivializations ) A set of trivializations is called admissible, in case ${\varphi}_{\sigma}$ for $\sigma={s}_i\tau$ is given by ${\varphi}_{\sigma}={{s}^i}^*({\varphi}_{\tau})$.

\[6.4\] Admissible trivializations always exist.

Now, let us construct the transition functions for a simplex $\sigma\in{S_p}$ before giving the following proposition:

\[6.5\]

Given a bundle and a set of trivializations, we get for each face ${\tau}$ of say dim$\tau=q<p$ in $\sigma$, a transition function $v_{\sigma,\tau}:{\Delta}^q\to{G}$. E.g., if $\tau={d}_i\sigma$ then the transition function $v_{\sigma,{d}_i\sigma}:{\Delta}^{p-1}\to{G}$ is given by the diagram $$\xymatrix{
{\Delta}^{p-1}\times{({d}_i\sigma)}\times{G} \ar[d]
\ar[r]^-{\Theta}
&{\Delta}^p\times{(\sigma)}\times{G}   \ar[d] \\
{\Delta}^{p-1}\times{{d}_i\sigma} \ar[r]^-{{\varepsilon}^i}
& {\Delta}^p\times{\sigma}
}$$ where ${d}_i\sigma=\tau$ and $\Theta={\varphi}_{\sigma}\circ{{\bar{\varepsilon}}^i}\circ{{{\varphi}_{{d}_i\sigma}}^{-1}}$. So $$\{v_{\sigma,\tau}|\sigma\in{S_p}\text{   }\text{and} \text{   }\tau\text{   }\text{is a face of}\text{   }\sigma\}$$ are [**the transition functions**]{} for the bundle over $|S|$.

[**Remark :**]{} The transitions functions are generalized lattice gauge fields. Classically Lattice gauge fields are defined only on 1-skeletons but one can extend them to $p-1$ simplices for all $p$, given rise to transition functions on $\Delta^p$, as above.

We now list a number of propositions stating the properties of these. The proofs are straight forward. For details see Akyar [@B].

\[6.6\]

Given a bundle 