$P_p(f)=P_pS$ is called the $p$-th prismatic subdivision of $S$ and for each $t\in{\stackrel{\circ}{\Delta}^p}$ the map $\lambda_p(t,-):|P_pS|\to{|S_.|}$ is a homeomorphism. In this case, Theorem \[3.4\] gives a homeomorphism $\Lambda:\|\text{
  }|P_.S|\text{   }\|\stackrel{\approx}\to{\|*\|}\times{|S_.|}$, here $\|*\|=\bigcup_n\Delta^n/\partial\Delta^n$. In particular $\lambda:\|\text{   }|P_.S|\text{   }\|\to{|S_.|}$ is a homotopy equivalence. We shall call $P_.S$ the prismatic triangulation of $S$.

For later use, let us give the explicit construction of the $p+1$-prismatic set $P_.S_.$ and its realization:

$$P_pS_{q_0,...,q_p}=S_{q_0+...+q_p+p}.$$

The face operators $${d_j}^{(i)}:P_pS_{q_0,...,q_i,...,q_p}=S_{q+p}\to{P_pS_{q_0,...,q_i-1,...,q_p}}=S_{q+p-1}$$ are defined by $$d_j^{(i)}:=d_{q_0+...+q_{i-1}+i+j},$$ $j=0,...,q_i$. Similarly, the degeneracy operators $${s_j}^{(i)}:P_pS_{q_0,...,q_i,...,q_p}=S_{q+p}\to{P_pS_{q_0,...,q_i+1,...,q_i}}=S_{q+p+1}$$ can be defined by $$s_j^{(i)}:=s_{q_0+...+q_{i-1}+i+j},$$ $j=0,...,q_i$. The face maps $$d_{(i)}:P_pS_{q_0,...,q_p}\to{P_{p-1}S_{q_0,...,{\hat{q}}_i,...q_p}}$$ are the operators corresponding to $${\varepsilon}^{(i)}:{\Delta}^{q_0+...+{\hat{q}}_i+...+q_p+p-1}\to{{\Delta}^{q_0+...+...+q_p+p}}$$ take $(e_0,...,e_{q_0+...+{\hat{q}}_i+...+q_p+p-1})$ to $(e_0,...,e_{q_0+...+q_p+p})$, deleting the elements $q_0+...+q_{i-1}+i,...,q_0+...+q_i+i$. It deletes $(q_i+1)$- elements. In contrary to this, there is no degeneracy operator.

Now we turn to the realizations. For the sequences of spaces $\{|P_.S_.|\}$, we obtain the fat realization: 0.3 cm $$||\text{   }|P_.S_.|\text{
}||=\bigsqcup_{p\geq{0}}{\Delta}^p\times{|P_pS_.|}/_{\sim},$$ where $$|P_pS_.|=\bigsqcup{\Delta}^{q_0...q_p}\times{S_{q_0+...+q_p+p}}/_{\sim}$$ and the face operators $\pi_i:|P_pS_.|\to{|P_{p-1}S_.|}$ are given by $\pi_i={\text{proj}}_i\times{d_{(i)}}$ with ${\text{proj}}_i:\Delta^{q_0...q_p}\to{\Delta^{q_0...{\hat{q}}_i...q_p}}$ beeing the natural projection.

Note that $\lambda_p:\Delta^p\times{|P_pS_.|}\to{|S_.|}$ satisfies $$\lambda_p\circ{(\varepsilon^i\times{\text{id}})}=\lambda_{p-1}\circ{(\text{id}\times{\pi_i})}.$$ Thus $\lambda_p$ induces the map $\lambda$ on the fat realization.

Let $\|\text{   }|P_.S_.|\text{   }\|^p$ respectively $\|\text{
}|S_.|\text{   }\|^p$ denote the subcomplexes generated by $\Delta^p\times{|P_pS|}$ respectively $\Delta^p\times{|S_.|}$. Then the restriction of $\Lambda$ to $\|\text{   }|P_.S_.|\text{   }\|^p$ is given 