given by $$aw(x)=\sum_{q_0+...+q_p=n}s_{q_0+...+q_{p-1}+p-1}\circ{...}\circ{s_{q_0}}(x)_{(q_0,...,q_p)},$$ where $x\in{S_n}$.

**[Prismatic Sets and Stars of Simplicial Complexes]{}\[four\]**
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For a simplicial set $S$ and the prismatic triangulation $P_.S$ there is another closely related prismatic set ${\bar{P}}_pS_.$ which as we shall see for a simplicial complex is the nerve of the covering by stars of vertices considered as a prismatic set.

\[3.7\]

For $S$ a simplicial set let $\bar{P_.}S$ be the prismatic set given by $${\bar{P}}_pS_{q_0,...,q_p}:=S_{q_0+...+q_p+2p+1}.$$ where face and degeneracy operators on $\bar{P_p}S_{q_0,...,q_p}$ are inherited from the ones of $S_{q+2p+1}$ as follows:

Let $q=q_0+...+q_p$, the face operators $$d_j^{(i)}:S_{q+2p+1}={\bar{P}}_pS_{q_0,...,q_p}\to{S_{q+2p}}={\bar{P}}_pS_{q_0,...,q_i-1,...,q_p}$$ are defined by $$d_j^{(i)}:=d_{q_0+...+q_{i-1}+2i+j},\text{   }j=0,...,q_i\text{   }\text{but}\text{   }j\not={2i+1+\sum_{k=0}^iq_k}.$$ So $\bar{P_p}S_{q_0,...,q_p}$ has only $q+p$-face operators, i.e., we skip the following $p+1$ face operators $$\{d_{q_0+1},d_{q_0+q_1+3},...,d_{q+2p+1}\}.$$

Similarly the degeneracy operators $$s_j^{(i)}:S_{q+2p+1}\to{S_{q+2p+2}}$$ can be defined by $$s_j^{(i)}:=s_{q_0+...+q_{i-1}+2i+j},\text{   }j=0,...,q_i,\text{   }\text{but}\text{   }j\not={2i+1+\sum_k^iq_k}.$$ Furthermore the face operators are $$d_{(i)}:S_{q+2p+1}={\bar{P}}_pS_{q_0,...,q_p}\to{S_{q+2p-q_i-1}}={\bar{P}}_{p-1}S_{q_0,...,{\hat{q}}_i,...,q_p}$$ corresponding to $$\varepsilon^{(i)}:{\Delta}^{q+2p-q_i-1}\to{{\Delta}^{q+2p+1}}$$ which take $(e_0,...,e_{q_0+...+{\hat{q}}_i+...+q_p+2p-1})$ to $(e_0,...,e_{q+2p+1})$, by deleting the vectors with indices $(q_0+...+q_{i-1}+2i,...,q_0+...+q_i+2i+1)$. So it deletes $q_i+2$ elements. That is, $$d_{(i)}=d_{q_0+...+q_{i-1}+2i}\circ{...}\circ{d_{q_0+...+q_i+2i+1}},\text{   }i=0,...,p.$$

[**Remark:**]{} As $P_.S$, ${\bar{P}}_.S$ is a prismatic set but in general not a strong prismatic set.

[**Realization of ${\bar{P}_.S_.}$:**]{}

Notice that $$\|\text{   }|\bar{P_.}S_.|\text{
}\|=\bigsqcup_{p\geq{0}}{\Delta}^p\times{{\Delta}^{q_0...q_p}}
\times{{\bar{P}}_pS_{q_0,...,q_p}}/_{\sim}$$ where the equivalence relation apart from the internal relations in $|{\bar{P}}_pS|$ using $d_j^{(i)}$ and $s_j^{(i)}$, include the relations 