one.

\[3.4\] There is a pullback diagram

$$\xymatrix{ \|\text{   }|P_.(f)|\text{   }\| \ar[d]^{\|f\|}
\ar[r]^{\phantom{12}\lambda}
&|S_.|  \ar[d]^{|f|} \\
\|\bar{S}_.\| \ar[r]^q & |\bar{S}_.| }$$

In particular $\lambda$ is a homotopy equivalence.

[**Proof:**]{} The map $\lambda:\Delta^p\times{\Delta^{q_0...q_p}}\times{P_p(f)_{q_0...q_p}}\to{\Delta^{q+p}\times{S_{q+p}}}$ is given by $\lambda(t,s,\sigma,\bar{\sigma})=(t_0s^0,...,t_ps^p,\sigma).$ The commutativity of the diagram follows from the definition of $P_.(f)$ since $$P_p(f)_{q_0,...,q_p}\subseteq{S_{q+p}\times{{\bar{S}}_p}}$$ consists of pairs $(\sigma,\bar{\sigma})\in{S_{q+p}}\times{{\bar{S}}_p}$ such that $f(\sigma)=\mu_{q_0,...,q_p}(\bar{\sigma})\in{{\bar{S}}_p}$.

By the commutativity of the diagram, $\lambda$ factors over the pullback $|S_.|\times_{|\bar{S}_.|}{\|\bar{S}_.\|}$ in the diagram

$$\xymatrix{ |S_.|\times_{|\bar{S}_.|}{\|\bar{S}_.\|} \ar[d]^{\|f\|}
\ar[r]^{\phantom{1234}{\text{pr}}_1}
&|S_.|  \ar[d]^{|f|} \\
\|\bar{S}_.\|
\ar[r]^q & |\bar{S}_.| }$$

Here elements in the pullback $|S_.|\times_{|\bar{S}|}{\|\bar{S}_.\|}$ are represented by pairs $((t,\sigma),(\bar{t},\bar{\sigma}))$ such that $f(\sigma)=\mu_{q_0,...,q_p}(\bar{\sigma})$ and $\bar{t}=\mu^{q_0,...,q_p}(t)$, where $\sigma\in{S_{q+p}}$, $\bar{\sigma}\in{{\bar{S}}_q}$. Therefore $\lambda\times{\|f\|}:\|\text{
  }|P_.(f)|\text{   }\|\to{|S|\times{\|\bar{S}\|}}$ induces $\Lambda$ in the diagram

$$\xymatrix{ \|\text{   }|P_.(f)|\text{   }\| \ar[r]^{\Lambda} \ar[d]
& |S_.|\times_{|\bar{S}_.|}{\|\bar{S}_.\|}
\ar[r]^{\phantom{1234}\text{pr}_1} \ar[d]^{{\text{pr}}_2}
& |S_.| \ar[d]^{|f|}\\
\|\bar{S}_.\| \ar[r]^{\text{id}} & \|\bar{S}_.\| \ar[r] &
|\bar{S}_.|}$$

Now $\Lambda$ is a homotopy equivalence. Indeed, an argument similar to the note following (\[3.1\]) gives a homeomorphism of the preimage $\|f\|$ of an open simplex in $\|\bar{S}_.\|$. Hence $\Lambda$ is shown to by a homeomorphism by induction over skeleton of $\|\bar{S}_.\|$.

$\hfill \Box$

\[3.5\] [*Prismatic triangulation of a simplicial set*]{}. Let $S_.$ be a simplicial set and $\bar{S_.}=*$ the simplicial set with one element in each degree. Here 