the projection and the diagonal, respectively so they induce a structure of a simplicial set on $|P_p|$. In other words, the projection $\pi_i:\Delta^{q_0...q_p}\to{\Delta^{q_0...{\hat{q}}_i...q_p}}$ deletes the $i$-th coordinate and the diagonal map $\Delta_i:\Delta^{q_0...q_p}\to{\Delta^{q_0...q_iq_i...q_p}}$ repeats the $i$-th factor. Then the further equivalence relation on $|P_.|$ given in (\[2.10\*\]) is generated by $$\begin{aligned}
(\varepsilon^it,s,\sigma)\sim(t,\pi_is,d_i\sigma),\text{   }t\in{\Delta^{p-1}},\text{   }s\in{\Delta^{q_0...q_p}},\text{   }\sigma\in{P_{p,q_0,...,q_p}}.\\\end{aligned}$$

If $P_.$ is strong then we also have a thin realization $$|P_.|=\||P_.|\|/\sim$$ given by the above and the further relation $$\begin{aligned}
(\eta^it,s,\sigma)\sim(t,\Delta_is,s_i\sigma),\text{
}t\in{\Delta^{p+1}},\text{   }s\in{\Delta^{q_0,...,q_p}},\text{
}\sigma\in{P_{p,q_0,...,q_p}}.\end{aligned}$$

Similarly, we can define for each $p$, the fat realization $||P_.||$, that is $$\|P_p\|=\bigsqcup_{q_0,...,q_p}\Delta^{q_0...q_p}\times{P_{p,q_0,...,q_p}}/\sim$$ with equivalence relation given by only the face maps $d_j^i$.

Moreover, we have the very fat realization $$\|\text{   }\|P_.\|\text{
}\|=\bigsqcup_{p\geq{0}}\Delta^p\times{\|P_{p,.}\|}/\sim$$ using only face operators.

For a given simplicial set $S$ and $E_pS$ as in Example \[ES\] we have $\|\text{   }|E_.S|\text{   }\|$ as the fat realization of the space which maps $p$-th term to $\underbrace{|S_.|\times{...}\times{|S_.|}}_{p+1-times}$.

Define for a space $X$, $E_pX=\underbrace{X_.\times{...}\times{X_.}}_{p+1-times}$ Let us say $X=|S|$ then $\|\text{   }|E_.S|\text{   }\|$ is contractible.

**[Prismatic Triangulation]{}\[three\]**
========================================

Let us return to the case of a triangulated fibre bundle $|K|\to{|L|}$. In this case the natural map $$P_p(K/L)_{q_0,...q_p}\to{K_{q_0+...+q_p+p}}$$ induces a homeomorphism

$$\xymatrix{ |P_.(K/L)| \ar[d]^{|\pi^{\prime}|} \ar[r]^{\phantom{123}\approx}
&|K|  \ar[d]^\pi \\
|L| \ar[r]^{=} & |L| }$$

