2.8. is a prismatic set. However we shall need another model constructed as follows: For a continuous map $a\in{\text{Map}(\Delta^p\times{\Delta^{q_0...q_p}},G^{p+1})}$. Then we define $$a(t,s^0,...,s^p)=(a_0(t,s^0),a_1(t,s^0,s^1),...,a_p(t,s^0,...,s^p)),$$ where $(t,s^0,...,s^p)\in{\Delta^p\times{\Delta^{q_0...q_p}}}$. $S_.G$ acts on this prismatic set and we define $$\begin{aligned}
P_pEG_{q_0,...,q_p}&=&\{a:\Delta^p\times{\Delta^{q_0...q_p}}\to{G^{p+1}}
\text{
  }|\text{   }
a_j(\varepsilon^it,s)\text{   }{\text{is independent of}}\\
& &\text{   } s^i\text{   }
 \text{for all}\text{   }j\text{   }\text{different from}\text{    }i
\}.\end{aligned}$$ $P_.BG=P_.EG/S_.G$, that is, $$P_pBG_{q_0,...,q_p}=P_pEG_{q_0,...,q_p}/S_pG.$$

\[5.1\] The evaluation maps give horizontal homotopy equivalences in the diagram

$$\xymatrix{ \|\text{   }|P_.EG_.|\text{   }\| \ar[d]^{\|\text{
}|\gamma|\text{   }\|} \ar[r]^{\phantom{123}ev}
&EG  \ar[d]^\gamma \\
\|\text{   }|P_.BG_.|\text{   }\| \ar[r]^{\phantom{123}ev} & EG/G }$$

Furthermore the top map is equivariant with respect to the homomorphism $ev:|S_.G|\to{G}$.

First notice that the evaluation map $ev:|S_.G|\to{G}$ is a homotopy equivalence. Also the equivariance is obvious by the commutative diagram $$\xymatrix{\|\text{   }|P_.EG_.|\text{   }\| \times{|S_.G|}
\ar[d]^{{\text{pr}_1}} \ar[r]^{\phantom{1234567}ev\times{ev}}
&EG\times{G}  \ar[d]^{{\text{pr}_1}} \\
\|\text{   }|P_.EG_.|\text{   }\| \ar[r]^{\phantom{12}ev}
 & EG }$$ Since $\|\text{   }|P_.EG_.|\text{   }\|$ and $EG$ are both contractible, the evaluation map induces a homotopy equivalence on the quotient.

**[Lattice Gauge Theory, Parallel Transport Function]{}\[six\]**
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In Lattice gauge theory in the sense of Phillips and Stone [@PS1] they construct for a given Lie group $G$ and a simplicial complex $K$ a $G$-bundle with connection on $|K|$ associated to a set of $G$-valued continuous 