}\sigma\cup\tau\preccurlyeq\sigma^\prime\}\subseteq{K\times{K}}.$$ This is equivalent to say that $$\text{St}(K)=\{\text{faces of}\text{
  }\sigma^\prime\times{\sigma^\prime}\subseteq{K\times{K}}\}.$$

[**Remark 1:**]{} For each $\sigma\in{K}$, $(\{\sigma\}\times{K})\cap{\text{St}}(K)$ is the closure of the usual open star of $\sigma$, i.e., the union of the open simplices having $\sigma$ as a face. Whence the name $\text{St}(K)$. Note that $\text{St}(K)\subseteq{K\times{K}}$ is a subcomplex.

Let $K^s$ denote the simplicial set associated to the simplicial complex $K$. That is, $${K_n}^s=\{(a_{i_0},...,a_{i_n})\text{ }|\text{
  }\{a_{i_0},...,a_{i_n}\}\text{   }\text{a simplex of}\text{   }K\text{   }\text{(with repetitions)}\text{   }i_0\leq{...}\leq{i_n}\}.$$

$\operatorname{St}(K)_n ^s$ are the following:

Let $(\sigma,\tau)\in{K\times{K}}$, where $\sigma=(a_{i_0},...,a_{i_p})$, $\tau=(b_{j_0},...,b_{j_q})$. For $(\sigma,\tau)\in{\text{St}(K)}$, let $\sigma^\prime=\sigma\cup\tau=(c_{k_0},...,c_{k_n})$. By allowing repetitions in Definition \[4.1\], i.e., by taking $\sigma^\prime\in{K^s}$, we can assume $n=p+q$ so that either $c_{k_s}=a_{i_t}$ or $c_{k_s}=b_{j_u}$, where $t=0,...,p$, $u=0,...,q$. Also we can assume $c_{k_n}=a_{i_p}$, and if $a_{i_t}=b_{j_u}$ then $b_{j_u}$ comes before $a_{i_t}$. In other words $(\sigma,\tau)$ is of the form $$\sigma=d_{\nu_1...\nu_q}\sigma^\prime,\text{   }\tau=d_{\mu_1...\mu_p}\sigma^\prime,$$ where $0\leq{\nu_1}<...<\nu_q<n$ and $0\leq{\mu_1}<...<\mu_p\leq{n}$ and $\mu_i\neq{\nu_j}$, $\forall{i,j}$. Therefore we introduce for a general simplicial set $S$ the following.

\[4.2\] Let $\text{St}(S)$ be the simplicial subset of the diagonal $\delta(S\times{S})$ containing all simplices of the form

$$(s_{\nu_q...\nu_1}\circ{d_{\nu_1...\nu_q}}\sigma^\prime,s_{\mu_p...\mu_1}\circ{d_{\mu_1...\mu_p}}\sigma^\prime),$$

where $0\leq{\nu_1}<...<\nu_q<n$ and $0\leq{\mu_1}<...<\mu_p\leq{n}$ with $\mu_i\neq{\nu_j}$, $\forall{i,j}$ as above. Here $s_{\nu_q...\nu_1}=s_{\nu_q}\circ{...}\circ{s_{\nu_1}}$ , $d_{\nu_1...\nu_q}=d_{\nu_1}\circ{...}\circ{d_{\nu_q}}$, $s_{\mu_p...\mu_1}=s_{\mu_p}\circ{...}\circ{s_{\mu_1}}$ and $d_{\mu_1...\mu_p}=d_{\mu_1}\circ{...}\circ{d_{\mu_p}}$.

\[4.3\] For $K$ a simplicial complex, there is a map $$\operatorname{st}:\operatorname{St}(K)^s\to{\operatorname{St}(K^s)}$$ which is an isomorphism.

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