set.

Let us denote $U_{i_0}\cap{...}\cap{U_{i_p}}$ by $U_{i_0,...,i_p}$. $N\mathcal{U}$ is a simplicial manifold, where the face and degeneracy maps come from the followings

$$\begin{aligned}
d_j:U_{i_0,...,i_p}&\to&{U_{i_0,...,{\hat{i}}_j,...,i_p}}\\
s_j:U_{i_0,...,i_p}&\to&{U_{i_0,...,i_j,i_j,...,i_p}}\end{aligned}$$

That is, $N\mathcal{U}(p)$ is a smooth manifold for each $p$ and the face and degeneracy maps are smooth. There is also a corresponding simplicial set $N_d\mathcal{U}=\{N_d\mathcal{U}(p)\}$ called the discrete nerve of the covering. Here $N_d\mathcal{U}(p)$ is simply the set consisting of an element for each non-empty intersection of $p+1$ open sets from $\mathcal{U}$. So there is a natural forgetful map $N\mathcal{U}\to{N_d\mathcal{U}}$.

[**Note:**]{} If $S_.$ has only face operators, then it is called a $\Delta$-set.

\[2.5\]

Given $p\geq{0}$, a $(p+1)$-multi-simplicial set is a sequence $\{S_{q_0,...,q_p}\}$ which is a simplicial set in each variable $q_i$, $i=0,...,p$.

\[2.6\] A prismatic set $P=\{P_{p,.}\}$ is a sequence $P_{p,.}=\{P_{p,q_0,...,q_p}\}$ of $(p+1)$-multi-simplicial sets, i.e., with face and degeneracy operators

$$\begin{aligned}
d_j^i:P_{p,q_0,...,q_p}&\to&{P_{p,q_0,...,q_i-1,...,q_p}}\\
s_j^i:P_{p,q_0,...,q_p}&\to&{P_{p,q_0,...,q_i+1,...,q_p}}\end{aligned}$$

such that $d_j^i$, $s_j^i$ commute with $d_l^k$, $s_l^k$ for $i\neq{k}$, and such that $d_j^i$, $s_j^i$ for fixed $i$ satisfy the identities in Definition \[2.2\].

Furthermore there are face operators $$d_k:P_{p,q_0,...,q_p}\to{P_{p-1,q_0,...,\hat{q}_k,...,q_p}}$$ commuting with $d_j^i$ and $s_j^i$ (interpreting $d_j^k=s_j^k=\text{id}$ on the right) such that $\{P_{p,.}\}$ is a $\Delta$-set.

\[2.7\] If similarly $P$ has degeneracy operators $$s_k:P_{p,q_0,...,q_p}\to{P_{p+1,q_0,...,q_k,q_k,...,q_p}}$$ then $P$ is called a strong 