prismatic set.

[**Remark:**]{} In this case $(P_p,d_k,s_k)$ is a usual simplicial set.

In general, degeneracy operators given in Definition \[2.7\] do not exist naturally so in this case $(P_{p,.},d_k)$ is only a $\Delta$-set.

Example \[2.1\] continued: A triangulated fibre bundle $$\pi:|K|\to{|L|}$$ gives a strong prismatic set $P_.(K/L)$ by letting $$P_p(K/L)_{q_0...q_p}\subseteq{K_{p+q_0+...+q_p}\times{L_p}}$$ be the subset of pairs of simplices $(\tau,\sigma)$ so that $q_i+1$ of the vertices in $\tau$ lies over the $i$-th vertex in $\sigma$. Then we have face and degeneracy operators defined in the obvious way. It is now straight forward to check that this is a strong prismatic set.

\[ES\] For a given simplicial set $S$, consider $E_pS=\underbrace{S_.\times{...}\times{S_.}}_{p+1-times}$.$\pi_i:E_pS\to{E_{p-1}S}$ is the projection which deletes the $i$-th factor. Similarly, the diagonal map $\delta_i:E_pS\to{E_{p+1}}S$ repeats the $i$-th factor. This is a strong prismatic set.

Prismatic sets have various geometric realizations.

\[2.9\] First, we have for each $p$ the thin (geometric) realization $$\begin{aligned}
\label{2.10}
|P_{p,.}|=\bigsqcup_{q_0,...,q_p}\Delta^{q_0...q_p}\times{P_{p,q_0,...,q_p}}/\sim\end{aligned}$$ with equivalence relation “$\sim$” generated by the face and degeneracy maps $$\begin{aligned}
\varepsilon_j^i&:&\Delta^{q_0...q_i...q_p}\to{\Delta^{q_0...q_i+1...q_p}}\text{   }\text{   }\text{and}\\
\eta_j^i&:&\Delta^{q_0...q_i...q_p}\to{\Delta^{q_0...q_i-1...q_p}},\end{aligned}$$ respectively. $\{|P_{p,.}|\}$ is a $\Delta$-space hence it gives a fat realization $$\begin{aligned}
\label{2.10*} \|\text{   }|P_.|\text{
}\|=\bigsqcup_{p\geq{0}}\Delta^p\times{|P_{p,.}|}/\sim\end{aligned}$$ by only using face operators $d_k$.

The face and degeneracy operators $d_k$, $s_k$ act on $\Delta^{q_0...q_p}$ as 