by $$\Lambda_p(t,s,\sigma)=(t,\lambda_p(t,s,\sigma)).$$

The map $\Lambda_p$ induce a homeomorphism $$\Lambda:\|\text{   }|P_.S_.|\text{   }\|\to{\|\text{   }|S_.|\text{
}\|}\approx{\|*\|\times{|S_.|}}.$$

\[composition\] The composition map ${\text{proj}}_2\circ{\Lambda}=\lambda$ $$\|\text{   }|P_.S_.|\text{   }\|\to{\|\text{   }|S_.|\text{
}\|}\to{|S_.|}$$ is a homotopy equivalence.

[**Remark 1:**]{} We can calculate the homology of the geometric realization of a prismatic set as follows:

A prismatic set $P_{.,.}$ has a double complex $(C_{p,n}(PS),\partial_F,\partial_H)$. Here $$C_{p,n}(PS)=\bigoplus_{q_0+...+q_p=n}C_{p,q_0,...,q_p}(PS)$$ is the associated chain complex $C_p(PS)$ generated by $P_{p,q_0,...,q_p}$. The vertical boundary map is defined by using boundary maps in the fibre direction $${\partial_F}^i:PC_{p,{q_0,...,q_p}}\to{PC_{p,{q_0,...,q_i-1,...,q_p}}}$$ defined by ${\partial_F}^i=\sum{(-1)}^jd_j^i$, where, if $q_i=0$ then ${\partial^i}_F=0$. The total vertical boundary map is then $$\begin{aligned}
\label{vertical}
\partial_V={\partial^0}_F+{(-1)}^{q_0+1}{\partial^1}_F+...+{(-1)}^{q_0+...+q_{p-1}+p}{\partial^p}_F.\end{aligned}$$ There is also a horizontal boundary map $$\begin{aligned}
\label{horizontal}
\partial_H=\partial_0+{(-1)}^{q_0+1}\partial_1+...+{(-1)}^{q_0+...+q_{p-1}+p}\partial_p,\end{aligned}$$ where $$\begin{aligned}
\partial_k= \left\{ \begin{array}{r@{\quad: \quad}l}0 & \text{   }\text{if}\text{   }q_k>0  \\
d_k & \text{   }\text{if}\text{   }q_k=0, \end{array} \right.\end{aligned}$$ so that $\partial=\partial_V+\partial_H$ is a boundary map in the total complex $PC_*$ which is the cellular chain complex for the geometric realization. Hence it calculates the homology. In the case of $P_.(f)$ for $f:S\to{\bar{S}}$ a simplicial map, the double complex gives rise to a spectral sequence which for a triangulated fibre bundle is the usual Leray-Serre spectral sequence.

[**Remark 2:**]{} For each $p$ and each $t\in{{\stackrel{\circ}{\Delta}}^p}$, $\lambda_p(t)^{-1}:|S|\to{\{t\}\times|P_pS|}$ induces a map of cellular chain complexes $$aw:C_*(S)\to{C_{*,*}(PS)}$$ 