\tau&=&(b_{j_0},\dots,b_{j_{q_0}},\dots,b_{j_{q_0+...+q_{p-1}+p}},\dots,b_{j_{q+p}})\in{{K^s}_{q+p}}.\end{aligned}$$ By the definition $\bar{P}_p(K^s)_{q_0,\dots,q_p}=P_p(K^s)_{q_0+1,\dots,q_p+1}$. Then $\gamma$ in ${K^s}_{q+2p+1}$ given by $\gamma=(c_0,\dots,c_{q_0+1}|...|c_{q_0+...+q_{p-1}+2p},\dots,c_{q+2p+1})\in{K^s_{q+2p+1}}$ is uniquely determined by $\sigma$ and $\tau$.

Explicitly the inverse map ${\bar{p}}^{-1}:P_p\text{St}(K^s)_{q_0,\dots,q_p}\to{\bar{P}_p{K^s}_{q_0,\dots,q_p}}$ is defined by ${\bar{p}}^{-1}(\sigma,\tau)=\gamma$, where

$$\begin{aligned}
\sigma&=&(a_{i_0},\dots,a_{i_p}),\\
\tau&=&(b_{j_0},\dots,b_{j_{q+p}})\text{   }\text{and}\\
\gamma&=&({j_0}^\prime,\dots,j_{q_0}^\prime,i_0|{j_{q_0+1}}^\prime,\dots,j_{q_0+q_1+1}^\prime,i_1|...|j_{q_0+...+q_{p-1}+p}^\prime,\dots,j_{q+p}^\prime,i_p)\end{aligned}$$

such that for $\sum_{i=0}^{k-1}q_i+p\leq{s}\leq{\sum_{i=0}^kq_i+p}$

$$j_s^\prime= \left\{
\begin{array}{r@{\quad: \quad}l}i_{k-1} &
j_s\le{i_{k-1}} \\ j_s & i_{k-1}<j_s<i_k \\
i_k  & i_k\le{j_s},  \end{array} \right.$$

$k=1,...,p$. Hence $\gamma\in{\bar{P_p}}{K^s}_{q_0,...,q_p}$ exists and is uniquely determined by ${(\sigma,\tau)}\in{\text{St}}(K^s)_{q+2p+1}$.

Therefore $\bar{p}:\bar{P_.}K^s\to{P_.\text{St}(K^s)}$ is an isomorphism.

[**Remark:**]{} Note that $\bar{p}$ is not injective for a simplicial set in general since for constructing the inverse map $P\text{St}(S)\to{\bar{P}S}$, there is no unique choice for the element $\gamma$ in $\bar{P}_.S_.$. In fact, we do not know which degeneracy operators we will use in order to define $\gamma$, so in general the inverse is not well-defined.

**[The Classifying Space and Lattice Gauge Theory]{}\[six\]**
=============================================================

For the definition of a classifying map we need a prismatic version of the standard construction of the classifying space.

Let $G$ be a topological group and the usual classifying space $BG=EG/G$ which is constructed as a simplicial space $EG_p=\underbrace{G_.\times{...}\times{G_.}}_{p+1-\text{times}}$, $BG_p=(G\times{...}\times{G})/G$.

In order to make this simplicial set discrete we can replace $G$ by the singular simplicial set of continuous maps $S_qG=\text{Map}(\Delta^q,G)$ and $E_.S_.G$ as in Example 