$$(\varepsilon^it,(s,y))\sim(t,\pi_i(s,y)),$$ with $\pi_i=({\text{proj}}_i)\times{d_{(i)}}$ the face operators on $|\bar{P_p}S_.|$.

The relation of ${\bar{P}}_.S$ with $S_.$ and $P_.S$ is as follows:

\[3.9\]

Let $i:\|S_.\|\hookrightarrow{\|\text{   }|{\bar{P}}_pS_.|\text{   }\|}$ be an inclusion defined for $(t,x)\in{\Delta^p\times{S_p}}$ by $$i(t,x)=(t,1,s_0\circ{...}\circ{s_p}x)\in{\Delta^p\times{{(\Delta^0)}^{p+1}}\times{S_{2p+1}}}\subseteq{\Delta^p\times{|{\bar{P}}_pS_.|}},$$ and $r:\|\text{   }|{\bar{P}}_.S_.|\text{   }\|\to{\|S_.\|}$ be the retraction defined for $(t,s,y)\in{\Delta^p\times{\Delta^{q_0...q_p}}\times{S_{q+2p+1}}}$ $$r(t,s,y)=(t,d_{0...q_0}\circ{{\hat{d}}_{q_0+1}}\circ{...}\circ{d_{(q_0+...+q_{p-1}+2p)...(q+2p)}}\circ{{\hat{d}}_{q+2p+1}}y),$$ where the ${\hat{d}}_i$ are left out and $d_{(q_0+...+q_{i-1}+2i)...(q_0+...+q_i+2i)}=d_{q_0+...+q_{i-1}+2i}\circ{...}\circ{d_{q_0+...+q_i+2i}}$, $i=0,...,p$.

1\) $i$ is a deformation retract with the retraction $r$.

2\) There is a commutative diagram of homotopy equivalences

$$\begin{aligned}
\xymatrix{
||S_.||\ar[r]^{i}\ar[dr] \ar[ddr]
&\|\text{   }|{\bar{P}}_.S_.|\text{   }\| \ar[d]^{f}\\
&\|\text{   }|P_.S_.|\text{   }\| \ar[d]^{\Lambda}\\
&\|\text{   }|S_.|\text{   }\|
}\end{aligned}$$

where $f:\Delta^p\times{\Delta^{q_0...q_p}\times{S_{q+2p+1}}}\to{\Delta^p\times{\Delta^{q_0...q_p}}}\times{S_{q+p}}$ takes $(t,s^0,...,s^p,x)$ to $(t,s^0,...,s^p,d_{q_0+1}\circ{d_{q_0+q_1+3}}\circ{...}\circ{d_{q+2p+1}}x)$, $x\in{S_{q+2p+1}}$.

The proof is straight forward see [@B] for details.

For a simplicial complex $K$ there is another prismatic complex defined using the stars of simplices. That is, let $K_0=\{a_i|i\in{I}\}$, where $I=\{1,...,N\}$, be the set of vertices and let $K_n=\{\sigma=(a_{i_0},...,a_{i_n})|i_0<...<i_n\}$ be the set of $n$-simplices such that if $\sigma\in{K_n}$ then any face $\tau=(a_{i_{j_0}},...,a_{i_{j_k}})$ lies in $K_k$. We shall write $\tau\preccurlyeq\sigma$ in this case. Now $K\times{K}$ is also a simplicial complex with the lexicographical order of the vertices $$(a_i,b_j)<(a_{i^\prime},b_{j^\prime})
\Leftrightarrow\text{either}\text{   }i<{i^\prime} \text{
  }\text{or}\text{   }i=i^\prime\text{   }\text{and}\text{  }j<j^\prime,$$ where $\{(a_{i_0},b_{j_0}),...,(a_{i_n},b_{j_n})\}\in{K\times{K}}$.

\[4.1\] Let $K$ be a simplicial complex. The Star of $K$ is defined as $$\text{St}(K)=\{(\sigma,\tau)\in{K\times{K}}\text{
  }|\text{   }\exists\text{   }{\sigma^\prime}\text{   }\text{such
that}\text{   